Grounding Investor Views: Neural Predicates in the Black-Litterman Model

# Grounding Investor Views: Neural Predicates in the Black-Litterman Model
Source: [https://arxiv.org/html/2607.20533](https://arxiv.org/html/2607.20533)
###### Abstract

Portfolio construction under the Black\-Litterman model requires investors to specify views on asset returns alongside explicit uncertainty estimates—a process that remains largely subjective and difficult to scale\. We propose a formal approach in which neural predicates serve as a structured, probabilistic mechanism for view generation\. In our formulation, structured financial analysis data is processed through a compositional hierarchy of neural predicates whose outputs—probability distributions over market stances—are mapped to the pick matrix𝐏\\mathbf\{P\}, the view return vector𝐪\\mathbf\{q\}, and the view uncertainty matrix𝛀\\boldsymbol\{\\Omega\}of the Black\-Litterman model\. View confidence is derived from predicate output distributions, providing a data\-driven alternative to subjective uncertainty elicitation\. The resulting approach is interpretable, in the sense that any portfolio weight can be traced back through the predicate’s logical chain to the underlying data, and fully differentiable, enabling end\-to\-end learning\.

Keywords:neural predicates, Black\-Litterman, portfolio optimization, expert knowledge integration

## 1Introduction

Portfolio construction involves decision\-making under uncertainty\. The mean\-variance approach ofMarkowitz[28](https://arxiv.org/html/2607.20533#bib.bib43)established the importance of portfolio covariance, but its sensitivity to expected return estimates often produces unstable and implausible allocations\. The Black\-Litterman model[4](https://arxiv.org/html/2607.20533#bib.bib5),[5](https://arxiv.org/html/2607.20533#bib.bib6)addresses this issue by combining equilibrium returns implied by the CAPM with investor views held at explicit confidence levels, yielding more stable and diversified portfolios\.111The CAPM[37](https://arxiv.org/html/2607.20533#bib.bib57),[23](https://arxiv.org/html/2607.20533#bib.bib36)is used for exposition\. The approach developed here is agnostic to this choice\.

However, Black\-Litterman specifies how to incorporate views without explaining how they should be generated\. In practice, directions, magnitudes, and confidence levels are typically assigned through subjective judgment, limiting reproducibility and scalability[19](https://arxiv.org/html/2607.20533#bib.bib29)\. Rich analytical information from valuation models, earnings analysis, and balance\-sheet diagnostics is often reduced to ad hoc point estimates with arbitrary uncertainty assumptions\.

This paper proposes a theoretical approach in which*neural predicates*[27](https://arxiv.org/html/2607.20533#bib.bib40)generate Black\-Litterman views\. Their output distributions over discrete stances \(e\.g\., bullish, neutral, bearish\) naturally provide view direction, magnitude, and uncertainty\. We make three contributions: \(i\) a formal mapping from neural predicate outputs to the Black\-Litterman components𝐏\\mathbf\{P\},𝐪\\mathbf\{q\}, and𝛀\\boldsymbol\{\\Omega\}; \(ii\) an entropy\-based measure of uncertainty that replaces fixed covariance heuristics; and \(iii\) a compositional method for combining multiple predicates into a coherent asset view\.

## 2Background and Related Work

### 2\.1Portfolio Optimization and the Limits of Mean\-Variance

Modern portfolio theory originates withMarkowitz[28](https://arxiv.org/html/2607.20533#bib.bib43), who cast portfolio selection as a quadratic program: given expected returns𝝁\\boldsymbol\{\\mu\}and covariance𝚺\\boldsymbol\{\\Sigma\}, the investor seeks weights𝐰\\mathbf\{w\}that minimize variance for a given expected return, tracing the efficient frontier\.

