Learning to Bias: Machine Learning-Enhanced Particle Filters

# Learning to Bias: Machine Learning-Enhanced Particle Filters
Source: [https://arxiv.org/html/2609.30498](https://arxiv.org/html/2609.30498)
Apoorv Srivastava††thanks:Corresponding authorEric DarveAffiliation:Stanford UniversityAffiliation:Stanford, CA 94305Email:[darve@stanford\.edu](mailto:)

###### Abstract

Sequential inference estimates latent states from noisy and incomplete observations\. Particle Filters \(PFs\), a class of Monte Carlo methods based on importance sampling, provide a flexible framework for this task, but often suffer from poor sample efficiency and unfavorable scaling with dimension, partly due to suboptimal proposal distributions\. We address these challenges by integrating learned proposals into the PF framework\. We introduce Neural Optimal Particle Filters \(NOPFs\), which learn an amortized approximation to the optimal proposal from offline simulated one\-step conditioning tuples\. The learned proposal is used as a drop\-in replacement in standard PF updates, with samples corrected by standard importance weights so that the method asymptotically targets the same filtering distribution under standard support and density\-evaluation assumptions\. Across stochastic nonlinear benchmarks of varying inference complexity, NOPFs improve sample efficiency and distributional accuracy over standard PF baselines with modest computational overhead\. The approach integrates data\-driven proposal learning into classical inference without altering the underlying filtering objective\.

## 1Introduction

Sequential inference aims to estimate latent states from noisy and incomplete observations and is central to applications such as robotics, tracking, weather forecasting, and data assimilation[Wills and Schön \(2023\)](https://arxiv.org/html/2609.30498#bib.bib2);[Fearnhead and Künsch \(2018\)](https://arxiv.org/html/2609.30498#bib.bib3)\. Particle Filters \(PFs\), also known as sequential Monte Carlo \(SMC\) methods, provide a flexible Monte Carlo approximation of the posterior distribution over the current state using weighted particles\.

PFs rely on importance sampling, and their sample efficiency depends critically on the proposal distribution\. The number of particles required can grow sharply with state dimension, leading to poor scaling in high\-dimensional settings[van Leeuwen et al\. \(2019\)](https://arxiv.org/html/2609.30498#bib.bib4)\. Naive proposals often place particles in regions unlikely under the observations, so most particles receive negligible weights and contribute little to the posterior estimate\. Effective proposals should therefore use the current observation to guide particles toward plausible posterior regions while still allowing proper importance weighting\.

In this work, we introduce Neural Optimal Particle Filters \(NOPFs\), which learn proposal distributions that serve as plug\-and\-play replacements in standard PF updates\. Samples from the learned proposal are corrected using the standard importance\-weighting update, thereby preserving the underlying filtering objective while improving particle placement\.

This work makes three contributions\. First, we formulate proposal learning for known state\-space models as an offline amortized approximation to the locally optimal one\-step proposal, rather than as end\-to\-end model learning or trajectory\-level SMC adaptation\. Second, we introduce a residual Gaussian proposal around the deterministic transition skeleton, enabling efficient sampling and tractable density evaluation for valid SIS weight correction\. Third, we evaluate NOPFs on nonlinear localization benchmarks spanning approximately Gaussian, multimodal, and data\-association regimes, comparing against Kalman\-based and proposal\-enhanced PF baselines\.

## 2Background

Given a latent state\-space model with state𝐱t\\mathbf\{x\}\_\{t\}and observation𝐲t\\mathbf\{y\}\_\{t\}at timett, the system evolves according to

𝐱t=𝐟⁡\(𝐱t−1,𝐯t\),𝐲t=𝐡⁡\(𝐱t,𝐰t\),\\mathbf\{x\}\_\{t\}=\\mathbf\{f\}\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{v\}\_\{t\}\),\\qquad\\mathbf\{y\}\_\{t\}=\\mathbf\{h\}\(\\mathbf\{x\}\_\{t\},\\mathbf\{w\}\_\{t\}\),\(1\)where𝐯t\\mathbf\{v\}\_\{t\}and𝐰t\\mathbf\{w\}\_\{t\}denote process and observation noise\. The transition model𝐟\\mathbf\{f\}and observation model𝐡\\mathbf\{h\}induce the transition densityp⁡\(𝐱t∣𝐱t−1\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)and likelihoodp⁡\(𝐲t∣𝐱t\)p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\), respectively\. The objective is to recursively approximate the filtering distributionp\(𝐱t∣𝐲1:t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{y\}\_\{1:t\}\)using the previous posteriorp\(𝐱t−1∣𝐲1:t−1\)p\(\\mathbf\{x\}\_\{t\-1\}\\mid\\mathbf\{y\}\_\{1:t\-1\}\)and the new observation𝐲t\\mathbf\{y\}\_\{t\}\.

For linear\-Gaussian models, the Kalman filter provides an analytic solution, with variants such as the EKF, UKF, and EnKF extending this idea to nonlinear settings under approximate Gaussian assumptions[Evensen \(2009\)](https://arxiv.org/html/2609.30498#bib.bib5);[Wan and Van Der Merwe \(2000\)](https://arxiv.org/html/2609.30498#bib.bib6);[Gustafsson et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib7)\. PFs, in contrast, represent the filtering distribution nonparametrically as

p\(𝐱t∣𝐲1:t\)≈∑i=1Nwt\(i\)δ\(𝐱t−𝐱t\(i\)\),p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{y\}\_\{1:t\}\)\\approx\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}\\delta\(\\mathbf\{x\}\_\{t\}\-\\mathbf\{x\}\_\{t\}^\{\(i\)\}\),\(2\)where\{𝐱t\(i\),wt\(i\)\}i=1N\\\{\\mathbf\{x\}\_\{t\}^\{\(i\)\},w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}are particles with non\-negative normalized weights satisfying∑i=1Nwt\(i\)=1\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}=1\.

PFs approximate the filtering update using Sequential Importance Sampling \(SIS\)\. At timett, each particle is sampled from a proposal,𝐱t\(i\)∼q⁡\(𝐱t∣𝐱t−1\(i\),𝐲t\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim q\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\), and assigned the importance weight

wt\(i\)∝wt−1\(i\)​p⁡\(𝐲t∣𝐱t\(i\)\)​p​\(𝐱t\(i\)∣𝐱t−1\(i\)\)q⁡\(𝐱t\(i\)∣𝐱t−1\(i\),𝐲t\),w\_\{t\}^\{\(i\)\}\\propto w\_\{t\-1\}^\{\(i\)\}\\frac\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i\)\}\)p\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)\}\{q\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)\},\(3\)followed by normalization\. This update follows from the standard SIS factorization[Doucet et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib22);[Arulampalam et al\. \(2002\)](https://arxiv.org/html/2609.30498#bib.bib16)\. The normalized weights determine each particle’s contribution to the posterior approximation\. When the proposal places many particles in regions with low likelihood under the new observation, most weights become negligible, and only a few particles dominate the approximation\. This weight degeneracy reduces the effective number of useful particles, even when the nominal ensemble size is large\. A standard way to stabilize PFs is to resample after normalization\.

Resampling mitigates weight degeneracy by duplicating high\-weight particles, discarding low\-weight particles, and resetting weights to1/N1/N\. However, resampling acts only after the particles have already been proposed and weighted\. If most particles are proposed in low\-likelihood regions, it can only replicate the few survivors, reducing diversity and causing sample impoverishment[Doucet et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib22);[Arulampalam et al\. \(2002\)](https://arxiv.org/html/2609.30498#bib.bib16)\. Thus, proposal design remains central to PF efficiency\.

For a fixed ancestor particle𝐱t−1\\mathbf\{x\}\_\{t\-1\}and new observation𝐲t\\mathbf\{y\}\_\{t\}, the locally optimal proposal is

q⋆​\(𝐱t∣𝐱t−1,𝐲t\)=p⁡\(𝐱t∣𝐱t−1,𝐲t\)\.\\displaystyle q^\{\\star\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)=p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\.\(4\)The term local reflects the recursive SIS setting in which𝐲t\\mathbf\{y\}\_\{t\}is used to sample only the new state𝐱t\\mathbf\{x\}\_\{t\}and does not alter the ancestor locations𝐱t−1\(i\)\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\. Within this class, the optimal proposal gives the least\-degenerate benchmark[Snyder et al\. \(2015\)](https://arxiv.org/html/2609.30498#bib.bib26)\. It also minimizes the conditional variance of the incremental importance weight given the ancestor particle and current observation[Doucet et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib22)\. Under this choice,wt\(i\)∝wt−1\(i\)​p​\(𝐲t∣𝐱t−1\(i\)\)w\_\{t\}^\{\(i\)\}\\propto w\_\{t\-1\}^\{\(i\)\}p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\), so the updated weight is independent of the newly sampled state𝐱t\(i\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}\. It therefore provides a natural target for learned proposal models\. By approximatingp⁡\(𝐱t∣𝐱t−1,𝐲t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\), the learned proposal biases particles toward dynamically plausible and observation\-consistent regions while preserving valid importance weighting\.

In practice, the locally optimal proposal is rarely available in closed form\. The Bootstrap Particle Filter \(BPF\) uses the transition densityp⁡\(𝐱t∣𝐱t−1\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)as the proposal, which is simple but ignores the current observation during sampling\. Several PF variants, therefore, construct more informative proposals or alter the sampling procedure\. Auxiliary PFs use predictive\-likelihood approximations to select promising ancestors[Pitt and Shephard \(1999\)](https://arxiv.org/html/2609.30498#bib.bib11), Unscented PFs build local Gaussian proposals using unscented transforms[Van Der Merwe et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib9), Implicit PFs move particles through particle\-wise optimization[Chorin et al\. \(2010\)](https://arxiv.org/html/2609.30498#bib.bib10), and Nudged PFs use likelihood gradients to push particles toward high\-likelihood regions[Akyildiz and Míguez \(2019\)](https://arxiv.org/html/2609.30498#bib.bib13)\. More recent score\-based filters use learned score models to generate posterior\-aligned samples[Bao et al\. \(2024\)](https://arxiv.org/html/2609.30498#bib.bib8)\.

A related line of work uses SMC not only for filtering with known models, but also within broader parameter\-learning and joint model\-state inference frameworks\. Nested particle filters estimate static model parameters together with latent states using a two\-layer particle approximation[Crisan and Míguez \(2018\)](https://arxiv.org/html/2609.30498#bib.bib1)\. Filtering Variational Objectives[Maddison et al\. \(2017\)](https://arxiv.org/html/2609.30498#bib.bib24), Variational Sequential Monte Carlo[Naesseth et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib25), and Auto\-Encoding Sequential Monte Carlo[Le et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib23)use SMC\-based variational objectives or likelihood estimators to learn sequential latent\-variable models and associated proposal distributions\. Differentiable particle filtering methods modify resampling or the PF computation graph so that model components, such as the forward model and observation operator, can be trained end\-to\-end[Corenflos et al\. \(2021\)](https://arxiv.org/html/2609.30498#bib.bib17);[Jonschkowski et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib20);[Karkus et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib21), while adaptive SMC methods tune proposal mechanisms using online criteria[Cornebise et al\. \(2008\)](https://arxiv.org/html/2609.30498#bib.bib18)\. More broadly, learned generative models have also been used to construct importance densities for complex target distributions[Müller et al\. \(2019\)](https://arxiv.org/html/2609.30498#bib.bib12)\.

Several prior methods are closer to learned or approximate proposal construction\. Inference networks for SMC learn amortized stochastic inverses for directed graphical models and use them as proposal distributions[Paige and Wood \(2016\)](https://arxiv.org/html/2609.30498#bib.bib27)\. Computational Doobhh\-transforms approximate fully adapted proposals for discretely observed diffusions using neural approximations of backward Kolmogorov equations[Chopin et al\. \(2023\)](https://arxiv.org/html/2609.30498#bib.bib28)\. Neural Adaptive Sequential Monte Carlo \(NASMC\) learns neural proposal distributions within an adaptive SMC framework[Gu et al\. \(2015\)](https://arxiv.org/html/2609.30498#bib.bib19)\. NASMC is closest in spirit to the present work, but its learning problem is different\. It optimizes an inclusive trajectory\-level KL objective, with gradients typically estimated from SMC particles, weights, and ancestry variables, and uses proposal models that may depend on latent and observation histories\.

In contrast, this work assumes known transition and observation models and focuses specifically on learning an amortized approximation to the one\-step Markov proposalp⁡\(𝐱t∣𝐱t−1,𝐲t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\. NOPF trains this proposal offline from simulated one\-step conditioning tuples, using an ELBO based on analytic model densities, parameterizes a residual Gaussian proposal using the deterministic transition skeleton, and inserts the learned proposal directly into the standard PF update\. At inference time, particles sampled from the learned proposal are corrected using the standard importance\-weight update in Eq\.[3](https://arxiv.org/html/2609.30498#S2.E3), so no online proposal optimization, model learning, differentiable resampling, or weighted SMC training trajectories are required\.

## 3Method

The standard PF update at timettis shown in Alg\.[1](https://arxiv.org/html/2609.30498#alg1)\. Different proposal choices yield different PF variants discussed in Section[2](https://arxiv.org/html/2609.30498#S2)\. Although richer proposal classes may depend on the particle indexii, particle histories𝐱0:t−1\(i\)\\mathbf\{x\}\_\{0:t\-1\}^\{\(i\)\}, or the full previous weighted ensemble\{\(𝐱t−1\(j\),wt−1\(j\)\)\}j=1N\\\{\(\\mathbf\{x\}\_\{t\-1\}^\{\(j\)\},w\_\{t\-1\}^\{\(j\)\}\)\\\}\_\{j=1\}^\{N\}, the Markov structure implies that the locally optimal one\-step proposal for theiith particle reduces top⁡\(𝐱t∣𝐱t−1\(i\),𝐲t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)[Doucet et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib22);[van Leeuwen et al\. \(2019\)](https://arxiv.org/html/2609.30498#bib.bib4)\. We therefore learn a conditional densityqθ​\(𝐱t∣𝐱t−1\(i\),𝐲t,t\)q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\},t\)to approximate this local proposal, yielding what we call the Neural Optimal Particle Filter \(NOPF\)\.

Algorithm 1Standard Particle Filter Update \(Timett\)1:Sample:Draw

xt\(i\)∼q⁡\(xt∣xt−1\(i\),yt\)x\_\{t\}^\{\(i\)\}\\sim q\(x\_\{t\}\\mid x\_\{t\-1\}^\{\(i\)\},y\_\{t\}\)for

i=1,…,Ni=1,\\ldots,N\.

2:Weight:Update weight using Eq\.[3](https://arxiv.org/html/2609.30498#S2.E3)then normalize to get

\{wt\(i\)\}i=1N\\\{w\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}such that

∑i=1Nwt\(i\)=1\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}=1\.

3:Resample:Resample using weight

wt\(i\)w\_\{t\}^\{\(i\)\}and reset

wt\(i\)=1/Nw\_\{t\}^\{\(i\)\}=1/N\.

### 3\.1Learned Proposal Density

To support importance sampling, the proposal model must allow both efficient sampling and tractable density evaluation\. This excludes implicit generative models whose densities are unavailable, and motivates tractable conditional density models such as Gaussian distributions, mixtures of Gaussians, or normalizing flows\. Our objective is to learn a family of distributions parameterized by\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\), such thatqθ​\(𝐱t∣𝐱t−1,𝐲t\)≈p⁡\(𝐱t∣𝐱t−1,𝐲t\)q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\approx p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\. Timettis also provided as an input to the proposal network in all implementations, but we suppress this dependence in the notation for readability\.

Although expressive models such as normalizing flows could be used, we adopt a simpler Gaussian proposal\. For a fixed ancestor particle𝐱t−1\(i\)\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}, the local proposal only needs to describe the region reachable under the transition density, reweighted by the current likelihood\. While this conditional proposal can in principle be multimodal due to ambiguities in the observation likelihood, in many practical settings such as robotics, the transition densityp⁡\(𝐱t∣𝐱t−1\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)is typically unimodal and concentrated around deterministic dynamics\([Thrun et al\., 2005](https://arxiv.org/html/2609.30498#bib.bib14), Chapter 5\), which often induces a dominant mode in the conditional proposal\. The full filtering distribution can still be multimodal, since it is represented by a weighted mixture over many particles\. We therefore parameterize the per\-particle proposal as a Gaussian density,

qθ​\(𝐱t∣𝐱t−1,𝐲t\)=𝒩⁡\(𝐱t,𝝁θ1​\(𝐱t−1,𝐲t,t\),𝚺θ2​\(𝐱t−1,𝐲t,t\)\),\\displaystyle q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)=\\mathcal\{N\}\\\!\\left\(\\mathbf\{x\}\_\{t\};\\boldsymbol\{\\mu\}\_\{\\theta\_\{1\}\}\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\),\\boldsymbol\{\\Sigma\}\_\{\\theta\_\{2\}\}\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\\right\),\(5\)where𝝁θ1\\boldsymbol\{\\mu\}\_\{\\theta\_\{1\}\}and𝚺θ2\\boldsymbol\{\\Sigma\}\_\{\\theta\_\{2\}\}are neural networks, andθ=\(θ1,θ2\)\\theta=\(\\theta\_\{1\},\\theta\_\{2\}\)denotes the trainable parameters\. This parameterization makes sampling and density evaluation inexpensive, requiring only a forward pass through the network and a Gaussian density evaluation\.

Directly learning the map from\(𝐱t−1\(i\),𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\},t\)to the proposal parameters would require the network to also learn the deterministic component of the transition model\. To reduce this burden, we first propagate each particle through the deterministic skeleton of the dynamics to obtain a nominal prediction𝐱^t\(i\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}111For example,𝐱^t\(i\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}may be chosen as𝔼⁡\[𝐱t∣𝐱t−1\(i\)\]\\mathbb\{E\}\[\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\]orarg⁡max𝐱t⁡p⁡\(𝐱t∣𝐱t−1\(i\)\)\\arg\\max\_\{\\mathbf\{x\}\_\{t\}\}p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)\. This deterministic prediction is analogous to the summary statistic used in Auxiliary Particle Filters to evaluate predictive likelihoods\.\. The neural networks then use𝐱^t\(i\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\},𝐲t\\mathbf\{y\}\_\{t\}, andttto output the mean and covariance of the proposal\. This lets the model learn an observation\-informed correction around the predicted state rather than relearning the forward dynamics\. Details of the mean and covariance parameterization and network architecture are provided in Appendix[B\.1](https://arxiv.org/html/2609.30498#A2.SS1)\.

### 3\.2Training and Inference

Rather than learning a single proposal distribution, NOPF learns a family of conditional proposals indexed by the conditioning tuple\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\. The proposal is trained offline by minimizing the expected KL divergence

𝔼\(𝐱t−1,𝐲t,t\)DKL\(qθ\(𝐱t∣𝐱t−1,𝐲t\)∥p\(𝐱t∣𝐱t−1,𝐲t\)\),\\displaystyle\\mathbb\{E\}\_\{\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\}D\_\{\\mathrm\{KL\}\}\\\!\\left\(q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\,\\\|\\,p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\right\),\(6\)where the expectation is taken over conditioning tuples generated from simulated trajectories\.

For a single tuple\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\), the ELBO identity derived in Appendix[B\.2](https://arxiv.org/html/2609.30498#A2.SS2)gives the loss

ℒELBO​\(θ,𝐱t−1,𝐲t,t\)=−𝔼qθ​\[log⁡p⁡\(𝐲t∣𝐱t\)\+log⁡p⁡\(𝐱t∣𝐱t−1\)−log⁡qθ​\(𝐱t∣𝐱t−1,𝐲t\)\],\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)=\-\\mathbb\{E\}\_\{q\_\{\\theta\}\}\\left\[\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)\+\\log p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\-\\log q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\right\],\(7\)which is equivalent, up to an additive constant independent ofθ\\theta, to the KL divergence in Eq\.[6](https://arxiv.org/html/2609.30498#S3.E6)\. The objective minimized in practice is therefore

ℒ⁡\(θ\)=𝔼\(𝐱t−1,𝐲t,t\)​\[ℒELBO​\(θ,𝐱t−1,𝐲t,t\)\]\.\\displaystyle\\mathcal\{L\}\(\\theta\)=\\mathbb\{E\}\_\{\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\}\\left\[\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\\right\]\.\(8\)This objective encourages the proposal to place probability mass on states that both explain the current observation and remain likely under the transition dynamics, while penalizing mismatch with the proposal density\.

Training tuples\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)are generated by simulating trajectories from the known forward model\. This decouples proposal learning from the online filtering procedure, in contrast to learning\-based and differentiable PF methods that train through SMC rollouts, weighted particle trajectories, or resampling operations[Maddison et al\. \(2017\)](https://arxiv.org/html/2609.30498#bib.bib24);[Naesseth et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib25);[Le et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib23);[Corenflos et al\. \(2021\)](https://arxiv.org/html/2609.30498#bib.bib17);[Jonschkowski et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib20);[Karkus et al\. \(2018\)](https://arxiv.org/html/2609.30498#bib.bib21);[Gu et al\. \(2015\)](https://arxiv.org/html/2609.30498#bib.bib19)\. NOPF training, therefore, does not require running a particle filter, differentiating through resampling, or using weighted particle trajectories as training targets\. Once trained, the proposal is used as a plug\-in replacement for the proposal step in a standard PF\. During inference, each particle𝐱t−1\(i\)\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}and the new observation𝐲t\\mathbf\{y\}\_\{t\}are passed through the proposal network to produce Gaussian proposal parameters, from which𝐱t\(i\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}is sampled\. The resulting particles are then weighted using Eq\.[3](https://arxiv.org/html/2609.30498#S2.E3), normalized, and resampled as usual\. The training and inference procedures are summarized in Alg\.[2](https://arxiv.org/html/2609.30498#alg2)and Alg\.[3](https://arxiv.org/html/2609.30498#alg3), with full algorithmic details provided in Appendix[B\.3](https://arxiv.org/html/2609.30498#A2.SS3)\.

Algorithm 2Training1:foreach iterationdo

2:simulate a batch of trajectories

\{\(𝐱0:T\(b\),𝐲0:T\(b\)\)\}b=1B\\\{\(\\mathbf\{x\}\_\{0:T\}^\{\(b\)\},\\mathbf\{y\}\_\{0:T\}^\{\(b\)\}\)\\\}\_\{b=1\}^\{B\}
3:foreach

\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)in the batchdo

4:

\(𝝁,𝚺\)←qθ​\(𝐱t−1,𝐲t,t\)\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\Sigma\}\)\\leftarrow q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)
5:draw

SSsamples from

𝐱t∼𝒩⁡\(𝝁,𝚺\)\\mathbf\{x\}\_\{t\}\\sim\\mathcal\{N\}\(\\boldsymbol\{\\mu\},\\boldsymbol\{\\Sigma\}\)and

6:use them to estimate

ℒELBO​\(θ,𝐱t−1,𝐲t,t\)\\mathcal\{L\}\_\{\\text\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)
7:endfor

8:estimate

ℒ⁡\(θ\)\\mathcal\{L\}\(\\theta\)over the batch

9:

θ←θ−η​∇θℒ​\(θ\)\\theta\\leftarrow\\theta\-\\eta\\nabla\_\{\\theta\}\\mathcal\{L\}\(\{\\theta\}\)
10:endfor

Algorithm 3Inference1:initialize

𝐱0\(i\)∼p0\\mathbf\{x\}\_\{0\}^\{\(i\)\}\\sim p\_\{0\},

w0\(i\)=1/Nw\_\{0\}^\{\(i\)\}=1/N
2:for

t=1,…,Tt=1,\\ldots,Tdo

3:

\(𝝁t\(i\),𝚺t\(i\)\)←qθ​\(𝐱t−1\(i\),𝐲t,t\)\(\\boldsymbol\{\\mu\}\_\{t\}^\{\(i\)\},\\boldsymbol\{\\Sigma\}\_\{t\}^\{\(i\)\}\)\\leftarrow q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\},t\)
4:sample

𝐱t\(i\)∼𝒩⁡\(𝝁t\(i\),𝚺t\(i\)\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim\\mathcal\{N\}\(\\boldsymbol\{\\mu\}\_\{t\}^\{\(i\)\},\\boldsymbol\{\\Sigma\}\_\{t\}^\{\(i\)\}\)
5:update weights using Eq\.[3](https://arxiv.org/html/2609.30498#S2.E3)

6:normalize weights to get

∑i=1Nwt\(i\)=1\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}=1
7:systematic resample and reset

wt\(i\)=1/Nw\_\{t\}^\{\(i\)\}=1/N
8:endfor

### 3\.3Evaluation Metrics

Since only a single realized trajectory\(𝐱t⋆,𝐲t\)\(\\mathbf\{x\}\_\{t\}^\{\\star\},\\mathbf\{y\}\_\{t\}\)from the true system is available during rollout, evaluating the full filtering distribution is challenging\. We therefore use several complementary metrics to evaluate different aspects of the particle approximation\.

To assess state\-estimation accuracy, we report normalized mean squared error \(NMSE\), which is most informative when the filtering distribution is approximately unimodal\. To evaluate the quality of the full posterior approximation, particularly in multimodal settings where point estimates can be misleading, we use the one\-step\-ahead predictive negative log\-likelihood \(PNLL\), derived fromp\(𝐲t\+1∣𝐲1:t\)p\(\\mathbf\{y\}\_\{t\+1\}\\mid\\mathbf\{y\}\_\{1:t\}\), and the energy score \(ES\)[Gneiting and Raftery \(2007\)](https://arxiv.org/html/2609.30498#bib.bib15)\. PNLL measures how well the inferred posterior predicts future observations, while the energy score provides a bandwidth\-free distributional score for comparing the full particle approximation:

ESt=𝔼𝐱t∼p^t​‖𝐱t−𝐱t⋆‖2−12​𝔼𝐱t,𝐱t′∼p^t​‖𝐱t−𝐱t′‖2\.\\displaystyle\\mathrm\{ES\}\_\{t\}=\\mathbb\{E\}\_\{\\mathbf\{x\}\_\{t\}\\sim\\widehat\{p\}\_\{t\}\}\\\|\\mathbf\{x\}\_\{t\}\-\\mathbf\{x\}\_\{t\}^\{\\star\}\\\|\_\{2\}\-\\frac\{1\}\{2\}\\mathbb\{E\}\_\{\\mathbf\{x\}\_\{t\},\\mathbf\{x\}\_\{t\}^\{\\prime\}\\sim\\widehat\{p\}\_\{t\}\}\\\|\\mathbf\{x\}\_\{t\}\-\\mathbf\{x\}\_\{t\}^\{\\prime\}\\\|\_\{2\}\.\(9\)Herep^t\\widehat\{p\}\_\{t\}denotes the empirical filtering distribution represented by the ensemble\. The first term favors concentration near the realized state, while the second penalizes excessive ensemble spread\.

To assess proposal efficiency, we report the pre\-resampling effective sample size \(ESS\),Nteff=1/∑i=1N\(wt\(i\)\)2N\_\{t\}^\{\\mathrm\{eff\}\}=1/\\sum\_\{i=1\}^\{N\}\(w\_\{t\}^\{\(i\)\}\)^\{2\}, which quantifies particle\-weight degeneracy[Arulampalam et al\. \(2002\)](https://arxiv.org/html/2609.30498#bib.bib16)\. Finally, we report wall time \(WT\) in seconds for the full run to quantify computational cost\.

For each example, we therefore report five time\-averaged metrics: NMSE for estimation accuracy, PNLL and ES for distributional quality, ESS for proposal efficiency, and WT for computational cost\. All computations, including forward passes through the learned proposal network, are performed on a CPU to maintain consistency across particle filters\. Details of metric computation, additional diagnostics, and the computational setup are provided in Appendices[C](https://arxiv.org/html/2609.30498#A3)and[D](https://arxiv.org/html/2609.30498#A4)\.

## 4Experiments

We evaluate NOPF on robotics localization benchmarks of increasing dimensionality and inference complexity, from approximately unimodal settings to ambiguous multimodal problems\. These benchmarks assess estimation accuracy, proposal efficiency, robustness to posterior multimodality, and computational cost\. All filters are evaluated on trajectories simulated from the true stochastic model, with results reported as mean±\\pmstandard deviation across 10 random seeds\. Angular state components use wrapped arithmetic throughout\.

We compare against Kalman\-based baselines when applicable, including KF for the linear\-Gaussian benchmark and EnKF for nonlinear benchmarks\. For the linear\-Gaussian benchmark, we also include an Optimal Particle Filter \(OPF\), which samples from the closed\-form locally optimal proposal\. We further compare against standard and proposal\-enhanced PF baselines, including the Bootstrap Particle Filter \(BPF\), Auxiliary Particle Filter \(APF\), Unscented Particle Filter \(UPF\), Nudged Particle Filter \(NuPF\), and Inner\-Sampling Proposal Filter \(ISPF\)\. ISPF is an expensive local proposal baseline that uses an inner Monte Carlo approximation of the locally optimal proposal to sample the next particle for each ancestor\. Implementation details for all filters are provided in Appendix[E](https://arxiv.org/html/2609.30498#A5)\. NOPF is evaluated using the metrics in Section[3\.3](https://arxiv.org/html/2609.30498#S3.SS3), with additional results, particle\-count sensitivity studies, and supplementary metrics provided in Appendix[A](https://arxiv.org/html/2609.30498#A1)and[C](https://arxiv.org/html/2609.30498#A3)\.

### 4\.1Linear System with Biased Sensors

The first benchmark is a linear\-Gaussian sensor\-bias model for a robot moving along one dimension, used as a controlled setting in which the optimal Bayesian update is analytically tractable, and the Kalman filter provides an exact reference\. The 3\-dimensional latent state is𝐱t=\[pt,vt,bt\]⊤\\mathbf\{x\}\_\{t\}=\[p\_\{t\},v\_\{t\},b\_\{t\}\]^\{\\top\}, whereptp\_\{t\}is position,vtv\_\{t\}velocity, andbtb\_\{t\}an additive bias in the velocity observations\. The dynamics and observation model are

𝐱t=\[1Δ​t0010001\]​𝐱t−1\+ϵt,𝐲t=\[100011\]​𝐱t\+𝜼t,\\displaystyle\\mathbf\{x\}\_\{t\}=\\begin\{bmatrix\}1&\\Delta t&0\\\\ 0&1&0\\\\ 0&0&1\\end\{bmatrix\}\\mathbf\{x\}\_\{t\-1\}\+\\boldsymbol\{\\epsilon\}\_\{t\},\\qquad\\mathbf\{y\}\_\{t\}=\\begin\{bmatrix\}1&0&0\\\\ 0&1&1\\end\{bmatrix\}\\mathbf\{x\}\_\{t\}\+\\boldsymbol\{\\eta\}\_\{t\},\(10\)withΔ​t=0\.1\\Delta t=0\.1,ϵt∼𝒩⁡\(0,σ2​𝐈3\)\\boldsymbol\{\\epsilon\}\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\\mathbf\{I\}\_\{3\}\), and𝜼t∼𝒩⁡\(0,σ2​𝐈2\)\\boldsymbol\{\\eta\}\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\\mathbf\{I\}\_\{2\}\), whereσ=0\.1\\sigma=0\.1\. The system is simulated fromt=1t=1tot=60t=60with𝐱0=\[1,0,0\]⊤\\mathbf\{x\}\_\{0\}=\[1,0,0\]^\{\\top\}\.

Figure 1:Linear system with biased sensors\. Top: true and estimated trajectories with 90% credible intervals for a representative seed\. Bottom: aggregate metrics across 10 seeds\. Arrows indicate whether larger or smaller values are preferred, and darker shading denotes better performance\.State trajectories for a representative seed and aggregate results over 10 seeds are shown in Fig\.[1](https://arxiv.org/html/2609.30498#S4.F1)\. All PFs and EnKF useN=2000N=2000particles\. As expected in this linear\-Gaussian setting, all methods achieve NMSE and PNLL values comparable to those of the Kalman Filter\. OPF samples from the closed\-form locally optimal proposal and therefore provide a direct reference for optimal\-proposal behavior\. NOPF closely matches OPF and ISPF in estimation accuracy and ESS, while maintaining wall time comparable to simple PF baselines\. The strong overlap of the 90% credible intervals and agreement across metrics indicate that NOPF closely approximates optimal\-proposal behavior in this setting, substantially improving sample efficiency with minimal computational overhead\.

### 4\.22D Robot Localization with Range and Bearing Observations

The second benchmark is a 2D nonlinear localization problem with four known landmarks\. The 7\-dimensional state is𝐱t=\[ptx,pty,θt,vt,ωt,btv,btω\]⊤\\mathbf\{x\}\_\{t\}=\[p\_\{t\}^\{x\},p\_\{t\}^\{y\},\\theta\_\{t\},v\_\{t\},\\omega\_\{t\},b\_\{t\}^\{v\},b\_\{t\}^\{\\omega\}\]^\{\\top\}, whereptxp\_\{t\}^\{x\}andptyp\_\{t\}^\{y\}denote robot position,θt\\theta\_\{t\}heading,vtv\_\{t\}andωt\\omega\_\{t\}linear and angular velocities, andbtv,btωb\_\{t\}^\{v\},b\_\{t\}^\{\\omega\}odometry biases\. The transition dynamics follow a nonlinear unicycle\-type motion with a first\-order velocity response to time\-varying controls\. The observations are represented as a 10\-dimensional vector comprising noisy range and bearing measurements for all four landmarks, along with biased odometry data\. Full model details are provided in Appendix[F\.2](https://arxiv.org/html/2609.30498#A6.SS2)\.

Representative snapshots at selected times from a single trajectory, together with aggregate results over 10 seeds, are shown in Fig\.[2](https://arxiv.org/html/2609.30498#S4.F2)\. All filters useN=10,000N=10,000particles\. Because the range and bearing observations produce approximately unimodal filtering distributions, EnKF performs particularly well and provides a strong Gaussian baseline\. NOPF achieves NMSE, PNLL, and ES comparable to EnKF while yielding substantially higher ESS than the other PF baselines\. ISPF also improves over standard PF proposals but is considerably more expensive due to its inner Monte Carlo sampling\. These results show that NOPF improves proposal efficiency in a nonlinear but approximately unimodal setting, without the computational cost of inner sampling or sigma\-point\-based proposals\.

![Refer to caption](https://arxiv.org/html/2609.30498v1/ex2_crop.png)Figure 2:2D robot localization with range and bearing observations\. Top: initial state and later\-time posterior summaries for a representative seed with NOPF particles and the EnKF covariance approximation\. Bottom: aggregate metrics across 10 seeds\.
### 4\.32D Robot Localization with Biased Range\-Only Observations

This benchmark employs dynamics similar to the previous example, but observations are now limited to a 3\-dimensional vector comprising a noisy, biased range measurement from a landmark at the origin, together with noisy, biased odometry measurements for linear and angular velocity\. The range biasbtrb\_\{t\}^\{r\}follows a random walk and is incorporated into the state, yielding the 8\-dimensional state𝐱t=\[ptx,pty,θt,vt,ωt,btr,btv,btω\]⊤\\mathbf\{x\}\_\{t\}=\[p\_\{t\}^\{x\},p\_\{t\}^\{y\},\\theta\_\{t\},v\_\{t\},\\omega\_\{t\},b\_\{t\}^\{r\},b\_\{t\}^\{v\},b\_\{t\}^\{\\omega\}\]^\{\\top\}\. This reduced observation information induces the multimodal ambiguity absent in the previous example\. Full forward and observation model details are provided in Appendix[F\.3](https://arxiv.org/html/2609.30498#A6.SS3)\.

Because a single range measurement constrains position to a circle, the initial filtering prior is specified as a mirrored two\-component mixture prior along thexx\-coordinate\. As the system evolves, the transition dynamics eventually make one branch more consistent with the observation sequence\. Therefore, this benchmark tests whether a filter can preserve and subsequently resolve a discrete multimodal ambiguity rather than prematurely collapsing onto an incorrect mode\.

![Refer to caption](https://arxiv.org/html/2609.30498v1/ex3_crop.png)Figure 3:2D robot localization with biased range\-only observations\. Top: posterior snapshots showing whether filters preserve and resolve posterior branch ambiguity over time for a representative seed\. Bottom: aggregate metrics across 10 seeds\.Representative snapshots at selected times from a single trajectory, together with aggregate results over 10 seeds usingN=10,000N=10\{,\}000particles, are shown in Fig\.[3](https://arxiv.org/html/2609.30498#S4.F3)\. This example directly probes multimodal posterior tracking\. The EnKF Gaussian approximation spreads probability mass across separated modes, producing an overly diffuse approximation that does not faithfully represent the bimodal posterior\. NOPF preserves and resolves the multimodal structure more reliably than the competing filters, improving distributional accuracy as measured by PNLL and ES while also achieving high ESS\. ISPF provides a strong inner\-sampling approximation to the locally optimal proposal, but is substantially more expensive\. These results indicate that the learned proposal improves both sample efficiency and multimodal posterior tracking when observations are weak and ambiguous\.

### 4\.43D Robot Localization with Unordered Range\-Only Observations

The final benchmark is a 20\-dimensional nonlinear 3D localization problem with four fixed anchors lying on the planez=0z=0\. Unlike the previous range\-only benchmark, where observations consist of labeled ranges, the filter now observes an unordered set of four range returns and does not know which anchor generated which measurement\. This introduces data\-association ambiguity in addition to state uncertainty\. The observation also includes three proprioceptive channels, consisting of biased speed, yaw\-rate, and pitch\-rate measurements, yielding a 7\-dimensional observation vector\. The latent state includes 3D position, orientation, motion states, proprioceptive and range biases, slowly varying drift terms representative of wind disturbances, and multiplicative calibration errors\. Full model details are provided in Appendix[F\.4](https://arxiv.org/html/2609.30498#A6.SS4)\. In the PF likelihood, anchor identity is handled by marginalizing over all4\!4\!measurement\-to\-anchor assignments\.

The filtering prior is specified as a two\-component Gaussian mixture\. Together with the unlabeled measurements, this results in a challenging, high\-dimensional, multimodal posterior with multiple plausible branches\. This benchmark, therefore, tests whether a filter can preserve and eventually resolve multimodal ambiguities rather than collapse prematurely in high\-dimensional settings\.

Representative snapshots at selected times from a single trajectory, together with aggregate results over 10 seeds usingN=50,000N=50,000particles, are shown in Fig\.[4](https://arxiv.org/html/2609.30498#S4.F4)\. This is the most challenging benchmark considered, jointly stressing proposal quality, multimodal posterior tracking, and robustness to latent data association\. NOPF improves PNLL and ES over standard PF baselines while maintaining high ESS and moderate computational cost\. ISPF achieves comparable distributional accuracy and higher ESS, suggesting that the local proposal structure is more complex than a single Gaussian in this benchmark\. Nevertheless, NOPF achieves similar performance with substantially lower wall time, highlighting the robustness of the Gaussian\-learned proposal as a computationally efficient approximation\.

![Refer to caption](https://arxiv.org/html/2609.30498v1/ex4_crop.png)Figure 4:3D robot localization with unordered range\-only observations\. Top: snapshots showing the evolution of multimodal filtering distribution for a representative seed\. Bottom: aggregate metrics across 10 seeds\.

## 5Discussion and Limitations

We introduced Neural Optimal Particle Filters \(NOPFs\), which combine classical PFs with learned proposal mechanisms while preserving the sequential importance sampling framework\. NOPF learns an amortized approximation to the locally optimal proposal\. This provides a principled local target because, within recursive one\-step PF updates where the current observation is used to sample the new particle but does not alter the previous particle locations, the locally optimal proposal gives the least\-degenerate benchmark[Snyder et al\. \(2015\)](https://arxiv.org/html/2609.30498#bib.bib26)\. This remains true even when richer proposal classes are allowed to condition on particle histories or the previous ensemble[Doucet et al\. \(2000\)](https://arxiv.org/html/2609.30498#bib.bib22);[van Leeuwen et al\. \(2019\)](https://arxiv.org/html/2609.30498#bib.bib4)\. By biasing particles toward states that are both dynamically plausible and observation\-consistent, NOPF reduces weight degeneracy with modest additional cost, primarily from a neural network forward pass and Gaussian density evaluation\. Because the learned proposal is used in the same place as any PF proposal and is still corrected by the standard importance weights, NOPF improves particle placement without changing the target filtering distribution\.

Across the benchmarks, NOPF achieves a favorable tradeoff between estimation accuracy, sample efficiency, and computational cost\. It matches Kalman\-based and Gaussian baselines in approximately unimodal settings and outperforms standard PF baselines in multimodal problems\. The gains from learned proposals become more pronounced as observational ambiguity and posterior multimodality increase\. Since the proposal is amortized offline, inference requires no online optimization or sigma\-point construction beyond a neural network forward pass, avoiding computational bottlenecks associated with methods such as UPF and ISPF\.

The Gaussian conditional proposal used here does not prevent the overall filtering distributionp\(𝐱t∣𝐲1:t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{y\}\_\{1:t\}\)from representing multimodal beliefs\. The Gaussian assumption applies only to the proposal conditioned on a single ancestor particle𝐱t−1\\mathbf\{x\}\_\{t\-1\}\. After combining these conditional proposals over the weighted particle ensemble, the resulting posterior approximation is effectively a mixture and can remain highly non\-Gaussian\. Conditioning on𝐱t−1\\mathbf\{x\}\_\{t\-1\}often makes the local proposal substantially simpler than the full posterior, motivating the Gaussian parameterization as a useful inductive bias\.

The method has three main limitations\. First, the proposal network is trained offline using trajectories generated from the forward model\. This can lead to train\-test mismatch if the filtering distribution encountered during inference drifts outside the training distribution, or if the simulator does not faithfully represent the true system\. In such cases, proposal quality may degrade\. However, as long as the proposal has adequate support, the standard importance\-weight update preserves the target distribution in the usual importance\-sampling sense, and poor proposals primarily appear through increased weight variance rather than asymptotic bias\.

Second, the current training implementation uses reparameterized samples from the learned proposal to optimize the ELBO\. This requires differentiable evaluations of the transition density and likelihood with respect to the proposed state\. Systems with non\-differentiable simulators or observation operators would require alternative gradient estimators or differentiable surrogate likelihood models\.

Third, the Gaussian conditional proposal may be restrictive when the locally optimal proposal for a single ancestor particle is strongly multimodal, as under severe data\-association ambiguity\. In such cases, a unimodal proposal may average across modes and place samples in low\-probability regions, thereby increasing the variance of the importance weights\. The results suggest that the Gaussian approximation is robust in the benchmarks considered, but richer tractable proposal families, such as mixtures or normalizing flows, may be beneficial in more strongly multimodal local proposal settings\.

Overall, these results suggest that learned proposal mechanisms can improve the practicality of PFs without sacrificing principled Bayesian weighting\. Future work includes more expressive proposal families and online adaptation mechanisms to reduce the train\-test distribution shift\.

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## Appendix AAdditional Results

This section reports supplementary results supporting the main text\. We examine sensitivity to the number of particles and include additional diagnostics not shown in the main paper\.

### A\.1Metrics vs Number of Particles

We report the variation of the metrics introduced in Section[3\.3](https://arxiv.org/html/2609.30498#S3.SS3)and Appendix[C\.6](https://arxiv.org/html/2609.30498#A3.SS6)as the number of particlesNNis varied, for all benchmarks in Fig\.[5](https://arxiv.org/html/2609.30498#A1.F5)–[8](https://arxiv.org/html/2609.30498#A1.F8)\. All results are averaged over 10 random seeds\. Across all benchmarks, NOPF maintains consistently higher effective sample sizes and competitive or improved accuracy across a wide range of particle counts\.

Figure 5:Linear system with biased sensors: Metrics as a function of the number of particlesNN\.Figure 6:2D robot localization with range and bearing observations: Metrics as a function of the number of particlesNN\.Figure 7:2D robot localization with biased range\-only observations: Metrics as a function of the number of particlesNN\.Figure 8:3D robot localization with unordered range\-only observations: Metrics as a function of the number of particlesNN\.
### A\.2Tabulated Results

This section reports complete per\-filter results for each of the four benchmarks introduced in Section[4](https://arxiv.org/html/2609.30498#S4)\. For each benchmark, we evaluate the applicable filters described in Appendix[E](https://arxiv.org/html/2609.30498#A5)\. For the linear\-Gaussian sensor\-bias benchmark, we additionally include the Kalman Filter and the Optimal Particle Filter\. Each scalar metric is computed per seed using the procedure in Appendix[C](https://arxiv.org/html/2609.30498#A3)and aggregated across1010independent random seeds; we report the across\-seed mean±\\pmsample standard deviation in the formatμ±σ\\mu\\pm\\sigma\.

The metric set in each table extends the five summary metrics highlighted in the main text with the two distributional diagnostics from Appendix[C\.6](https://arxiv.org/html/2609.30498#A3.SS6), giving seven metrics per filter:NMSE\(↓\\downarrow\),PNLL\(↓\\downarrow\),ES\(↓\\downarrow\),ESS\(↑\\uparrow\),WT\(↓\\downarrow\),MADE\(↓\\downarrow\), andJoint PICP\(→0\.9\\to 0\.9\)\. The arrow next to each header in Tables[1](https://arxiv.org/html/2609.30498#A1.T1)–[4](https://arxiv.org/html/2609.30498#A1.T4)indicates the preferred direction\. For Joint PICP, the target value is0\.90\.9\.

Table 1:Mean±\\pmstandard deviation across 10 seeds for the linear system with biased sensors benchmark \(Section[4\.1](https://arxiv.org/html/2609.30498#S4.SS1)\)\. Bold marks the top two per column \(closest to0\.90\.9for Joint PICP\)\.Table 2:Mean±\\pmstandard deviation across 10 seeds for the 2D robot localization benchmark with range and bearing observations \(Section[4\.2](https://arxiv.org/html/2609.30498#S4.SS2)\)\. Bold marks the top two per column \(closest to0\.90\.9for Joint PICP\)\.Table 3:Mean±\\pmstandard deviation across 10 seeds for the 2D robot localization benchmark with biased range\-only observations \(Section[4\.3](https://arxiv.org/html/2609.30498#S4.SS3)\)\. Bold marks the top two per column \(closest to0\.90\.9for Joint PICP\)\.Table 4:Mean±\\pmstandard deviation across 10 seeds for the 3D robot localization benchmark with unordered range\-only observations \(Section[4\.4](https://arxiv.org/html/2609.30498#S4.SS4)\)\. Bold marks the top two per column \(closest to0\.90\.9for Joint PICP\)\.

## Appendix BNeural Proposal: Architecture and Training

This appendix expands on the implementation of the learned proposal density introduced in Sec\.[3](https://arxiv.org/html/2609.30498#S3)\. We give the explicit conditional\-Gaussian parameterization used in all experiments \(Appendix[B\.1](https://arxiv.org/html/2609.30498#A2.SS1)\), state the training objective \(Appendix[B\.2](https://arxiv.org/html/2609.30498#A2.SS2)\), and describe the offline training and online filtering algorithms \(Appendix[B\.3](https://arxiv.org/html/2609.30498#A2.SS3)\)\. All choices are common across benchmarks unless explicitly noted; benchmark\-specific overrides \(network sizes, clamping ranges, training hyperparameters\) are summarized in Table[5](https://arxiv.org/html/2609.30498#A2.T5)and in Appendix[F](https://arxiv.org/html/2609.30498#A6)\. The metrics used to evaluate the resulting filter and the hardware on which they are reported are deferred to Appendices[C](https://arxiv.org/html/2609.30498#A3)and[D](https://arxiv.org/html/2609.30498#A4), respectively\.

### B\.1Parameterization of Mean and Covariance Matrices for the Learned Proposal

The NeuralPF proposal is parameterized as the conditional Gaussian

qθ​\(𝐱t∣𝐱t−1,𝐲t,t\)=𝒩⁡\(𝐱t,𝝁θ​\(𝐱^t,𝐲t,t\),Lθ,x​\(𝐱^t,𝐲t,t\)​Lθ,x​\(𝐱^t,𝐲t,t\)⊤\),\\displaystyle q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)=\\mathcal\{N\}\\\!\\left\(\\mathbf\{x\}\_\{t\};\\boldsymbol\{\\mu\}\_\{\\theta\}\(\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\),L\_\{\\theta,x\}\(\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\)L\_\{\\theta,x\}\(\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\)^\{\\top\}\\right\),\(11\)where𝐱^t=f⁡\(𝐱t−1,t−1\)\\hat\{\\mathbf\{x\}\}\_\{t\}=f\(\\mathbf\{x\}\_\{t\-1\},t\-1\)is the deterministic noise\-free propagation of the previous state\. The network input is

𝐮t=\[𝐱^t,𝐲t,t\]∈ℝdx\+dy\+1\.\\displaystyle\\mathbf\{u\}\_\{t\}=\[\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\]\\in\\mathbb\{R\}^\{d\_\{x\}\+d\_\{y\}\+1\}\.\(12\)Using𝐱^t\\hat\{\\mathbf\{x\}\}\_\{t\}avoids requiring the network to relearn the forward model and instead focuses learning on the observation\-informed proposal correction\.

The proposal is parameterized in whitened residual coordinates\. LetQQdenote the forward\-noise covariance and let𝝈f=diag⁡\(Q\)\\boldsymbol\{\\sigma\}\_\{f\}=\\sqrt\{\\operatorname\{diag\}\(Q\)\}denote the vector of forward\-noise standard deviations\. Since all active benchmarks use diagonalQQ, the proposal\-whitening scale is computed elementwise as

\[𝐬white\]j=max\(cwhite\[𝝈f\]j,ϵwhite\),j=1,…,dx,\\displaystyle\[\\mathbf\{s\}\_\{\\mathrm\{white\}\}\]\_\{j\}=\\max\\\!\\left\(c\_\{\\mathrm\{white\}\}\[\\boldsymbol\{\\sigma\}\_\{f\}\]\_\{j\},\\epsilon\_\{\\mathrm\{white\}\}\\right\),\\qquad j=1,\\ldots,d\_\{x\},\(13\)wherecwhitec\_\{\\mathrm\{white\}\}andϵwhite\\epsilon\_\{\\mathrm\{white\}\}are fixed hyperparameters\. In all experiments,cwhite=1c\_\{\\mathrm\{white\}\}=1andϵwhite=10−3\\epsilon\_\{\\mathrm\{white\}\}=10^\{\-3\}\. DefiningD=diag⁡\(𝐬white\)D=\\operatorname\{diag\}\(\\mathbf\{s\}\_\{\\mathrm\{white\}\}\), the whitened residual coordinate is

𝐳t=D−1​\(𝐱t−𝐱^t\)\.\\displaystyle\\mathbf\{z\}\_\{t\}=D^\{\-1\}\(\\mathbf\{x\}\_\{t\}\-\\hat\{\\mathbf\{x\}\}\_\{t\}\)\.\(14\)This scaling normalizes proposal residuals to the scale of the forward uncertainty, thereby improving numerical conditioning during training\.

The mean head predicts a residual mean in whitened coordinates\. Ifgθμ​\(𝐮t\)g^\{\\mu\}\_\{\\theta\}\(\\mathbf\{u\}\_\{t\}\)denotes the raw mean output, the proposal mean in state coordinates is

𝝁θ​\(𝐱^t,𝐲t,t\)=𝐱^t\+D​gθμ​\(𝐮t\)\.\\displaystyle\\boldsymbol\{\\mu\}\_\{\\theta\}\(\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\)=\\hat\{\\mathbf\{x\}\}\_\{t\}\+Dg^\{\\mu\}\_\{\\theta\}\(\\mathbf\{u\}\_\{t\}\)\.\(15\)The covariance head predicts thedx​\(dx\+1\)/2d\_\{x\}\(d\_\{x\}\+1\)/2entries of a lower\-triangular Cholesky factorLθ,zL\_\{\\theta,z\}in whitened coordinates\. The corresponding Cholesky factor in state coordinates is

Lθ,x=D​Lθ,z\.\\displaystyle L\_\{\\theta,x\}=DL\_\{\\theta,z\}\.\(16\)Positive definiteness is enforced through the Cholesky parameterization\. The diagonal entries ofLθ,zL\_\{\\theta,z\}are parameterized as

\[Lθ,z\]j​j=softplus⁡\(clamp⁡\(aj,−5,20\)\),\\displaystyle\[L\_\{\\theta,z\}\]\_\{jj\}=\\operatorname\{softplus\}\\\!\\left\(\\operatorname\{clamp\}\(a\_\{j\},\-5,20\)\\right\),\(17\)whereaja\_\{j\}is the corresponding raw diagonal output\.

##### Residual network backbone\.

The default backbone in all experiments is a residual multilayer perceptron\. The input layer maps𝐮t\\mathbf\{u\}\_\{t\}to a hidden representation𝐫0∈ℝH\\mathbf\{r\}\_\{0\}\\in\\mathbb\{R\}^\{H\}\. Each residual block computes

𝐮ℓ\\displaystyle\\mathbf\{u\}\_\{\\ell\}=σ⁡\(BN1⁡\(W1,ℓ​𝐫ℓ\+𝐛1,ℓ\)\),\\displaystyle=\\sigma\(\\operatorname\{BN\}\_\{1\}\(W\_\{1,\\ell\}\\mathbf\{r\}\_\{\\ell\}\+\\mathbf\{b\}\_\{1,\\ell\}\)\),𝐯ℓ\\displaystyle\\mathbf\{v\}\_\{\\ell\}=BN2⁡\(W2,ℓ​𝐮ℓ\+𝐛2,ℓ\),\\displaystyle=\\operatorname\{BN\}\_\{2\}\(W\_\{2,\\ell\}\\mathbf\{u\}\_\{\\ell\}\+\\mathbf\{b\}\_\{2,\\ell\}\),\(18\)𝐫ℓ\+1\\displaystyle\\mathbf\{r\}\_\{\\ell\+1\}=σ⁡\(𝐯ℓ\+𝐫ℓ\),\\displaystyle=\\sigma\(\\mathbf\{v\}\_\{\\ell\}\+\\mathbf\{r\}\_\{\\ell\}\),whereσ\\sigmais ReLU andBN\\operatorname\{BN\}denotes batch normalization\. Separate linear heads map the final hidden vector to the residual mean and Cholesky\-vector outputs\.

Table 5:Neural proposal hyperparameters used in the experiments\.All proposal networks are trained using Adam with learning rate10−310^\{\-3\}\. The expectation with respect toqθq\_\{\\theta\}inℒELBO​\(θ,𝐱t−1,𝐲t,t\)\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)is estimated using 100 Monte Carlo samples per training tuple\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\(see Section[B\.2](https://arxiv.org/html/2609.30498#A2.SS2)\)\.

### B\.2Loss Function

For a fixed conditioning tuple\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\), where dependence onttis henceforth left implicit in the proposal parameterization, the optimal one\-step proposal isp⁡\(𝐱t∣𝐱t−1,𝐲t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\. For any candidate proposalqθ​\(𝐱t∣𝐱t−1,𝐲t\)q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\),

DKL\(qθ\(𝐱t∣𝐱t−1,𝐲t\)∥p\(𝐱t∣𝐱t−1,𝐲t\)\)=𝔼qθ\[logqθ​\(𝐱t∣𝐱t−1,𝐲t\)p⁡\(𝐱t∣𝐱t−1,𝐲t\)\]\.\\displaystyle D\_\{\\mathrm\{KL\}\}\\\!\\left\(q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\,\\\|\\,p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\right\)=\\mathbb\{E\}\_\{q\_\{\\theta\}\}\\left\[\\log\\frac\{q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\}\{p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\}\\right\]\.\(19\)
Using Bayes’ rule,

p⁡\(𝐱t∣𝐱t−1,𝐲t\)=p⁡\(𝐲t∣𝐱t\)​p​\(𝐱t∣𝐱t−1\)p⁡\(𝐲t∣𝐱t−1\),\\displaystyle p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)=\\frac\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\}\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\},\(20\)which gives

DKL\(qθ∥p\)=\\displaystyle D\_\{\\mathrm\{KL\}\}\(q\_\{\\theta\}\\\|p\)=𝔼qθ​\[log⁡qθ​\(𝐱t∣𝐱t−1,𝐲t\)−log⁡p⁡\(𝐲t∣𝐱t\)−log⁡p⁡\(𝐱t∣𝐱t−1\)\]\\displaystyle\\mathbb\{E\}\_\{q\_\{\\theta\}\}\\left\[\\log q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\-\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)\-\\log p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\\right\]\(21\)\+log⁡p⁡\(𝐲t∣𝐱t−1\)\.\\displaystyle\+\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\.\(22\)
Rearranging yields the ELBO identity

log⁡p⁡\(𝐲t∣𝐱t−1\)=\\displaystyle\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)=𝔼qθ​\[log⁡p⁡\(𝐲t∣𝐱t\)\+log⁡p⁡\(𝐱t∣𝐱t−1\)−log⁡qθ​\(𝐱t∣𝐱t−1,𝐲t\)\]\\displaystyle\\mathbb\{E\}\_\{q\_\{\\theta\}\}\\left\[\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)\+\\log p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\-\\log q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\right\]\(23\)\+DKL\(qθ∥p\)\.\\displaystyle\+D\_\{\\mathrm\{KL\}\}\(q\_\{\\theta\}\\\|p\)\.\(24\)
Sincelog⁡p⁡\(𝐲t∣𝐱t−1\)\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)is constant with respect toθ\\theta, minimizing the KL divergence is equivalent to maximizing the ELBO, or equivalently minimizing the negative ELBO loss used in Eq\.[7](https://arxiv.org/html/2609.30498#S3.E7):

ℒELBO​\(θ,𝐱t−1,𝐲t,t\)=−𝔼qθ​\[log⁡p⁡\(𝐲t∣𝐱t\)\+log⁡p⁡\(𝐱t∣𝐱t−1\)−log⁡qθ​\(𝐱t∣𝐱t−1,𝐲t\)\]\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)=\-\\mathbb\{E\}\_\{q\_\{\\theta\}\}\\Big\[\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)\+\\log p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)\-\\log q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\}\)\\Big\]\.\(25\)The expectation with respect toqθq\_\{\\theta\}is approximated by Monte Carlo using multiple samples drawn fromqθq\_\{\\theta\}for each conditioning tuple\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\.

The full training objective is

ℒ⁡\(θ\)=𝔼\(𝐱t−1,𝐲t,t\)​\[ℒELBO​\(θ,𝐱t−1,𝐲t,t\)\],\\displaystyle\\mathcal\{L\}\(\\theta\)=\\mathbb\{E\}\_\{\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\}\\left\[\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)\\right\],\(26\)where the expectation over\(𝐱t−1,𝐲t,t\)\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)is approximated by averagingℒELBO\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}over minibatches of one\-step transitions\.

### B\.3Training and Inference Algorithms

The training data is generated online during training; the data generation process is a simple rollout of the forward model

Algorithm 4Training Data Generation via Simulated Trajectories1:Forward model

𝐟\\mathbf\{f\}, Observation operator

𝐡\\mathbf\{h\}, initial distribution

p0p\_\{0\}, time grid

\{tk\}k=0T\\\{t\_\{k\}\\\}\_\{k=0\}^\{T\}
2:Number of trajectories

Ntraj=1000N\_\{\\text\{traj\}\}=1000
3:for

j=1,…,Ntrajj=1,\\dots,N\_\{\\text\{traj\}\}do

4:

𝐱0\(j\)∼p0\\mathbf\{x\}\_\{0\}^\{\(j\)\}\\sim p\_\{0\}⊳\\trianglerightSample initial state from prior

5:for

k=1,…,Tk=1,\\dots,Tdo

6:

𝐱k\(j\)←𝐟⁡\(𝐱k−1\(j\),𝐯k\)\\mathbf\{x\}\_\{k\}^\{\(j\)\}\\leftarrow\\mathbf\{f\}\(\\mathbf\{x\}\_\{k\-1\}^\{\(j\)\},\\mathbf\{v\}\_\{k\}\)⊳\\trianglerightStochastic Propagation

7:

𝐲k\(j\)←𝐡⁡\(𝐱k\(j\),𝐰k\)\\mathbf\{y\}\_\{k\}^\{\(j\)\}\\leftarrow\\mathbf\{h\}\(\\mathbf\{x\}\_\{k\}^\{\(j\)\},\\mathbf\{w\}\_\{k\}\)⊳\\trianglerightGenerate noisy observation

8:endfor

9:Return sliding\-window pairs:

10:

𝒟\(j\)←\{\(𝐱k−1\(j\),𝐲k\(j\),tk\)\}k=1T\\mathcal\{D\}^\{\(j\)\}\\leftarrow\\left\\\{\\left\(\\mathbf\{x\}\_\{k\-1\}^\{\(j\)\},\\;\\mathbf\{y\}\_\{k\}^\{\(j\)\},\\;t\_\{k\}\\right\)\\right\\\}\_\{k=1\}^\{T\}
11:endfor

12:return

𝒟=⋃j=1Ntraj𝒟\(j\)\\mathcal\{D\}=\\bigcup\_\{j=1\}^\{N\_\{\\text\{traj\}\}\}\\mathcal\{D\}^\{\(j\)\}

Algorithm 5Proposal Network Training for NOPFs1:Neural proposal network

qθ​\(𝐱t∣𝐱^t,𝐲t,t\)q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\hat\{\\mathbf\{x\}\}\_\{t\},\\mathbf\{y\}\_\{t\},t\)parameterized by

θ\\theta
2:Forward model

𝐟\\mathbf\{f\}, Observation operator

𝐡\\mathbf\{h\}, initial distribution

p0p\_\{0\}, time grid

\{tk\}k=0T\\\{t\_\{k\}\\\}\_\{k=0\}^\{T\}
3:Number of MC samples

SS, learning rate

η\\eta, batch size

BB
4:Initialize proposal network parameters

θ\\theta
5:foreach epochdo

6:Generate training dataset

𝒟\\mathcal\{D\}via Algorithm[4](https://arxiv.org/html/2609.30498#alg4)

7:foreach mini\-batch

\{\(𝐱t−1\(b\),𝐲t\(b\),t\(b\)\)\}b=1B\\\{\(\\mathbf\{x\}\_\{t\-1\}^\{\(b\)\},\\;\\mathbf\{y\}\_\{t\}^\{\(b\)\},\\;t^\{\(b\)\}\)\\\}\_\{b=1\}^\{B\}from

𝒟\\mathcal\{D\}do

8:foreach sample

b=1,…,Bb=1,\\dots,Bdo

9:

𝐱^t\(b\)←𝐟⁡\(𝐱t−1\(b\)\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(b\)\}\\leftarrow\\mathbf\{f\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(b\)\}\)⊳\\trianglerightDeterministic state propagation using noise\-free model

10:

𝝁θ\(b\),vec​\(𝐋θ\(b\)\)←Netθ​\(𝐱^t\(b\),𝐲t\(b\),t\(b\)\)\\boldsymbol\{\\mu\}\_\{\\theta\}^\{\(b\)\},\\;\\text\{vec\}\(\\mathbf\{L\}\_\{\\theta\}^\{\(b\)\}\)\\leftarrow\\textsc\{Net\}\_\{\\theta\}\\\!\\left\(\\hat\{\\mathbf\{x\}\}^\{\(b\)\}\_\{t\},\\;\\mathbf\{y\}\_\{t\}^\{\(b\)\},\\;t^\{\(b\)\}\\right\)
11:

𝝁θ\(b\)←𝝁θ\(b\)\+𝐱^\(b\)\\boldsymbol\{\\mu\}\_\{\\theta\}^\{\(b\)\}\\leftarrow\\boldsymbol\{\\mu\}\_\{\\theta\}^\{\(b\)\}\+\\hat\{\\mathbf\{x\}\}^\{\(b\)\}⊳\\trianglerightResidual connection

12:Construct

𝐋θ\(b\)\\mathbf\{L\}\_\{\\theta\}^\{\(b\)\}\(lower\-triangular\) from

vec​\(𝐋θ\(b\)\)\\text\{vec\}\(\\mathbf\{L\}\_\{\\theta\}^\{\(b\)\}\)⊳\\trianglerightSee Section[B\.1](https://arxiv.org/html/2609.30498#A2.SS1)

13:for

s=1,…,Ss=1,\\dots,Sdo⊳\\trianglerightMonte Carlo Estimate ofℒELBO\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}

14:

𝝃\(s\)∼𝒩⁡\(𝟎,𝐈dx\)\\boldsymbol\{\\xi\}^\{\(s\)\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\_\{x\}\}\)
15:

𝐱t\(b,s\)←𝝁θ\(b\)\+𝐋θ\(b\)​𝝃\(s\)\\mathbf\{x\}\_\{t\}^\{\(b,s\)\}\\leftarrow\\boldsymbol\{\\mu\}\_\{\\theta\}^\{\(b\)\}\+\\mathbf\{L\}\_\{\\theta\}^\{\(b\)\}\\,\\boldsymbol\{\\xi\}^\{\(s\)\}⊳\\trianglerightReparameterization trick

16:endfor

17:Estimate

ℒELBO​\(θ,𝐱t−1\(b\),𝐲t\(b\),t\(b\)\)\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\}^\{\(b\)\},\\;\\mathbf\{y\}\_\{t\}^\{\(b\)\},\\;t^\{\(b\)\}\)using

SSsamples

18:endfor

19:

ℒ⁡\(θ\)←1B​∑b=1BLELBO​\(θ,𝐱t−1\(b\),𝐲t\(b\),t\(b\)\)\\mathcal\{L\}\(\\theta\)\\leftarrow\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}L\_\{\\mathrm\{ELBO\}\}\(\\theta;\\mathbf\{x\}\_\{t\-1\}^\{\(b\)\},\\;\\mathbf\{y\}\_\{t\}^\{\(b\)\},\\;t^\{\(b\)\}\)⊳\\trianglerightMonte Carlo estimate over batch

20:

θ←θ−η​∇θℒ​\(θ\)\\theta\\leftarrow\\theta\-\\eta\\,\\nabla\_\{\\theta\}\\mathcal\{L\}\(\\theta\)
21:endfor

22:endfor

Algorithm 6Inference with the Learned Particle Filter1:Trained proposal

qθq\_\{\\theta\},Forward model

𝐟\\mathbf\{f\}, Observation operator

𝐡\\mathbf\{h\}, true observations

\{𝐲k\}k=1T\\\{\\mathbf\{y\}\_\{k\}\\\}\_\{k=1\}^\{T\}, initial prior

p0p\_\{0\}, number of particles

NN
2:Weighted particle approximation of the filtering distribution at each step

3:Initialize:

\{𝐱0\(i\)\}i=1N∼p0\\\{\\mathbf\{x\}\_\{0\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\sim p\_\{0\},

w0\(i\)←1/Nw\_\{0\}^\{\(i\)\}\\leftarrow 1/N
4:for

k=1,…,Tk=1,\\dots,Tdo

5:for

i=1,…,Ni=1,\\dots,Ndo

6:

𝐱^k\(i\)←f⁡\(𝐱k−1\(i\)\)\\hat\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\}\\leftarrow f\(\\mathbf\{x\}\_\{k\-1\}^\{\(i\)\}\)⊳\\trianglerightDeterministic state propagation using noise\-free model

7:

𝝁\(i\),𝐋\(i\)←qθ​\(𝐱^k\(i\),𝐲k,tk\)\\boldsymbol\{\\mu\}^\{\(i\)\},\\,\\mathbf\{L\}^\{\(i\)\}\\leftarrow q\_\{\\theta\}\(\\hat\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\},\\,\\mathbf\{y\}\_\{k\},\\,t\_\{k\}\)
8:

𝐱~k\(i\)∼𝒩⁡\(𝝁\(i\),𝐋\(i\)​𝐋\(i\)⊤\)\\tilde\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\}\\sim\\mathcal\{N\}\\\!\\left\(\\boldsymbol\{\\mu\}^\{\(i\)\},\\,\\mathbf\{L\}^\{\(i\)\}\{\\mathbf\{L\}^\{\(i\)\}\}^\{\\top\}\\right\)
9:

log⁡w~k\(i\)←log⁡p⁡\(𝐲k∣𝐱~k\(i\)\)\+log⁡p⁡\(𝐱~k\(i\)∣𝐱k−1\(i\)\)−log⁡qθ​\(𝐱~k\(i\)∣𝐱^k\(i\),𝐲k,tk\)\\log\\tilde\{w\}\_\{k\}^\{\(i\)\}\\leftarrow\\log p\(\\mathbf\{y\}\_\{k\}\\mid\\tilde\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\}\)\+\\log p\(\\tilde\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{k\-1\}^\{\(i\)\}\)\-\\log q\_\{\\theta\}\(\\tilde\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\}\\mid\\hat\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\},\\,\\mathbf\{y\}\_\{k\},\\,t\_\{k\}\)
10:endfor

11:

wk\(i\)←softmax​\(\{log⁡w~k\(j\)\}j=1N\)iw\_\{k\}^\{\(i\)\}\\leftarrow\\mathrm\{softmax\}\\\!\\left\(\\\{\\log\\tilde\{w\}\_\{k\}^\{\(j\)\}\\\}\_\{j=1\}^\{N\}\\right\)\_\{i\}⊳\\trianglerightNormalize Weights

12:

\{𝐱k\(i\),wk\(i\)\}←Resample​\(\{𝐱~k\(i\),wk\(i\)\}\)\\\{\\mathbf\{x\}\_\{k\}^\{\(i\)\},\\,w\_\{k\}^\{\(i\)\}\\\}\\leftarrow\\textsc\{Resample\}\\\!\\left\(\\\{\\tilde\{\\mathbf\{x\}\}\_\{k\}^\{\(i\)\},\\,w\_\{k\}^\{\(i\)\}\\\}\\right\);

wk\(i\)←1/Nw\_\{k\}^\{\(i\)\}\\leftarrow 1/N⊳\\trianglerightResample

13:endfor

14:Output:

\{\(𝐱k\(i\),wk\(i\)\)i=1N\}k=1T\\Big\\\{\\left\(\\mathbf\{x\}\_\{k\}^\{\(i\)\},\\,w\_\{k\}^\{\(i\)\}\\right\)\_\{i=1\}^\{N\}\\Big\\\}\_\{k=1\}^\{T\}

## Appendix CEvaluation Metrics

All particle metrics are reported as time averages over an evaluation index set𝒯=\{1,…,T\}\\mathcal\{T\}=\\\{1,\\dots,T\\\}, whereTTis the trajectory length and𝒯\\mathcal\{T\}excludes the initial transient \(and, where applicable, the final step at which a metric is undefined, for e\.g\., PNLL\)\. The cardinality of this set is\|𝒯\|\|\\mathcal\{T\}\|\. We denote the truth at timettby𝐱t⋆\\mathbf\{x\}\_\{t\}^\{\\star\}\(state\) and𝐲t⋆\\mathbf\{y\}\_\{t\}^\{\\star\}\(observation\), the state dimension bydxd\_\{x\}, and the number of Monte Carlo replicates used by sampling\-based estimators byMM\.

### C\.1Normalized Mean Squared Error

At every time step we form the weighted posterior mean𝝁^t\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}using the linear average on non\-angular coordinates and the circular \(atan2\) average on angular coordinates\. The per\-step mean squared error is the average squared component of the wrapped residual between𝝁^t\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}and the truth,

MSEt=1dx​‖Δ𝒜​\(𝝁^t,𝐱t⋆\)‖22,\\displaystyle\\mathrm\{MSE\}\_\{t\}=\\tfrac\{1\}\{d\_\{x\}\}\\,\\\|\\Delta\_\{\\mathcal\{A\}\}\(\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\},\\mathbf\{x\}\_\{t\}^\{\\star\}\)\\\|\_\{2\}^\{2\},\(27\)which is then time\-averaged and normalized by the time\-and\-coordinate\-averaged squared truth on the same window:

NMSE=1\|𝒯\|​∑t∈𝒯1dx​‖Δ𝒜​\(𝝁^t,𝐱t⋆\)‖221\|𝒯\|​dx​∑t∈𝒯‖𝐱t⋆‖22\+ε,\\mathrm\{NMSE\}=\\frac\{\\dfrac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\dfrac\{1\}\{d\_\{x\}\}\\,\\\|\\Delta\_\{\\mathcal\{A\}\}\(\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\},\\mathbf\{x\}\_\{t\}^\{\\star\}\)\\\|\_\{2\}^\{2\}\}\{\\dfrac\{1\}\{\|\\mathcal\{T\}\|\\,d\_\{x\}\}\\sum\_\{t\\in\\mathcal\{T\}\}\\\|\\mathbf\{x\}\_\{t\}^\{\\star\}\\\|\_\{2\}^\{2\}\+\\varepsilon\},\(28\)whereε\\varepsilonis a small numerical floor \(set to10310^\{3\}times machine epsilon in the code\) that prevents division by zero when the truth norm is vanishingly small\. NMSE is a point\-estimate metric: it collapses the entire particle cloud to its mean and is therefore not informative in multimodal regimes where𝝁^t\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}falls in a region of low posterior mass\.

### C\.2Predictive Negative Log Likelihood

The Predictive NLL \(PNLL\) scores how well the filter predicts the next observation𝐲t\+1⋆\\mathbf\{y\}\_\{t\+1\}^\{\\star\}given everything observed up to timett\. Each particle is propagated one step through the stochastic dynamics \(with randomly sample noise\)𝐱t\+1\(i\)=f⁡\(𝐱t\(i\),𝐯t\)\\mathbf\{x\}\_\{t\+1\}^\{\(i\)\}=f\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{v\}\_\{t\}\)\. The one\-step predictive density is approximated by the weighted particle mixture

p^\(𝐲t\+1∣𝐲1:t\)=∑i=1Nwt\(i\)p\(𝐲t\+1\|𝐱t\+1\(i\)\),\\widehat\{p\}\(\\mathbf\{y\}\_\{t\+1\}\\mid\\mathbf\{y\}\_\{1:t\}\)=\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}\\,p\\\!\\left\(\\mathbf\{y\}\_\{t\+1\}\\,\\big\|\\,\\mathbf\{x\}\_\{t\+1\}^\{\(i\)\}\\right\),\(29\)and the metric is the time\-averaged negative log of this quantity evaluated at the true next observation,

PNLL=1\|𝒯\|∑t∈𝒯\[−logp^\(𝐲t\+1⋆∣𝐲1:t⋆\)\]\.\\mathrm\{PNLL\}=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\Big\[\-\\log\\widehat\{p\}\(\\mathbf\{y\}\_\{t\+1\}^\{\\star\}\\mid\\mathbf\{y\}\_\{1:t\}^\{\\star\}\)\\Big\]\.\(30\)The score is defined only fort<Tt<T, so the indext=Tt=Tis excluded from𝒯\\mathcal\{T\}for this metric\. Unlike state\-space scores, PNLL is well\-defined even when the posterior is symmetric or label\-ambiguous: any two states that produce the same observation distribution receive identical predictive scores\.

### C\.3Energy Score

The energy score, considering wrapping of the angular states, is defined as

ESt=𝔼𝐗∼p^t​\[d𝒜​\(𝐗,𝐱t⋆\)\]−12​𝔼𝐗,𝐗′∼p^t​\[d𝒜​\(𝐗,𝐗′\)\]\.\\displaystyle\\mathrm\{ES\}\_\{t\}=\\mathbb\{E\}\_\{\\mathbf\{X\}\\sim\\widehat\{p\}\_\{t\}\}\[d\_\{\\mathcal\{A\}\}\(\\mathbf\{X\},\\mathbf\{x\}\_\{t\}^\{\\star\}\)\]\-\\frac\{1\}\{2\}\\mathbb\{E\}\_\{\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\\sim\\widehat\{p\}\_\{t\}\}\[d\_\{\\mathcal\{A\}\}\(\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\)\]\.\(31\)
The first expectation,𝔼𝐗∼p^t​\[d𝒜​\(𝐗,𝐱t⋆\)\]\\mathbb\{E\}\_\{\\mathbf\{X\}\\sim\\widehat\{p\}\_\{t\}\}\[d\_\{\\mathcal\{A\}\}\(\\mathbf\{X\},\\mathbf\{x\}\_\{t\}^\{\\star\}\)\], is computed under the weighted empirical distribution as∑i=1Nwt\(i\)​d𝒜​\(𝐱t\(i\),𝐱t⋆\)\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}\\,d\_\{\\mathcal\{A\}\}\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{x\}\_\{t\}^\{\\star\}\), since this requires only one wrapped\-distance evaluation per particle\.

The second expectation,𝔼𝐗,𝐗′∼p^t​\[d𝒜​\(𝐗,𝐗′\)\]\\mathbb\{E\}\_\{\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\\sim\\widehat\{p\}\_\{t\}\}\[d\_\{\\mathcal\{A\}\}\(\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\)\], would otherwise be anO⁡\(N2\)O\(N^\{2\}\)double sum and is replaced with an unbiased Monte Carlo estimator\. We draw2​M2Mparticle indicesI1,…,IM,J1,…,JMI\_\{1\},\\dots,I\_\{M\},J\_\{1\},\\dots,J\_\{M\}i\.i\.d\. from the categorical distributionCat⁡\(wt\(1\),…,wt\(N\)\)\\mathrm\{Cat\}\(w\_\{t\}^\{\(1\)\},\\dots,w\_\{t\}^\{\(N\)\}\)\(with replacement, using the same weights for both halves and resampled afresh at everytt\) and use

ES^t=∑i=1Nwt\(i\)​d𝒜​\(𝐱t\(i\),𝐱t⋆\)−12​M​∑m=1Md𝒜​\(𝐱t\(Im\),𝐱t\(Jm\)\)\.\\widehat\{\\mathrm\{ES\}\}\_\{t\}=\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}\\,d\_\{\\mathcal\{A\}\}\\\!\\left\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{x\}\_\{t\}^\{\\star\}\\right\)\-\\frac\{1\}\{2M\}\\sum\_\{m=1\}^\{M\}d\_\{\\mathcal\{A\}\}\\\!\\left\(\\mathbf\{x\}\_\{t\}^\{\(I\_\{m\}\)\},\\mathbf\{x\}\_\{t\}^\{\(J\_\{m\}\)\}\\right\)\.\(32\)The reported scalar is the time averageES=\|𝒯\|−1​∑t∈𝒯ES^t\\mathrm\{ES\}=\|\\mathcal\{T\}\|^\{\-1\}\\sum\_\{t\\in\\mathcal\{T\}\}\\widehat\{\\mathrm\{ES\}\}\_\{t\}\. The number of Monte Carlo pairsMMis a hyperparameter\. We useM=2048M=2048, which makes the variance of the second term negligible relative to typical filter\-to\-filter differences\.

### C\.4Effective Sample Size

The effective sample size \(ESS\) at timettis reported in the standard inverse\-sum\-of\-squared\-weights form,

ESSt=\(∑i=1N\(wt\(i\)\)2\)−1,\\mathrm\{ESS\}\_\{t\}=\\left\(\\sum\_\{i=1\}^\{N\}\\bigl\(w\_\{t\}^\{\(i\)\}\\bigr\)^\{2\}\\right\)^\{\-1\},\(33\)wherewt\(i\)w\_\{t\}^\{\(i\)\}are the normalized importance weights satisfying∑i=1Nwt\(i\)=1\\sum\_\{i=1\}^\{N\}w\_\{t\}^\{\(i\)\}=1\. By construction1≤ESSt≤N1\\leq\\mathrm\{ESS\}\_\{t\}\\leq N: the lower bound is reached when a single particle carries unit weight \(full degeneracy\) and the upper bound when all weights are uniform\. The ESS is computed from the post\-update weights*before*the systematic resampling step at each time index, so it diagnoses weight degeneracy of the Sequential Importance Sampling step itself rather than the residual Monte Carlo variance left after resampling\. The reported scalar is the time average

ESS=1\|𝒯\|​∑t∈𝒯ESSt\.\\displaystyle\\mathrm\{ESS\}=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\mathrm\{ESS\}\_\{t\}\.\(34\)ESS does not by itself certify accuracy of the filter\. It is purely a diagnostic of how informative the weighted ensemble is relative to the raw particle count, and is reported alongside accuracy metrics rather than in place of them\.

### C\.5Wall Clock Time

Wall clock time \(WT\) measures the end\-to\-end runtime, in seconds, of running one full filter on a single trajectory of lengthTTon CPU\. Timing is recorded per seed and per filter and includes every operation invoked between the initialization of the particle ensemble att=0t=0and the final state estimate att=Tt=T\. It includes proposal evaluation \(including the forward pass through the learned proposal network for NOPF on CPU\), state propagation via the dynamics, importance\-weight updates, the systematic resampling step, and any per\-step bookkeeping required by the specific filter variant\.

It does*not*include offline costs such as proposal training, data simulation, or post\-hoc metric evaluation, which are amortized across runs and therefore not comparable on a per\-trajectory basis\. The reported scalar is the across\-seed mean\. To avoid hardware heterogeneity confounding the comparison, all filters in a given benchmark are timed on the same CPU node described in Appendix[D](https://arxiv.org/html/2609.30498#A4)\.

### C\.6Additional Metrics

The additional metric reported includes Mean Absolute Deviation Error \(MADE\) and Joint Particle In Credible\-region Probability \(Joint PICP\), which are presented for all benchmarks in Fig\.[9](https://arxiv.org/html/2609.30498#A3.F9)and are tabulated in Appendix[A\.2](https://arxiv.org/html/2609.30498#A1.SS2)\.

Figure 9:Additional metrics \(MADE and Joint PICP\) for benchmarks\. Each column presents the metrics for a benchmark\.#### C\.6\.1Mean Absolute Deviation Error

MADE captures whether*any*particle is close to the truth, regardless of its weight\. At each time step we compute the wrapped product\-metric distance between every particle and the true state and take the minimum,MADEt=min1≤i≤N⁡d𝒜​\(𝐱t\(i\),𝐱t⋆\)\\mathrm\{MADE\}\_\{t\}=\\min\_\{1\\leq i\\leq N\}\\,d\_\{\\mathcal\{A\}\}\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{x\}\_\{t\}^\{\\star\}\); the reported scalar is the time average

MADE=1\|𝒯\|​∑t∈𝒯min1≤i≤N⁡d𝒜​\(𝐱t\(i\),𝐱t⋆\)\.\\mathrm\{MADE\}=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\min\_\{1\\leq i\\leq N\}\\,d\_\{\\mathcal\{A\}\}\\\!\\left\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{x\}\_\{t\}^\{\\star\}\\right\)\.\(35\)Note that the importance weightswt\(i\)w\_\{t\}^\{\(i\)\}do not enter Eq\. \([35](https://arxiv.org/html/2609.30498#A3.E35)\) and a single, possibly negligibly weighted, particle near the truth is sufficient to keep MADE small\. MADE is therefore a support\-coverage diagnostic complementary to NMSE: a filter whose ensemble collapses onto the wrong mode will have small MADE only if at least one surviving particle still tracks the truth\.

#### C\.6\.2Joint Particle In Credible\-region Probability

The Joint PICP treatsp^t\\widehat\{p\}\_\{t\}as a Gaussian surrogate𝒩⁡\(𝝁^t,𝚺^t\)\\mathcal\{N\}\(\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\},\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}\)on the full state and tests whether the truth lies inside the corresponding\(1−α\)\(1\-\\alpha\)\-coverage Mahalanobis ellipsoid\. Concretely, we form weighted mean𝝁^t\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}and covariance𝚺^t\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}\(a small isotropic ridge10−8​I10^\{\-8\}Iis added for numerical stability\)\. We then evaluate the squared Mahalanobis distance of the truth from the weighted mean,

qt=Δ𝒜​\(𝐱t⋆,𝝁^t\)⊤​𝚺^t−1​Δ𝒜​\(𝐱t⋆,𝝁^t\),q\_\{t\}=\\Delta\_\{\\mathcal\{A\}\}\(\\mathbf\{x\}\_\{t\}^\{\\star\},\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}\)^\{\\top\}\\,\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}^\{\-1\}\\,\\Delta\_\{\\mathcal\{A\}\}\(\\mathbf\{x\}\_\{t\}^\{\\star\},\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}\),\(36\)and compare it to the\(1−α\)\(1\-\\alpha\)quantile of a chi\-squared distribution withdxd\_\{x\}degrees of freedom,χdx2​\(1−α\)\\chi^\{2\}\_\{d\_\{x\}\}\(1\-\\alpha\)\. The metric is the empirical coverage rate

JointPICPα=1\|𝒯\|∑t∈𝒯\[qt≤χdx2\(1−α\)\]\.\\mathrm\{JointPICP\}\_\{\\alpha\}=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}\\mathbf\{1\}\\\!\\left\[q\_\{t\}\\leq\\chi^\{2\}\_\{d\_\{x\}\}\(1\-\\alpha\)\\right\]\.\(37\)We useα=0\.1\\alpha=0\.1, so a perfectly calibrated filter would yieldJointPICP0\.1≈0\.9\\mathrm\{JointPICP\}\_\{0\.1\}\\approx 0\.9; values below indicate over\-confident posteriors and values above indicate under\-confident ones\. The Gaussian surrogate is only an assumption for the calibration test, not for the underlying particle representation:p^t\\widehat\{p\}\_\{t\}itself remains a weighted empirical distribution, but the chi\-squared threshold is what makes the per\-step decision computable inO⁡\(dx3\)O\(d\_\{x\}^\{3\}\)rather than via a kernel density tail integral\.

Becauseqtq\_\{t\}is compared againstχdx2​\(1−α\)\\chi^\{2\}\_\{d\_\{x\}\}\(1\-\\alpha\), joint PICP is calibrated to the case in which the residualΔ𝒜​\(𝐱t⋆,𝝁^t\)\\Delta\_\{\\mathcal\{A\}\}\(\\mathbf\{x\}\_\{t\}^\{\\star\},\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\}\)is Gaussian with covariance𝚺^t\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}\. When the underlying posterior departs from this assumption, the threshold is no longer the\(1−α\)\(1\-\\alpha\)quantile ofqtq\_\{t\}and the metric becomes biased in either direction\. Heavy\-tailed, skewed, or weight\-collapsed posteriors inflateqtq\_\{t\}aboveχdx2​\(1−α\)\\chi^\{2\}\_\{d\_\{x\}\}\(1\-\\alpha\)and cause systematic under\-coverage, while strongly multimodal posteriors inflate𝚺^t\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}with the inter\-mode spread and produce artificially high coverage even when the weighted mean lies in a low\-density region between modes\. Joint PICP should therefore be read as a calibration check on the Gaussian surrogate𝒩⁡\(𝝁^t,𝚺^t\)\\mathcal\{N\}\(\\widehat\{\\boldsymbol\{\\mu\}\}\_\{t\},\\widehat\{\\boldsymbol\{\\Sigma\}\}\_\{t\}\)rather than on the true particle distributionp^t\\widehat\{p\}\_\{t\}, and is best interpreted alongside shape\-aware scores such as the energy score, which remain meaningful for non\-Gaussian, multimodal, and heavy\-tailed posteriors\.

## Appendix DComputing Environment

All proposal networks were trained on a Linux cluster using PyTorch on NVIDIA A100 80GB PCIe GPUs\. Proposal training used a single GPU per benchmark together with CPU resources for data generation and batching\. Propoasal were trained once and then the same learned proposal was used during inference across all seeds\.

All inference filtering experiments and post\-processing analyses were performed separately on a CPU node equipped with two Intel Xeon Silver 4216 processors \(32 total cores; 16 cores per socket; 2\.10 GHz base clock; two NUMA nodes\), with one hardware thread per core\. Independent random seeds were distributed across CPU workers for parallel experiment execution\. All wall\-clock runtimes \(WT\) reported in the main text, including forward passes through the learned proposal network, were measured on this CPU architecture to maintain consistent comparisons across particle filters\.

Software was implemented in Python using PyTorch and NumPy under Linux\. All neural proposal evaluations during filtering were executed on CPU for fair runtime comparison with competing particle filters\. All forward\-model and observation operator implementations are vectorized across particles\.

## Appendix EParticle Filter Implementations

This section includes details of different filter implementations at time stepsttwhen the weighted ensemble\{𝐱t−1\(i\),w~t−1\(i\)\}i=1N\\\{\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\tilde\{w\}\_\{t\-1\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}is provided as input\.

### E\.1Bootstrap particle filter

The bootstrap particle filter \(BPF\) samples

𝐱t\(i\)∼p⁡\(𝐱t∣𝐱t−1\(i\)\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)\(38\)and updates weights according to

w~t\(i\)=wt−1\(i\)​p​\(𝐲t∣𝐱t\(i\)\),wt\(i\)=w~t\(i\)∑j=1Nw~t\(j\)\.\\tilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i\)\}\),\\qquad w\_\{t\}^\{\(i\)\}=\\frac\{\\tilde\{w\}\_\{t\}^\{\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}\\tilde\{w\}\_\{t\}^\{\(j\)\}\}\.\(39\)

### E\.2Nudged particle filter

NudgedPF first samples from the transition prior as

𝐱t\(i\)∼p⁡\(𝐱t∣𝐱t−1\(i\)\),\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\),\(40\)and then computes

𝐠t\(i\)=∇𝐱t​log​p​\(𝐲t∣𝐱t\)\|𝐱t=𝐱t\(i\)\.\\mathbf\{g\}\_\{t\}^\{\(i\)\}=\\nabla\_\{\\mathbf\{x\}\_\{t\}\}\\log p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)\\big\|\_\{\\mathbf\{x\}\_\{t\}=\\mathbf\{x\}\_\{t\}^\{\(i\)\}\}\.\(41\)Computing the gradient of the log\-likelihood is numerically more stable than evaluating the gradient of the likelihood,∇𝐱tp​\(𝐲t∣𝐱t\)\\nabla\_\{\\mathbf\{x\}\_\{t\}\}p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\), which can be very large or zero in flat regions\.

Each particle is nudged independently with probabilityN−1/2N^\{\-1/2\}:

𝐱¯t\(i\)=𝐱t\(i\)\+ηIt\(i\)𝐠t\(i\),It\(i\)∼Bernoulli\(N−1/2\)\.\\bar\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}=\\mathbf\{x\}\_\{t\}^\{\(i\)\}\+\\eta I\_\{t\}^\{\(i\)\}\\mathbf\{g\}\_\{t\}^\{\(i\)\},\\qquad I\_\{t\}^\{\(i\)\}\\sim\\operatorname\{Bernoulli\}\(N^\{\-1/2\}\)\.\(42\)The particles are then weighed using

w~t\(i\)=wt−1\(i\)​p​\(𝐲t∣𝐱¯t\(i\)\),wt\(i\)=w~t\(i\)∑j=1Nw~t\(j\)\.\\tilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}p\(\\mathbf\{y\}\_\{t\}\\mid\\bar\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}\),\\qquad w\_\{t\}^\{\(i\)\}=\\frac\{\\tilde\{w\}\_\{t\}^\{\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}\\tilde\{w\}\_\{t\}^\{\(j\)\}\}\.\(43\)without adding a proposal correction term for the nudging transformation\. The nudging strength parameterη\\etais selected as the highestη\\etafor which the likelihood of most particles increases\. The first benchmark usesη=10−2\\eta=10^\{\-2\}while the rest of the benchmarks useη=2×10−4\\eta=2\\times 10^\{\-4\}\.

### E\.3Oracle Particle Filter

The Oracle Particle Filter \(OPF\) is used only for the linear\-Gaussian sensor\-bias benchmark, where the locally optimal proposal is available in closed form\. Consider the linear\-Gaussian model

𝐱t=A​𝐱t−1\+ϵt,ϵt∼𝒩⁡\(0,Q\),\\mathbf\{x\}\_\{t\}=A\\mathbf\{x\}\_\{t\-1\}\+\\boldsymbol\{\\epsilon\}\_\{t\},\\qquad\\boldsymbol\{\\epsilon\}\_\{t\}\\sim\\mathcal\{N\}\(0,Q\),\(44\)𝐲t=H​𝐱t\+𝜼t,𝜼t∼𝒩⁡\(0,R\),\\mathbf\{y\}\_\{t\}=H\\mathbf\{x\}\_\{t\}\+\\boldsymbol\{\\eta\}\_\{t\},\\qquad\\boldsymbol\{\\eta\}\_\{t\}\\sim\\mathcal\{N\}\(0,R\),\(45\)whereAAis the transition matrix,HHis the observation matrix,QQis the process\-noise covariance, andRRis the observation\-noise covariance\. For each ancestor particle𝐱t−1\(i\)\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}, OraclePF samples from the locally optimal proposal

q⋆​\(𝐱t∣𝐱t−1\(i\),𝐲t\)=p⁡\(𝐱t∣𝐱t−1\(i\),𝐲t\)\.q^\{\\star\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)=p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)\.\(46\)This conditional distribution is Gaussian\. Let

S=H​Q​H⊤\+R,K=Q​H⊤​S−1\.S=HQH^\{\\top\}\+R,\\qquad K=QH^\{\\top\}S^\{\-1\}\.\(47\)Then

p⁡\(𝐱t∣𝐱t−1\(i\),𝐲t\)=𝒩⁡\(𝐱t,𝐦t\(i\),C\),p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)=\\mathcal\{N\}\\left\(\\mathbf\{x\}\_\{t\};\\mathbf\{m\}\_\{t\}^\{\(i\)\},C\\right\),\(48\)with

𝐦t\(i\)=A​𝐱t−1\(i\)\+K⁡\(𝐲t−H​A​𝐱t−1\(i\)\),C=Q−K​H​Q\.\\mathbf\{m\}\_\{t\}^\{\(i\)\}=A\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\+K\\left\(\\mathbf\{y\}\_\{t\}\-HA\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\\right\),\\qquad C=Q\-KHQ\.\(49\)Equivalently,C=\(Q−1\+H⊤​R−1​H\)−1C=\(Q^\{\-1\}\+H^\{\\top\}R^\{\-1\}H\)^\{\-1\}and𝐦t\(i\)=C⁡\(Q−1​A​𝐱t−1\(i\)\+H⊤​R−1​𝐲t\)\\mathbf\{m\}\_\{t\}^\{\(i\)\}=C\(Q^\{\-1\}A\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\+H^\{\\top\}R^\{\-1\}\\mathbf\{y\}\_\{t\}\)\. The Kalman\-update form above is used in implementation for numerical stability\.

After sampling𝐱t\(i\)∼𝒩⁡\(𝐦t\(i\),C\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim\\mathcal\{N\}\(\\mathbf\{m\}\_\{t\}^\{\(i\)\},C\), the outer weight is updated using the predictive likelihood

p⁡\(𝐲t∣𝐱t−1\(i\)\)=𝒩⁡\(𝐲t,H​A​𝐱t−1\(i\),S\)\.p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)=\\mathcal\{N\}\\left\(\\mathbf\{y\}\_\{t\};HA\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},S\\right\)\.\(50\)Thus,

w~t\(i\)=wt−1\(i\)​p​\(𝐲t∣𝐱t−1\(i\)\),wt\(i\)=w~t\(i\)∑j=1Nw~t\(j\)\.\\widetilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\),\\qquad w\_\{t\}^\{\(i\)\}=\\frac\{\\widetilde\{w\}\_\{t\}^\{\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}\\widetilde\{w\}\_\{t\}^\{\(j\)\}\}\.\(51\)
OPF provides a reference PF baseline for the linear\-Gaussian benchmark, measuring how closely NPF approaches the closed\-form locally optimal proposal in the setting where that proposal is available\.

### E\.4Unscented particle filter

UPF constructs a local Unscented Kalman Filter \(UKF\) proposal for each particle\. The augmented state dimension for the UKF is

daug=2​dx\+dy,d\_\{\\mathrm\{aug\}\}=2d\_\{x\}\+d\_\{y\},\(52\)corresponding to state, process noise, and observation noise\. The sigma points constructed according to[Wan and Van Der Merwe \(2000\)](https://arxiv.org/html/2609.30498#bib.bib6)with

λ=α2​\(daug\+κ\)−daug\.\\lambda=\\alpha^\{2\}\(d\_\{\\mathrm\{aug\}\}\+\\kappa\)\-d\_\{\\mathrm\{aug\}\}\.\(53\)induces a Gaussian proposal𝒩⁡\(𝐦t\(i\),Pt\(i\)\)\\mathcal\{N\}\(\\mathbf\{m\}\_\{t\}^\{\(i\)\},P\_\{t\}^\{\(i\)\}\)for each particle, and

𝐱t\(i\)∼𝒩⁡\(𝐦t\(i\),Pt\(i\)\)\.\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim\\mathcal\{N\}\(\\mathbf\{m\}\_\{t\}^\{\(i\)\},P\_\{t\}^\{\(i\)\}\)\.\(54\)The importance weight is evaluated as

w~t\(i\)=wt−1\(i\)​p⁡\(𝐲t∣𝐱t\(i\)\)​p​\(𝐱t\(i\)∣𝐱t−1\(i\)\)qUKF​\(𝐱t\(i\)∣𝐱t−1\(i\),𝐲t\),\\tilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}\\frac\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i\)\}\)p\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)\}\{q\_\{\\mathrm\{UKF\}\}\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\}\)\},\(55\)followed by a normalization step\.

The implementation adds diagonal jitter to the covariance matrix during sigma\-point construction if the estimated covariance matrix is not positive definite\. All UPFs useα=10−3\\alpha=10^\{\-3\},β=2\\beta=2, andκ=0\\kappa=0for sigma point construction\.

### E\.5Inner\-Sampling Proposal Filter

The Inner\-Sampling Proposal Filter \(ISPF\) is an expensive local proposal baseline that approximates the locally optimal proposal using inner Monte Carlo sampling\. For each outer particle𝐱t−1\(i\)\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}, we drawMinnerM\_\{\\mathrm\{inner\}\}transition candidates

𝐱t\(i,m\)∼p\(𝐱t∣𝐱t−1\(i\)\),m=1,…,Minner\.\\mathbf\{x\}\_\{t\}^\{\(i,m\)\}\\sim p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\),\\qquad m=1,\\ldots,M\_\{\\mathrm\{inner\}\}\.\(56\)Each candidate is scored using the current likelihood,

ℓt,m\(i\)=p⁡\(𝐲t∣𝐱t\(i,m\)\),\\ell\_\{t,m\}^\{\(i\)\}=p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i,m\)\}\),\(57\)and normalized to obtain inner weights

αt,m\(i\)=ℓt,m\(i\)∑r=1Minnerℓt,r\(i\)\.\\alpha\_\{t,m\}^\{\(i\)\}=\\frac\{\\ell\_\{t,m\}^\{\(i\)\}\}\{\\sum\_\{r=1\}^\{M\_\{\\mathrm\{inner\}\}\}\\ell\_\{t,r\}^\{\(i\)\}\}\.\(58\)These weights define an empirical approximation to the local observation\-informed proposal,

q^t\(i\)​\(d​𝐱\)=∑m=1Minnerαt,m\(i\)​δ𝐱t\(i,m\)​\(d​𝐱\)\.\\widehat\{q\}\_\{t\}^\{\(i\)\}\(d\\mathbf\{x\}\)=\\sum\_\{m=1\}^\{M\_\{\\mathrm\{inner\}\}\}\\alpha\_\{t,m\}^\{\(i\)\}\\delta\_\{\\mathbf\{x\}\_\{t\}^\{\(i,m\)\}\}\(d\\mathbf\{x\}\)\.\(59\)We then draw an index

Ki∼Categorical⁡\(αt,1\(i\),…,αt,Minner\(i\)\),K\_\{i\}\\sim\\mathrm\{Categorical\}\\left\(\\alpha\_\{t,1\}^\{\(i\)\},\\ldots,\\alpha\_\{t,M\_\{\\mathrm\{inner\}\}\}^\{\(i\)\}\\right\),\(60\)and propagate the outer particle as

𝐱t\(i\)=𝐱t\(i,Ki\)\.\\mathbf\{x\}\_\{t\}^\{\(i\)\}=\\mathbf\{x\}\_\{t\}^\{\(i,K\_\{i\}\)\}\.\(61\)
The locally optimal proposal weight update requires the predictive likelihood

p⁡\(𝐲t∣𝐱t−1\(i\)\)=∫p⁡\(𝐲t∣𝐱t\)​p​\(𝐱t∣𝐱t−1\(i\)\)​d​𝐱t\.p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)=\\int p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)d\\mathbf\{x\}\_\{t\}\.\(62\)ISPF approximates this quantity using the same inner samples,

Z^t\(i\)=1Minner​∑m=1Minnerp⁡\(𝐲t∣𝐱t\(i,m\)\)\.\\widehat\{Z\}\_\{t\}^\{\(i\)\}=\\frac\{1\}\{M\_\{\\mathrm\{inner\}\}\}\\sum\_\{m=1\}^\{M\_\{\\mathrm\{inner\}\}\}p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i,m\)\}\)\.\(63\)The outer weights are then updated as

w~t\(i\)=wt−1\(i\)​Z^t\(i\),wt\(i\)=w~t\(i\)∑j=1Nw~t\(j\)\.\\widetilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}\\widehat\{Z\}\_\{t\}^\{\(i\)\},\\qquad w\_\{t\}^\{\(i\)\}=\\frac\{\\widetilde\{w\}\_\{t\}^\{\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}\\widetilde\{w\}\_\{t\}^\{\(j\)\}\}\.\(64\)The selected candidate likelihood is not applied again in the outer weight update, since the inner categorical draw already samples candidates with probability proportional to their likelihood\. Thus, ISPF is a stochastic inner\-sampling approximation to the locally optimal proposal\. In all experiments, we useMinner=200M\_\{\\mathrm\{inner\}\}=200inner samples per outer particle\.

### E\.6Auxiliary particle filter

The auxiliary particle filter \(APF\) introduces an ancestor\-selection stage that uses the current observation𝐲t\\mathbf\{y\}\_\{t\}to focus subsequent propagation on particles likely to explain it\. Each particle is first advanced to its deterministic predictive mean

𝐱^t\(i\)=𝐟¯​\(𝐱t−1\(i\)\),\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\),\(65\)and assigned a first\-stage \(auxiliary\) weight that combines the prior normalized weight with the observation likelihood evaluated at this mean,

λt\(i\)∝wt−1\(i\)​p​\(𝐲t∣𝐱^t\(i\)\),∑i=1Nλt\(i\)=1\.\\lambda\_\{t\}^\{\(i\)\}\\propto w\_\{t\-1\}^\{\(i\)\}\\,p\(\\mathbf\{y\}\_\{t\}\\mid\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}\),\\qquad\\sum\_\{i=1\}^\{N\}\\lambda\_\{t\}^\{\(i\)\}=1\.\(66\)Ancestor indicesai∈\{1,…,N\}a\_\{i\}\\in\\\{1,\\dots,N\\\}are drawn with replacement fromCategorical⁡\(λt\(1\),…,λt\(N\)\)\\operatorname\{Categorical\}\(\\lambda\_\{t\}^\{\(1\)\},\\dots,\\lambda\_\{t\}^\{\(N\)\}\), and the selected ancestors are propagated through the full stochastic forward map,

𝐱t\(i\)=𝐟⁡\(𝐱t−1ai,𝐯t\)\\mathbf\{x\}\_\{t\}^\{\(i\)\}=\\mathbf\{f\}\(\\mathbf\{x\}\_\{t\-1\}^\{a\_\{i\}\},\\mathbf\{v\}\_\{t\}\)\(67\)The second\-stage weight removes the bias introduced by the auxiliary draw using the ratio of the true likelihood at the propagated particle to the predictive likelihood used to select its ancestor

w~t\(i\)∝p⁡\(𝐲t∣𝐱t\(i\)\)p⁡\(𝐲t∣𝐱^tai\),wt\(i\)=w~t\(i\)∑j=1Nw~t\(j\)\.\\tilde\{w\}\_\{t\}^\{\(i\)\}\\propto\\frac\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i\)\}\)\}\{p\(\\mathbf\{y\}\_\{t\}\\mid\\hat\{\\mathbf\{x\}\}\_\{t\}^\{a\_\{i\}\}\)\},\\qquad w\_\{t\}^\{\(i\)\}=\\frac\{\\tilde\{w\}\_\{t\}^\{\(i\)\}\}\{\\sum\_\{j=1\}^\{N\}\\tilde\{w\}\_\{t\}^\{\(j\)\}\}\.\(68\)

### E\.7Ensemble Kalman filter

The Ensemble Kalman Filter \(EnKF\) replaces the analytic covariance in the Kalman update with empirical estimates obtained from a Monte Carlo ensemble\. At timett, each ensemble member is propagated through the stochastic forward model and pushed through the observation operator with an independent observation\-noise sample to produce a paired state\-observation ensemble,

𝐱t\(i\)=𝐟⁡\(𝐱t−1\(i\),𝐯t\),𝐲t\(i\)=𝐡⁡\(𝐱t\(i\),𝐞t\)\.\\mathbf\{x\}\_\{t\}^\{\(i\)\}=\\mathbf\{f\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{v\}\_\{t\}\),\\qquad\\mathbf\{y\}\_\{t\}^\{\(i\)\}=\\mathbf\{h\}\(\\mathbf\{x\}\_\{t\}^\{\(i\)\},\\mathbf\{e\}\_\{t\}\)\.\(69\)The empirical means𝐱¯t=1N​∑i𝐱t\(i\)\\bar\{\\mathbf\{x\}\}\_\{t\}=\\tfrac\{1\}\{N\}\\sum\_\{i\}\\mathbf\{x\}\_\{t\}^\{\(i\)\}and𝐲¯t=1N​∑i𝐲t\(i\)\\bar\{\\mathbf\{y\}\}\_\{t\}=\\tfrac\{1\}\{N\}\\sum\_\{i\}\\mathbf\{y\}\_\{t\}^\{\(i\)\}define the centered deviationsδ​𝐱t\(i\)=𝐱t\(i\)−𝐱¯t\\delta\\mathbf\{x\}\_\{t\}^\{\(i\)\}=\\mathbf\{x\}\_\{t\}^\{\(i\)\}\-\\bar\{\\mathbf\{x\}\}\_\{t\}andδ​𝐲t\(i\)=𝐲t\(i\)−𝐲¯t\\delta\\mathbf\{y\}\_\{t\}^\{\(i\)\}=\\mathbf\{y\}\_\{t\}^\{\(i\)\}\-\\bar\{\\mathbf\{y\}\}\_\{t\}, from which the cross\-covariance and observation\-covariance are estimated as

Px​y=1N​∑i=1Nδ​𝐱t\(i\)​\(δ​𝐲t\(i\)\)⊤,Py​y=1N​∑i=1Nδ​𝐲t\(i\)​\(δ​𝐲t\(i\)\)⊤\.P\_\{xy\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\delta\\mathbf\{x\}\_\{t\}^\{\(i\)\}\(\\delta\\mathbf\{y\}\_\{t\}^\{\(i\)\}\)^\{\\top\},\\qquad P\_\{yy\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\delta\\mathbf\{y\}\_\{t\}^\{\(i\)\}\(\\delta\\mathbf\{y\}\_\{t\}^\{\(i\)\}\)^\{\\top\}\.\(70\)The ensemble Kalman gain isKt=Px​y​Py​y−1K\_\{t\}=P\_\{xy\}P\_\{yy\}^\{\-1\}, and each member is corrected toward the true observation𝐲t\\mathbf\{y\}\_\{t\}using its own synthetic observation𝐲t\(i\)\\mathbf\{y\}\_\{t\}^\{\(i\)\}as a stochastic innovation,

𝐱t\(i\)=𝐱^t\(i\)\+Kt​\(𝐲t−𝐲^t\(i\)\),\\mathbf\{x\}\_\{t\}^\{\(i\)\}=\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}\+K\_\{t\}\(\\mathbf\{y\}\_\{t\}\-\\hat\{\\mathbf\{y\}\}\_\{t\}^\{\(i\)\}\),\(71\)where𝐱^t\(i\)=𝐟¯​\(𝐱t−1\(i\)\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)and𝐲^t\(i\)=𝐡¯​\(𝐲^t\(i\)\)\\hat\{\\mathbf\{y\}\}\_\{t\}^\{\(i\)\}=\\bar\{\\mathbf\{h\}\}\(\\hat\{\\mathbf\{y\}\}\_\{t\}^\{\(i\)\}\)\. The updated ensemble\{𝐱t\(i\)\}i=1N\\\{\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\}carries no importance weights, and for downstream metric computation, each member is assigned a uniform weightwt\(i\)=1/Nw\_\{t\}^\{\(i\)\}=1/N\. EnKF recovers the Kalman update for the filtering distributionp\(𝐱t∣𝐲1:t\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{y\}\_\{1:t\}\)when\(𝐱t,𝐲t\)\(\\mathbf\{x\}\_\{t\},\\mathbf\{y\}\_\{t\}\)are jointly Gaussian in the large\-ensemble limit, and remains robust and widely used in practice\.

### E\.8Neural Optimal Particle filter

The Neural Optimal Particle Filter \(NOPF\) replaces the importance density with a learned conditional distributionqθ​\(𝐱t∣𝐱t−1,𝐲t,t\)q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\},\\mathbf\{y\}\_\{t\},t\)trained as described in Appendix[B\.2](https://arxiv.org/html/2609.30498#A2.SS2)\. To keep the network from having to relearn the dynamics, each particle is first advanced to its deterministic predictive mean

𝐱^t\(i\)=𝐟¯​\(𝐱t−1\(i\)\),\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\),\(72\)and𝐱^t\(i\)\\hat\{\\mathbf\{x\}\}\_\{t\}^\{\(i\)\}is supplied to the proposal network as an auxiliary conditioning input so the network only needs to model the observation\-informed correction to the deterministic skeleton\. A new particle is then drawn from the learned proposal,

𝐱t\(i\)∼qθ​\(𝐱t∣𝐱t−1\(i\),𝐲t,t\),\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\sim q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\},t\),\(73\)and the standard SIS weight is evaluated in log\-space using the observation likelihood, the transition density, and the proposal density,

w~t\(i\)=wt−1\(i\)​p⁡\(𝐲t∣𝐱t\(i\)\)​p​\(𝐱t\(i\)∣𝐱t−1\(i\)\)qθ​\(𝐱t\(i\)∣𝐱t−1\(i\),𝐲t,t\),\\tilde\{w\}\_\{t\}^\{\(i\)\}=w\_\{t\-1\}^\{\(i\)\}\\frac\{p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}^\{\(i\)\}\)p\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\}\)\}\{q\_\{\\theta\}\(\\mathbf\{x\}\_\{t\}^\{\(i\)\}\\mid\\mathbf\{x\}\_\{t\-1\}^\{\(i\)\},\\mathbf\{y\}\_\{t\},t\)\},\(74\)The forward pass throughqθq\_\{\\theta\}adds one network evaluation per particle on top of the cost of a bootstrap step, so the per\-step cost is𝒪⁡\(N​cθ\)\\mathcal\{O\}\(N\\,c\_\{\\theta\}\)wherecθc\_\{\\theta\}is the network cost of propagating a single particle\.

### E\.9Systematic resampling

Particle filters use systematic resampling\. Given normalized weightswt\(i\)w\_\{t\}^\{\(i\)\}, sampleu0∼Uniform⁡\(0,1/N\)u\_\{0\}\\sim\\operatorname\{Uniform\}\(0,1/N\)and define

ui=u0\+i−1N,i=1,…,N\.u\_\{i\}=u\_\{0\}\+\\frac\{i\-1\}\{N\},\\qquad i=1,\\ldots,N\.\(75\)Theiith resampled particle is the first one whose cumulative weight exceedsuiu\_\{i\}\. After resampling, all weights are reset to1/N1/N\.

### E\.10Computational Complexity of PFs

Letdxd\_\{x\}anddyd\_\{y\}denote the dimensions of the state and observation spaces, respectively, and letnpn\_\{p\}denote the number of outer particles in a particle filter\. For ISPF,nqn\_\{q\}denotes the number of inner samples per outer particle\. The cost of one evaluation of the forward model𝐟\\mathbf\{f\}and the observation operator𝐡\\mathbf\{h\}\(Eq\.[1](https://arxiv.org/html/2609.30498#S2.E1)\) is denoted by𝒪⁡\(𝐟\)\\mathcal\{O\}\(\\mathbf\{f\}\)and𝒪⁡\(𝐡\)\\mathcal\{O\}\(\\mathbf\{h\}\), and the cost of a single forward pass through the proposal network is𝒪⁡\(n​n\)\\mathcal\{O\}\(nn\)\. Per\-step asymptotic costs are summarized in Table[6](https://arxiv.org/html/2609.30498#A5.T6); systematic resampling, Gaussian density evaluations, and other operations linear innpn\_\{p\}are absorbed into the listed terms\.

Table 6:Per\-step computational complexity of the particle filters considered in this work\.We assume additive process noise, so the transition densityp⁡\(𝐱t∣𝐱t−1\)p\(\\mathbf\{x\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\-1\}\)can be evaluated from the deterministic forward map cached during the propagation step without an additional𝐟\\mathbf\{f\}call\.

## Appendix FBenchmark Specifications

Each benchmark specializes the state\-space model in Eq\. \([1](https://arxiv.org/html/2609.30498#S2.E1)\) with additive process noise𝐯t∼𝒩⁡\(𝟎,𝐐\)\\mathbf\{v\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{Q\}\), where𝐐\\mathbf\{Q\}is diagonal\. For the first three benchmarks, the observation noise is additive with diagonal covariance𝐑\\mathbf\{R\}, so𝐲t=𝐡¯​\(𝐱t\)\+𝐰t\\mathbf\{y\}\_\{t\}=\\bar\{\\mathbf\{h\}\}\(\\mathbf\{x\}\_\{t\}\)\+\\mathbf\{w\}\_\{t\}\. For the current positive\-prior 3D unordered\-range benchmark, the proprioceptive channels are additive, but independent range noise is applied to the four labeled raw ranges before sorting; the likelihood is therefore written as a permutation\-marginalized unordered\-range likelihood in Appendix[F\.4](https://arxiv.org/html/2609.30498#A6.SS4)\. We write𝐟¯\\bar\{\\mathbf\{f\}\}for the deterministic skeleton of𝐟\\mathbf\{f\}\(i\.e\.𝐟¯​\(𝐱t−1\)=𝐟⁡\(𝐱t−1,𝟎\)\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\}\)=\\mathbf\{f\}\(\\mathbf\{x\}\_\{t\-1\},\\mathbf\{0\}\)\)\. All angular dimensions are wrapped to\(−π,π\]\(\-\\pi,\\pi\]\. Throughout,𝝈f\\boldsymbol\{\\sigma\}\_\{f\}and𝝈h\\boldsymbol\{\\sigma\}\_\{h\}denote per\-coordinate process and observation standard deviations when those diagonal covariances are defined\. Table[7](https://arxiv.org/html/2609.30498#A6.T7)summarizes the key parameters for the four benchmarks\.

Table 7:Benchmark details\.### F\.1Benchmark 1: Linear System with Biased Sensors

The sensor\-bias benchmark has state

𝐱t=\[pt,vt,bt\]⊤∈ℝ3,\\mathbf\{x\}\_\{t\}=\[p\_\{t\},\\,v\_\{t\},\\,b\_\{t\}\]^\{\\top\}\\in\\mathbb\{R\}^\{3\},\(76\)whereptp\_\{t\}is position,vtv\_\{t\}velocity, andbtb\_\{t\}an additive sensor bias\. The time grid istk=0\.1​\(k\+1\)t\_\{k\}=0\.1\(k\+1\)fork=0,…,59k=0,\\ldots,59\. Specializing Eq\. \([1](https://arxiv.org/html/2609.30498#S2.E1)\) to a linear\-Gaussian state\-space model gives

𝐱t=𝐀𝐱t−1\+𝐯t,𝐲t=𝐇𝐱t\+𝐰t,\\mathbf\{x\}\_\{t\}=\\mathbf\{A\}\\mathbf\{x\}\_\{t\-1\}\+\\mathbf\{v\}\_\{t\},\\qquad\\mathbf\{y\}\_\{t\}=\\mathbf\{H\}\\mathbf\{x\}\_\{t\}\+\\mathbf\{w\}\_\{t\},\(77\)with

𝐀=\[1Δ​t0010001\],𝐇=\[100011\],Δ​t=0\.1,\\mathbf\{A\}=\\begin\{bmatrix\}1&\\Delta t&0\\\\ 0&1&0\\\\ 0&0&1\\end\{bmatrix\},\\qquad\\mathbf\{H\}=\\begin\{bmatrix\}1&0&0\\\\ 0&1&1\\end\{bmatrix\},\\qquad\\Delta t=0\.1,\(78\)and diagonal noise covariances

𝐐=diag⁡\(0\.12,0\.12,0\.12\),𝐑=diag⁡\(0\.12,0\.12\),𝐯t∼𝒩⁡\(𝟎,𝐐\),𝐰t∼𝒩⁡\(𝟎,𝐑\)\.\\mathbf\{Q\}=\\operatorname\{diag\}\(0\.1^\{2\},\\,0\.1^\{2\},\\,0\.1^\{2\}\),\\quad\\mathbf\{R\}=\\operatorname\{diag\}\(0\.1^\{2\},\\,0\.1^\{2\}\),\\quad\\mathbf\{v\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{Q\}\),\\ \\mathbf\{w\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{R\}\)\.\(79\)The prior is

p0​\(𝐱0\)=𝒩⁡\(𝝁0,𝚺0\),𝝁0=\[0\.8,0\.2,−0\.1\]⊤,𝚺0=diag⁡\(0\.5,0\.5,0\.5\)\.p\_\{0\}\(\\mathbf\{x\}\_\{0\}\)=\\mathcal\{N\}\(\\boldsymbol\{\\mu\}\_\{0\},\\boldsymbol\{\\Sigma\}\_\{0\}\),\\qquad\\boldsymbol\{\\mu\}\_\{0\}=\[0\.8,\\,0\.2,\\,\-0\.1\]^\{\\top\},\\qquad\\boldsymbol\{\\Sigma\}\_\{0\}=\\operatorname\{diag\}\(0\.5,\\,0\.5,\\,0\.5\)\.\(80\)

### F\.2Benchmark 2: 2D Robot Localization with Range and Bearing Observations

The known\-map localization benchmark estimates robot pose, motion states, and odometry biases with a fixed landmark map\. The state is

𝐱t=\[ptx,pty,θt,vt,ωt,btv,btω\]⊤∈ℝ7,\\mathbf\{x\}\_\{t\}=\[p\_\{t\}^\{x\},\\,p\_\{t\}^\{y\},\\,\\theta\_\{t\},\\,v\_\{t\},\\,\\omega\_\{t\},\\,b\_\{t\}^\{v\},\\,b\_\{t\}^\{\\omega\}\]^\{\\top\}\\in\\mathbb\{R\}^\{7\},\(81\)with\(ptx,pty\)\(p\_\{t\}^\{x\},p\_\{t\}^\{y\}\)position,θt\\theta\_\{t\}heading,vtv\_\{t\}forward speed,ωt\\omega\_\{t\}yaw rate, and\(btv,btω\)\(b\_\{t\}^\{v\},b\_\{t\}^\{\\omega\}\)slowly varying odometry biases\. The four landmarks are

ℓ1=\(−4\.5,1\.8\),ℓ2=\(3\.8,3\.1\),ℓ3=\(−2\.2,−3\.8\),ℓ4=\(4\.4,−2\.6\),\\boldsymbol\{\\ell\}\_\{1\}=\(\-4\.5,1\.8\),\\quad\\boldsymbol\{\\ell\}\_\{2\}=\(3\.8,3\.1\),\\quad\\boldsymbol\{\\ell\}\_\{3\}=\(\-2\.2,\-3\.8\),\\quad\\boldsymbol\{\\ell\}\_\{4\}=\(4\.4,\-2\.6\),\(82\)and the time grid istk=0\.1​kt\_\{k\}=0\.1kfork=0,…,50k=0,\\ldots,50withΔ​t=0\.1\\Delta t=0\.1,ρv=2\.0\\rho\_\{v\}=2\.0,ρω=2\.7\\rho\_\{\\omega\}=2\.7\. Specializing Eq\. \([1](https://arxiv.org/html/2609.30498#S2.E1)\) with additive process noise gives𝐱t=𝐟¯​\(𝐱t−1,t−1\)\+𝐯t\\mathbf\{x\}\_\{t\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\},t\-1\)\+\\mathbf\{v\}\_\{t\}, where𝐟¯\\bar\{\\mathbf\{f\}\}is defined component\-wise by

ptx\\displaystyle p\_\{t\}^\{x\}=pt−1x\+Δtvt−1cosθt−1,\\displaystyle=p\_\{t\-1\}^\{x\}\+\\Delta t\\,v\_\{t\-1\}\\cos\\theta\_\{t\-1\},pty\\displaystyle p\_\{t\}^\{y\}=pt−1y\+Δtvt−1sinθt−1,\\displaystyle=p\_\{t\-1\}^\{y\}\+\\Delta t\\,v\_\{t\-1\}\\sin\\theta\_\{t\-1\},θt\\displaystyle\\theta\_\{t\}=wrap⁡\(θt−1\+Δ​t​ωt−1\),\\displaystyle=\\operatorname\{wrap\}\(\\theta\_\{t\-1\}\+\\Delta t\\,\\omega\_\{t\-1\}\),vt\\displaystyle v\_\{t\}=vt−1\+ρv​\(vcmd​\(t−1\)−vt−1\)​Δ​t,\\displaystyle=v\_\{t\-1\}\+\\rho\_\{v\}\\bigl\(v\_\{\\mathrm\{cmd\}\}\(t\-1\)\-v\_\{t\-1\}\\bigr\)\\Delta t,ωt\\displaystyle\\omega\_\{t\}=ωt−1\+ρω​\(ωcmd​\(t−1\)−ωt−1\)​Δ​t,\\displaystyle=\\omega\_\{t\-1\}\+\\rho\_\{\\omega\}\\bigl\(\\omega\_\{\\mathrm\{cmd\}\}\(t\-1\)\-\\omega\_\{t\-1\}\\bigr\)\\Delta t,btv\\displaystyle b\_\{t\}^\{v\}=bt−1v,btω=bt−1ω,\\displaystyle=b\_\{t\-1\}^\{v\},\\qquad b\_\{t\}^\{\\omega\}=b\_\{t\-1\}^\{\\omega\},\(83\)so the bias states evolve as a Gaussian random walk through the additive process noise term𝐯t\\mathbf\{v\}\_\{t\}\. The command schedule is

vcmd​\(t\)\\displaystyle v\_\{\\mathrm\{cmd\}\}\(t\)=0\.85\+0\.20​sin⁡\(0\.33​t\)\+0\.08​cos⁡\(0\.07​t\),\\displaystyle=0\.85\+0\.20\\sin\(0\.33t\)\+0\.08\\cos\(0\.07t\),\(84\)ωcmd​\(t\)\\displaystyle\\omega\_\{\\mathrm\{cmd\}\}\(t\)=0\.02\+0\.42​sin⁡\(0\.21​t\)\+0\.12​cos⁡\(0\.09​t\)\.\\displaystyle=0\.02\+0\.42\\sin\(0\.21t\)\+0\.12\\cos\(0\.09t\)\.\(85\)Process noise is𝐯t∼𝒩⁡\(𝟎,𝐐\)\\mathbf\{v\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{Q\}\)with𝐐=diag⁡\(𝝈f2\)\\mathbf\{Q\}=\\operatorname\{diag\}\(\\boldsymbol\{\\sigma\}\_\{f\}^\{2\}\)and

𝝈f=\[0\.20,0\.20,1∘,0\.40,4∘,0\.30,0\.20∘\]⊤\.\\boldsymbol\{\\sigma\}\_\{f\}=\[0\.20,\\,0\.20,\\,1^\{\\circ\},\\,0\.40,\\,4^\{\\circ\},\\,0\.30,\\,0\.20^\{\\circ\}\]^\{\\top\}\.\(86\)For each landmarkℓj=\(ℓjx,ℓjy\)\\boldsymbol\{\\ell\}\_\{j\}=\(\\ell\_\{j\}^\{x\},\\ell\_\{j\}^\{y\}\), define the relative offsets, range, and bearing

δj,tx=ℓjx−ptx,δj,ty=ℓjy−pty,\\displaystyle\\delta\_\{j,t\}^\{x\}=\\ell\_\{j\}^\{x\}\-p\_\{t\}^\{x\},\\quad\\delta\_\{j,t\}^\{y\}=\\ell\_\{j\}^\{y\}\-p\_\{t\}^\{y\},\(87\)rj,t=\(δj,tx\)2\+\(δj,ty\)2,βj,t=wrap⁡\(atan2⁡\(δj,ty,δj,tx\)−θt\)\.\\displaystyle r\_\{j,t\}=\\sqrt\{\(\\delta\_\{j,t\}^\{x\}\)^\{2\}\+\(\\delta\_\{j,t\}^\{y\}\)^\{2\}\},\\quad\\beta\_\{j,t\}=\\operatorname\{wrap\}\\\!\\bigl\(\\operatorname\{atan2\}\(\\delta\_\{j,t\}^\{y\},\\delta\_\{j,t\}^\{x\}\)\-\\theta\_\{t\}\\bigr\)\.The observation𝐲t=𝐡¯​\(𝐱t\)\+𝐰t\\mathbf\{y\}\_\{t\}=\\bar\{\\mathbf\{h\}\}\(\\mathbf\{x\}\_\{t\}\)\+\\mathbf\{w\}\_\{t\}stacks the four range\-bearing pairs and the two biased odometry channels,

𝐡¯​\(𝐱t\)=\[r1,t,β1,t,r2,t,β2,t,r3,t,β3,t,r4,t,β4,t,vt\+btv,ωt\+btω\]⊤∈ℝ10,\\bar\{\\mathbf\{h\}\}\(\\mathbf\{x\}\_\{t\}\)=\[r\_\{1,t\},\\,\\beta\_\{1,t\},\\,r\_\{2,t\},\\,\\beta\_\{2,t\},\\,r\_\{3,t\},\\,\\beta\_\{3,t\},\\,r\_\{4,t\},\\,\\beta\_\{4,t\},\\,v\_\{t\}\+b\_\{t\}^\{v\},\\,\\omega\_\{t\}\+b\_\{t\}^\{\\omega\}\]^\{\\top\}\\in\\mathbb\{R\}^\{10\},\(88\)and𝐰t∼𝒩⁡\(𝟎,𝐑\)\\mathbf\{w\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{R\}\)with𝐑=diag⁡\(𝝈h2\)\\mathbf\{R\}=\\operatorname\{diag\}\(\\boldsymbol\{\\sigma\}\_\{h\}^\{2\}\)and

𝝈h=\[0\.08,1\.5∘,0\.08,1\.5∘,0\.08,1\.5∘,0\.08,1\.5∘⏟range, bearing per landmark,0\.05,1\.2∘\]⊤\.\\boldsymbol\{\\sigma\}\_\{h\}=\[\\,\\underbrace\{0\.08,\\,1\.5^\{\\circ\},\\,0\.08,\\,1\.5^\{\\circ\},\\,0\.08,\\,1\.5^\{\\circ\},\\,0\.08,\\,1\.5^\{\\circ\}\}\_\{\\text\{range, bearing per landmark\}\},\\;0\.05,\\,1\.2^\{\\circ\}\\,\]^\{\\top\}\.\(89\)The initial distribution is Gaussian,p0​\(𝐱0\)=𝒩⁡\(𝝁0,𝚺0\)p\_\{0\}\(\\mathbf\{x\}\_\{0\}\)=\\mathcal\{N\}\(\\boldsymbol\{\\mu\}\_\{0\},\\boldsymbol\{\\Sigma\}\_\{0\}\), with

𝝁0=\[−3\.0,−2\.0,0\.3,0\.7,0\.0,0\.08,−0\.05\]⊤,\\displaystyle\\boldsymbol\{\\mu\}\_\{0\}=\[\-3\.0,\\,\-2\.0,\\,0\.3,\\,0\.7,\\,0\.0,\\,0\.08,\\,\-0\.05\]^\{\\top\},\(90\)𝚺0=diag⁡\(0\.402,0\.402,\(10∘\)2,0\.122,\(6∘\)2,0\.052,\(2∘\)2\)\.\\displaystyle\\boldsymbol\{\\Sigma\}\_\{0\}=\\operatorname\{diag\}\\\!\\bigl\(0\.40^\{2\},\\,0\.40^\{2\},\\,\(10^\{\\circ\}\)^\{2\},\\,0\.12^\{2\},\\,\(6^\{\\circ\}\)^\{2\},\\,0\.05^\{2\},\\,\(2^\{\\circ\}\)^\{2\}\\bigr\)\.

### F\.3Benchmark 3: 2D Robot Localization with Biased Range\-Only Observations

The range\-only benchmark has state

𝐱t=\[ptx,pty,θt,vt,ωt,btr,btv,btω\]⊤∈ℝ8,\\mathbf\{x\}\_\{t\}=\[p\_\{t\}^\{x\},\\,p\_\{t\}^\{y\},\\,\\theta\_\{t\},\\,v\_\{t\},\\,\\omega\_\{t\},\\,b\_\{t\}^\{r\},\\,b\_\{t\}^\{v\},\\,b\_\{t\}^\{\\omega\}\]^\{\\top\}\\in\\mathbb\{R\}^\{8\},\(91\)with anchor𝐚=\(0,0\)\\mathbf\{a\}=\(0,0\),Δ​t=0\.1\\Delta t=0\.1, andtk=0\.1​kt\_\{k\}=0\.1kfork=0,…,59k=0,\\ldots,59\. Specializing Eq\. \([1](https://arxiv.org/html/2609.30498#S2.E1)\) with additive process noise gives𝐱t=𝐟¯​\(𝐱t−1,t−1\)\+𝐯t\\mathbf\{x\}\_\{t\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\},t\-1\)\+\\mathbf\{v\}\_\{t\}, with

ptx\\displaystyle p\_\{t\}^\{x\}=pt−1x\+Δtvt−1cosθt−1,\\displaystyle=p\_\{t\-1\}^\{x\}\+\\Delta t\\,v\_\{t\-1\}\\cos\\theta\_\{t\-1\},pty\\displaystyle p\_\{t\}^\{y\}=pt−1y\+Δtvt−1sinθt−1,\\displaystyle=p\_\{t\-1\}^\{y\}\+\\Delta t\\,v\_\{t\-1\}\\sin\\theta\_\{t\-1\},θt\\displaystyle\\theta\_\{t\}=wrap⁡\(θt−1\+Δ​t​ωt−1\),\\displaystyle=\\operatorname\{wrap\}\(\\theta\_\{t\-1\}\+\\Delta t\\,\\omega\_\{t\-1\}\),vt\\displaystyle v\_\{t\}=vt−1,ωt=ωt−1,\\displaystyle=v\_\{t\-1\},\\qquad\\omega\_\{t\}=\\omega\_\{t\-1\},btr\\displaystyle b\_\{t\}^\{r\}=bt−1r,\\displaystyle=b\_\{t\-1\}^\{r\},btv\\displaystyle b\_\{t\}^\{v\}=bt−1v,btω=bt−1ω\.\\displaystyle=b\_\{t\-1\}^\{v\},\\qquad b\_\{t\}^\{\\omega\}=b\_\{t\-1\}^\{\\omega\}\.\(92\)All bias and motion states therefore evolve as Gaussian random walks once the process noise𝐯t\\mathbf\{v\}\_\{t\}is added, with𝐯t∼𝒩⁡\(𝟎,𝐐\)\\mathbf\{v\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{Q\}\),𝐐=diag⁡\(𝝈f2\)\\mathbf\{Q\}=\\operatorname\{diag\}\(\\boldsymbol\{\\sigma\}\_\{f\}^\{2\}\), and

𝝈f=\[0\.035,0\.035,0\.8∘,0\.035,1\.2∘,0\.012,0\.012,0\.2∘\]⊤\.\\boldsymbol\{\\sigma\}\_\{f\}=\[0\.035,\\,0\.035,\\,0\.8^\{\\circ\},\\,0\.035,\\,1\.2^\{\\circ\},\\,0\.012,\\,0\.012,\\,0\.2^\{\\circ\}\]^\{\\top\}\.\(93\)The observation is𝐲t=𝐡¯​\(𝐱t\)\+𝐰t\\mathbf\{y\}\_\{t\}=\\bar\{\\mathbf\{h\}\}\(\\mathbf\{x\}\_\{t\}\)\+\\mathbf\{w\}\_\{t\}with

𝐡¯​\(𝐱t\)=\[‖𝐩t−𝐚‖2\+btr,vt\+btv,ωt\+btω\]⊤,𝐩t=\[ptx,pty\]⊤,\\bar\{\\mathbf\{h\}\}\(\\mathbf\{x\}\_\{t\}\)=\\bigl\[\\,\\\|\\mathbf\{p\}\_\{t\}\-\\mathbf\{a\}\\\|\_\{2\}\+b\_\{t\}^\{r\},\\;v\_\{t\}\+b\_\{t\}^\{v\},\\;\\omega\_\{t\}\+b\_\{t\}^\{\\omega\}\\,\\bigr\]^\{\\top\},\\qquad\\mathbf\{p\}\_\{t\}=\[p\_\{t\}^\{x\},p\_\{t\}^\{y\}\]^\{\\top\},\(94\)and𝐰t∼𝒩⁡\(𝟎,𝐑\)\\mathbf\{w\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{R\}\),𝐑=diag⁡\(𝝈h2\)\\mathbf\{R\}=\\operatorname\{diag\}\(\\boldsymbol\{\\sigma\}\_\{h\}^\{2\}\),

𝝈h=\[0\.025,0\.04,1∘\]⊤\.\\boldsymbol\{\\sigma\}\_\{h\}=\[0\.025,\\,0\.04,\\,1^\{\\circ\}\]^\{\\top\}\.\(95\)The filtering prior is a mirrored two\-component Gaussian mixture,

k∼Categorical\(0\.5,0\.5\),𝐱0\|k∼𝒩\(𝝁0\(k\),𝚺0mode\),k\\sim\\operatorname\{Categorical\}\(0\.5,0\.5\),\\qquad\\mathbf\{x\}\_\{0\}\\,\|\\,k\\sim\\mathcal\{N\}\\\!\\bigl\(\\boldsymbol\{\\mu\}\_\{0\}^\{\(k\)\},\\boldsymbol\{\\Sigma\}\_\{0\}^\{\\mathrm\{mode\}\}\\bigr\),\(96\)with

𝝁0\(1\)\\displaystyle\\boldsymbol\{\\mu\}\_\{0\}^\{\(1\)\}=\[3\.0,0\.6,π,0\.42,0\.02,0\.0,0\.0,0\.0\]⊤,\\displaystyle=\[3\.0,\\,0\.6,\\,\\pi,\\,0\.42,\\,0\.02,\\,0\.0,\\,0\.0,\\,0\.0\]^\{\\top\},\(97\)𝝁0\(2\)\\displaystyle\\boldsymbol\{\\mu\}\_\{0\}^\{\(2\)\}=\[−3\.0,0\.6,π,0\.42,0\.02,0\.0,0\.0,0\.0\]⊤,\\displaystyle=\[\-3\.0,\\,0\.6,\\,\\pi,\\,0\.42,\\,0\.02,\\,0\.0,\\,0\.0,\\,0\.0\]^\{\\top\},\(98\)𝚺0mode\\displaystyle\\boldsymbol\{\\Sigma\}\_\{0\}^\{\\mathrm\{mode\}\}=diag⁡\(0\.702,0\.252,\(8∘\)2,0\.162,\(4∘\)2,0\.402,0\.122,\(2∘\)2\)\.\\displaystyle=\\operatorname\{diag\}\\\!\\bigl\(0\.70^\{2\},\\,0\.25^\{2\},\\,\(8^\{\\circ\}\)^\{2\},\\,0\.16^\{2\},\\,\(4^\{\\circ\}\)^\{2\},\\,0\.40^\{2\},\\,0\.12^\{2\},\\,\(2^\{\\circ\}\)^\{2\}\\bigr\)\.\(99\)

### F\.4Benchmark 4: 3D Unordered\-Range Localization

The 3D unordered\-range benchmark has state

𝐱t=\[\\displaystyle\\mathbf\{x\}\_\{t\}=\\big\[ptx,pty,ptz,ψt,θt,vt,ωtψ,ωtθ,btv,btωψ,btωθ,\\displaystyle p\_\{t\}^\{x\},\\,p\_\{t\}^\{y\},\\,p\_\{t\}^\{z\},\\;\\psi\_\{t\},\\,\\theta\_\{t\},\\;v\_\{t\},\\,\\omega\_\{t\}^\{\\psi\},\\,\\omega\_\{t\}^\{\\theta\},\\;b\_\{t\}^\{v\},\\,b\_\{t\}^\{\\omega^\{\\psi\}\},\\,b\_\{t\}^\{\\omega^\{\\theta\}\},\(100\)btr1,btr2,btr3,btr4,dtx,dty,dtz,stv,str\]⊤∈ℝ20,\\displaystyle b\_\{t\}^\{r\_\{1\}\},\\,b\_\{t\}^\{r\_\{2\}\},\\,b\_\{t\}^\{r\_\{3\}\},\\,b\_\{t\}^\{r\_\{4\}\},\\;d\_\{t\}^\{x\},\\,d\_\{t\}^\{y\},\\,d\_\{t\}^\{z\},\\;s\_\{t\}^\{v\},\\,s\_\{t\}^\{r\}\\;\\big\]^\{\\top\}\\in\\mathbb\{R\}^\{20\},where𝐩t=\[ptx,pty,ptz\]⊤\\mathbf\{p\}\_\{t\}=\[p\_\{t\}^\{x\},p\_\{t\}^\{y\},p\_\{t\}^\{z\}\]^\{\\top\}is position,\(ψt,θt\)\(\\psi\_\{t\},\\theta\_\{t\}\)are yaw and pitch,vtv\_\{t\}is forward speed,\(ωtψ,ωtθ\)\(\\omega\_\{t\}^\{\\psi\},\\omega\_\{t\}^\{\\theta\}\)are angular rates,\(btv,btωψ,btωθ\)\(b\_\{t\}^\{v\},b\_\{t\}^\{\\omega^\{\\psi\}\},b\_\{t\}^\{\\omega^\{\\theta\}\}\)are proprioceptive biases,\(btr1,…,btr4\)\(b\_\{t\}^\{r\_\{1\}\},\\ldots,b\_\{t\}^\{r\_\{4\}\}\)are per\-anchor range biases,\(dtx,dty,dtz\)\(d\_\{t\}^\{x\},d\_\{t\}^\{y\},d\_\{t\}^\{z\}\)are slowly varying drift disturbances on position, and\(stv,str\)\(s\_\{t\}^\{v\},s\_\{t\}^\{r\}\)are multiplicative scale\-calibration factors on speed and range\. The four range anchors are

𝐚1=\(−4\.4,−2\.1,0\),𝐚2=\(4\.2,−1\.3,0\),𝐚3=\(−0\.8,3\.8,0\),𝐚4=\(4\.7,3\.4,0\)\.\\mathbf\{a\}\_\{1\}=\(\-4\.4,\-2\.1,0\),\\quad\\mathbf\{a\}\_\{2\}=\(4\.2,\-1\.3,0\),\\quad\\mathbf\{a\}\_\{3\}=\(\-0\.8,3\.8,0\),\\quad\\mathbf\{a\}\_\{4\}=\(4\.7,3\.4,0\)\.\(101\)The time grid istk=0\.1​kt\_\{k\}=0\.1kfork=0,…,34k=0,\\ldots,34, withΔ​t=0\.1\\Delta t=0\.1,ρv=1\.7\\rho\_\{v\}=1\.7,ρψ=2\.1\\rho\_\{\\psi\}=2\.1, andρθ=2\.3\\rho\_\{\\theta\}=2\.3\.\. Specializing Eq\. \([1](https://arxiv.org/html/2609.30498#S2.E1)\) with additive process noise gives𝐱t=𝐟¯​\(𝐱t−1\)\+𝐯t\\mathbf\{x\}\_\{t\}=\\bar\{\\mathbf\{f\}\}\(\\mathbf\{x\}\_\{t\-1\}\)\+\\mathbf\{v\}\_\{t\}\. The pose, attitude, and velocity components evolve as

ptx\\displaystyle p\_\{t\}^\{x\}=pt−1x\+Δt\(vt−1cosθt−1cosψt−1\+dt−1x\),\\displaystyle=p\_\{t\-1\}^\{x\}\+\\Delta t\\bigl\(v\_\{t\-1\}\\cos\\theta\_\{t\-1\}\\cos\\psi\_\{t\-1\}\+d\_\{t\-1\}^\{x\}\\bigr\),pty\\displaystyle p\_\{t\}^\{y\}=pt−1y\+Δt\(vt−1cosθt−1sinψt−1\+dt−1y\),\\displaystyle=p\_\{t\-1\}^\{y\}\+\\Delta t\\bigl\(v\_\{t\-1\}\\cos\\theta\_\{t\-1\}\\sin\\psi\_\{t\-1\}\+d\_\{t\-1\}^\{y\}\\bigr\),ptz\\displaystyle p\_\{t\}^\{z\}=pt−1z\+Δt\(vt−1sinθt−1\+dt−1z\),\\displaystyle=p\_\{t\-1\}^\{z\}\+\\Delta t\\bigl\(v\_\{t\-1\}\\sin\\theta\_\{t\-1\}\+d\_\{t\-1\}^\{z\}\\bigr\),ψt\\displaystyle\\psi\_\{t\}=wrap⁡\(ψt−1\+Δ​t​ωt−1ψ\),θt=wrap⁡\(θt−1\+Δ​t​ωt−1θ\),\\displaystyle=\\operatorname\{wrap\}\(\\psi\_\{t\-1\}\+\\Delta t\\,\\omega\_\{t\-1\}^\{\\psi\}\),\\qquad\\theta\_\{t\}=\\operatorname\{wrap\}\(\\theta\_\{t\-1\}\+\\Delta t\\,\\omega\_\{t\-1\}^\{\\theta\}\),vt\\displaystyle v\_\{t\}=vt−1\+ρv​\(vcmd​\(t−1\)−vt−1\)​Δ​t,\\displaystyle=v\_\{t\-1\}\+\\rho\_\{v\}\\bigl\(v\_\{\\rm cmd\}\(t\-1\)\-v\_\{t\-1\}\\bigr\)\\Delta t,ωtψ\\displaystyle\\omega\_\{t\}^\{\\psi\}=ωt−1ψ\+ρψ​\(ωcmdψ​\(t−1\)−ωt−1ψ\)​Δ​t,\\displaystyle=\\omega\_\{t\-1\}^\{\\psi\}\+\\rho\_\{\\psi\}\\bigl\(\\omega^\{\\psi\}\_\{\\rm cmd\}\(t\-1\)\-\\omega\_\{t\-1\}^\{\\psi\}\\bigr\)\\Delta t,ωtθ\\displaystyle\\omega\_\{t\}^\{\\theta\}=ωt−1θ\+ρθ​\(ωcmdθ​\(t−1\)−ωt−1θ\)​Δ​t,\\displaystyle=\\omega\_\{t\-1\}^\{\\theta\}\+\\rho\_\{\\theta\}\\bigl\(\\omega^\{\\theta\}\_\{\\rm cmd\}\(t\-1\)\-\\omega\_\{t\-1\}^\{\\theta\}\\bigr\)\\Delta t,\(102\)and the remaining 12 latent parameters persist as identity maps in𝐟¯\\bar\{\\mathbf\{f\}\}, so they evolve as Gaussian random walks once𝐯t\\mathbf\{v\}\_\{t\}is added,

btv=bt−1v,btωψ=bt−1ωψ,btωθ=bt−1ωθ,btrj=bt−1rj\(j=1,…,4\),dtξ=dt−1ξ\(ξ∈\{x,y,z\}\),stv=st−1v,str=st−1r\.\\begin\{gathered\}b\_\{t\}^\{v\}=b\_\{t\-1\}^\{v\},\\quad b\_\{t\}^\{\\omega^\{\\psi\}\}=b\_\{t\-1\}^\{\\omega^\{\\psi\}\},\\quad b\_\{t\}^\{\\omega^\{\\theta\}\}=b\_\{t\-1\}^\{\\omega^\{\\theta\}\},\\\\ b\_\{t\}^\{r\_\{j\}\}=b\_\{t\-1\}^\{r\_\{j\}\}\\ \(j=1,\\ldots,4\),\\quad d\_\{t\}^\{\\xi\}=d\_\{t\-1\}^\{\\xi\}\\ \(\\xi\\in\\\{x,y,z\\\}\),\\quad s\_\{t\}^\{v\}=s\_\{t\-1\}^\{v\},\\quad s\_\{t\}^\{r\}=s\_\{t\-1\}^\{r\}\.\\end\{gathered\}\(103\)The command schedule is

vcmd​\(t\)\\displaystyle v\_\{\\rm cmd\}\(t\)=0\.88\+0\.16​sin⁡\(0\.26​t\)\+0\.05​cos⁡\(0\.11​t\),\\displaystyle=0\.88\+0\.16\\sin\(0\.26t\)\+0\.05\\cos\(0\.11t\),\(104\)ωcmdψ​\(t\)\\displaystyle\\omega^\{\\psi\}\_\{\\rm cmd\}\(t\)=0\.020\+0\.16​sin⁡\(0\.31​t\)\+0\.04​cos⁡\(0\.13​t\),\\displaystyle=0\.020\+0\.16\\sin\(0\.31t\)\+0\.04\\cos\(0\.13t\),\(105\)ωcmdθ​\(t\)\\displaystyle\\omega^\{\\theta\}\_\{\\rm cmd\}\(t\)=0\.015\+0\.08​sin⁡\(0\.29​t\)\+0\.03​cos⁡\(0\.10​t\)\.\\displaystyle=0\.015\+0\.08\\sin\(0\.29t\)\+0\.03\\cos\(0\.10t\)\.\(106\)Process noise is𝐯t∼𝒩⁡\(𝟎,𝐐\)\\mathbf\{v\}\_\{t\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{Q\}\)with𝐐=diag⁡\(𝝈f2\)\\mathbf\{Q\}=\\operatorname\{diag\}\(\\boldsymbol\{\\sigma\}\_\{f\}^\{2\}\)and per\-coordinate standard deviations

𝝈f=\[\\displaystyle\\boldsymbol\{\\sigma\}\_\{f\}=\\big\[0\.070,0\.070,0\.060,0\.8∘,0\.7∘,0\.070,0\.9∘,0\.8∘,0\.006,0\.10∘,\\displaystyle 0\.070,\\,0\.070,\\,0\.060,\\;0\.8^\{\\circ\},\\,0\.7^\{\\circ\},\\;0\.070,\\,0\.9^\{\\circ\},\\,0\.8^\{\\circ\},\\;0\.006,\\,0\.10^\{\\circ\},\(107\)0\.10∘,0\.008,0\.008,0\.008,0\.008,0\.010,0\.010,0\.010,0\.006,0\.003\]⊤,\\displaystyle 0\.10^\{\\circ\},\\;0\.008,\\,0\.008,\\,0\.008,\\,0\.008,\\;0\.010,\\,0\.010,\\,0\.010,\\;0\.006,\\,0\.003\\;\\big\]^\{\\top\},ordered as in Eq\. \([100](https://arxiv.org/html/2609.30498#A6.E100)\)\. The raw biased and scale\-corrected ranges are

𝐫t​\(𝐱t\)=\[str​ρ1,t\+btr1,str​ρ2,t\+btr2,str​ρ3,t\+btr3,str​ρ4,t\+btr4\]⊤,ρj,t=‖𝐩t−𝐚j‖2\.\\mathbf\{r\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\)=\\bigl\[\\,s\_\{t\}^\{r\}\\rho\_\{1,t\}\+b\_\{t\}^\{r\_\{1\}\},\\;s\_\{t\}^\{r\}\\rho\_\{2,t\}\+b\_\{t\}^\{r\_\{2\}\},\\;s\_\{t\}^\{r\}\\rho\_\{3,t\}\+b\_\{t\}^\{r\_\{3\}\},\\;s\_\{t\}^\{r\}\\rho\_\{4,t\}\+b\_\{t\}^\{r\_\{4\}\}\\,\\bigr\]^\{\\top\},\\quad\\rho\_\{j,t\}=\\\|\\mathbf\{p\}\_\{t\}\-\\mathbf\{a\}\_\{j\}\\\|\_\{2\}\.\(108\)The proprioceptive observation channels are

𝐝t​\(𝐱t\)=\[stv​vt\+btv,ωtψ\+btωψ,ωtθ\+btωθ\]⊤\.\\mathbf\{d\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\)=\\bigl\[\\,s\_\{t\}^\{v\}v\_\{t\}\+b\_\{t\}^\{v\},\\;\\omega\_\{t\}^\{\\psi\}\+b\_\{t\}^\{\\omega^\{\\psi\}\},\\;\\omega\_\{t\}^\{\\theta\}\+b\_\{t\}^\{\\omega^\{\\theta\}\}\\,\\bigr\]^\{\\top\}\.\(109\)The current stored observation has four unordered range returns and three labeled proprioceptive measurements\. Equivalently, with independent range noiseϵtr∼𝒩⁡\(𝟎,0\.0552​𝐈4\)\\boldsymbol\{\\epsilon\}\_\{t\}^\{r\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},0\.055^\{2\}\\mathbf\{I\}\_\{4\}\)and proprioceptive noiseϵtd∼𝒩⁡\(𝟎,𝐑d\)\\boldsymbol\{\\epsilon\}\_\{t\}^\{d\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{R\}\_\{d\}\),

𝐲t=\[sort⁡\(𝐫t​\(𝐱t\)\+ϵtr\)𝐝t​\(𝐱t\)\+ϵtd\]∈ℝ7,𝐑d=diag⁡\(0\.0452,\(1∘\)2,\(1∘\)2\)\.\\mathbf\{y\}\_\{t\}=\\begin\{bmatrix\}\\operatorname\{sort\}\\\!\\bigl\(\\mathbf\{r\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\)\+\\boldsymbol\{\\epsilon\}\_\{t\}^\{r\}\\bigr\)\\\\ \\mathbf\{d\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\)\+\\boldsymbol\{\\epsilon\}\_\{t\}^\{d\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{7\},\\qquad\\mathbf\{R\}\_\{d\}=\\operatorname\{diag\}\\\!\\bigl\(0\.045^\{2\},\\,\(1^\{\\circ\}\)^\{2\},\\,\(1^\{\\circ\}\)^\{2\}\\bigr\)\.\(110\)Because sorting destroys the anchor\-to\-range correspondence, the range likelihood is permutation invariant\. Let𝐲t=\[\(𝐲tr\)⊤,\(𝐲td\)⊤\]⊤\\mathbf\{y\}\_\{t\}=\[\(\\mathbf\{y\}\_\{t\}^\{r\}\)^\{\\top\},\(\\mathbf\{y\}\_\{t\}^\{d\}\)^\{\\top\}\]^\{\\top\}with𝐲tr∈ℝ4\\mathbf\{y\}\_\{t\}^\{r\}\\in\\mathbb\{R\}^\{4\}and𝐲td∈ℝ3\\mathbf\{y\}\_\{t\}^\{d\}\\in\\mathbb\{R\}^\{3\}\. Let𝒮4\\mathcal\{S\}\_\{4\}denote the symmetric group of all4\!=244\!=24permutations of\{1,2,3,4\}\\\{1,2,3,4\\\}, and let𝐏π∈\{0,1\}4×4\\mathbf\{P\}\_\{\\pi\}\\in\\\{0,1\\\}^\{4\\times 4\}denote the permutation matrix associated withπ∈𝒮4\\pi\\in\\mathcal\{S\}\_\{4\}\. The likelihood used is

p⁡\(𝐲t∣𝐱t\)=\[1\|𝒮4\|​∑π∈𝒮4𝒩⁡\(𝐲tr,𝐏π​𝐫t​\(𝐱t\),0\.0552​𝐈4\)\]​𝒩​\(𝐲td,𝐝t​\(𝐱t\),𝐑d\),p\(\\mathbf\{y\}\_\{t\}\\mid\\mathbf\{x\}\_\{t\}\)=\\left\[\\frac\{1\}\{\|\\mathcal\{S\}\_\{4\}\|\}\\sum\_\{\\pi\\in\\mathcal\{S\}\_\{4\}\}\\mathcal\{N\}\\\!\\bigl\(\\mathbf\{y\}\_\{t\}^\{r\};\\;\\mathbf\{P\}\_\{\\pi\}\\,\\mathbf\{r\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\),\\;0\.055^\{2\}\\,\\mathbf\{I\}\_\{4\}\\bigr\)\\right\]\\mathcal\{N\}\\\!\\bigl\(\\mathbf\{y\}\_\{t\}^\{d\};\\;\\mathbf\{d\}\_\{t\}\(\\mathbf\{x\}\_\{t\}\),\\;\\mathbf\{R\}\_\{d\}\\bigr\),\(111\)evaluated numerically vialogsumexp\\operatorname\{logsumexp\}over the 24 assignments\. The filtering prior, truth\-generation prior, and proposal\-training initial distribution are the same two\-component positive\-zzGaussian mixture,

k∼Categorical\(0\.5,0\.5\),𝐱0\|k∼𝒩\(𝝁0\(k\),𝚺0mode\),𝐱0⋆∼p0\(𝐱0\),k\\sim\\operatorname\{Categorical\}\(0\.5,0\.5\),\\qquad\\mathbf\{x\}\_\{0\}\\,\|\\,k\\sim\\mathcal\{N\}\\\!\\bigl\(\\boldsymbol\{\\mu\}\_\{0\}^\{\(k\)\},\\boldsymbol\{\\Sigma\}\_\{0\}^\{\\mathrm\{mode\}\}\\bigr\),\\qquad\\mathbf\{x\}\_\{0\}^\{\\star\}\\sim p\_\{0\}\(\\mathbf\{x\}\_\{0\}\),\(112\)with means

𝝁0\(1\)=\[\\displaystyle\\boldsymbol\{\\mu\}\_\{0\}^\{\(1\)\}=\\big\[0\.10,−0\.75,1\.55,0\.14,0\.18,0\.92,0\.08,0\.04,0,0,\\displaystyle 0\.10,\\,\-0\.75,\\,1\.55,\\,0\.14,\\,0\.18,\\,0\.92,\\,0\.08,\\,0\.04,\\,0,\\,0,\(113\)0,0,0,0,0,0,0,0,1,1\]⊤,\\displaystyle 0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,1,\\,1\\,\\big\]^\{\\top\},𝝁0\(2\)=\[\\displaystyle\\boldsymbol\{\\mu\}\_\{0\}^\{\(2\)\}=\\big\[0\.65,−0\.30,2\.65,0\.14,0\.18,0\.92,0\.08,0\.04,0,0,\\displaystyle 0\.65,\\,\-0\.30,\\,2\.65,\\,0\.14,\\,0\.18,\\,0\.92,\\,0\.08,\\,0\.04,\\,0,\\,0,0,0,0,0,0,0,0,0,1,1\]⊤,\\displaystyle 0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,0,\\,1,\\,1\\,\\big\]^\{\\top\},and covariance

𝚺0mode=diag⁡\(CLOSE\\displaystyle\\boldsymbol\{\\Sigma\}\_\{0\}^\{\\mathrm\{mode\}\}=\\operatorname\{diag\}\\big\(0\.352,0\.322,0\.142,\(8∘\)2,\(8∘\)2,0\.162,\(2\.4∘\)2,\(2\.2∘\)2,0\.062,\(0\.8∘\)2,\\displaystyle 0\.35^\{2\},\\,0\.32^\{2\},\\,0\.14^\{2\},\\;\(8^\{\\circ\}\)^\{2\},\\,\(8^\{\\circ\}\)^\{2\},\\;0\.16^\{2\},\\,\(2\.4^\{\\circ\}\)^\{2\},\\,\(2\.2^\{\\circ\}\)^\{2\},\\;0\.06^\{2\},\\,\(0\.8^\{\\circ\}\)^\{2\},\(114\)OPEN\(0\.8∘\)2,0\.082,0\.082,0\.082,0\.082,0\.062,0\.062,0\.062,0\.032,0\.0202\),\\displaystyle\(0\.8^\{\\circ\}\)^\{2\},\\;0\.08^\{2\},\\,0\.08^\{2\},\\,0\.08^\{2\},\\,0\.08^\{2\},\\;0\.06^\{2\},\\,0\.06^\{2\},\\,0\.06^\{2\},\\;0\.03^\{2\},\\,0\.020^\{2\}\\;\\big\),ordered as in Eq\. \([100](https://arxiv.org/html/2609.30498#A6.E100)\)\.