The approach suffers a well\-documented fragility: optimal weights are extraordinarily sensitive to the inputs, particularly to𝝁\\boldsymbol\{\\mu\}[32](https://arxiv.org/html/2607.20533#bib.bib48)\. Because expected returns must be estimated from noisy data, estimation error propagates directly into weights\.Jobson and Korkie[20](https://arxiv.org/html/2607.20533#bib.bib31)showed that in\-sample optimality rarely survives out of sample, andBest and Grauer[3](https://arxiv.org/html/2607.20533#bib.bib4)demonstrated analytically that small perturbations in return estimates produce large, economically implausible swings in weights\. The optimizer thus acts as an*error maximizer*, concentrating weight in assets whose returns are most overestimated[32](https://arxiv.org/html/2607.20533#bib.bib48)\.

The literature has responded with shrinkage estimators[22](https://arxiv.org/html/2607.20533#bib.bib34), resampling[32](https://arxiv.org/html/2607.20533#bib.bib48), and robust optimization[14](https://arxiv.org/html/2607.20533#bib.bib22)\. These improve stability but share a limitation: they address estimation error statistically, without incorporating the domain knowledge needed to form return expectations—a requirement left to the implementation\. The Black\-Litterman model instead addresses the source of the fragility, replacing unconstrained estimation with a structured Bayesian approach anchored to a stable prior\.

### 2\.2Incorporating Qualitative Judgment in Quantitative Models

Research on incorporating qualitative judgment into portfolio construction generally follows three approaches: direct use of analyst forecasts, factor models, and Bayesian methods such as Black\-Litterman\. Analyst recommendations and price targets contain predictive information[2](https://arxiv.org/html/2607.20533#bib.bib3), but they provide point estimates without a principled measure of uncertainty and are difficult to aggregate systematically\. Factor models[10](https://arxiv.org/html/2607.20533#bib.bib16)offer a structured approach based on systematic risk premia, yet they are less suited to firm\-specific insights and rarely propagate estimation uncertainty into portfolio decisions\. The Black\-Litterman model[4](https://arxiv.org/html/2607.20533#bib.bib5),[5](https://arxiv.org/html/2607.20533#bib.bib6)addresses these issues by combining equilibrium returns with investor views and explicitly modeling view uncertainty; extensions have incorporated non\-normal returns[30](https://arxiv.org/html/2607.20533#bib.bib45)and non\-linear views[31](https://arxiv.org/html/2607.20533#bib.bib47)\. Despite their differences, all three approaches rely on an informal translation of qualitative judgment into quantitative inputs, sacrificing reproducibility and the structure of the underlying analysis\. As financial data become increasingly rich and high\-dimensional, a systematic mechanism for generating Black\-Litterman\-compatible views is needed\. We argue that neural predicates provide such a mechanism\.

### 2\.3Neuro\-Symbolic AI and the Rise of Neural Predicates

AI has historically been divided between symbolic approaches, which emphasize rule\-based reasoning and interpretability[33](https://arxiv.org/html/2607.20533#bib.bib49), and connectionist approaches, which learn distributed representations from data[35](https://arxiv.org/html/2607.20533#bib.bib54)\. Their complementary strengths and weaknesses—precision and compositionality on one side, robustness and scalability on the other—motivated the development of neuro\-symbolic systems[11](https://arxiv.org/html/2607.20533#bib.bib18)\. Early work showed that neural computation could implement logical operations[29](https://arxiv.org/html/2607.20533#bib.bib44), while later systems such as KBANN[39](https://arxiv.org/html/2607.20533#bib.bib61), CILP[13](https://arxiv.org/html/2607.20533#bib.bib20), and CILP\+\+[12](https://arxiv.org/html/2607.20533#bib.bib19)demonstrated how symbolic knowledge could guide neural architectures and be recovered from trained models\. Neural predicates, introduced in DeepProbLog byManhaeveet al\.[27](https://arxiv.org/html/2607.20533#bib.bib40), provide a prominent solution: logical predicates whose truth probabilities are generated by neural networks through neural annotated disjunctions\. This integration enables probabilistic inference, symbolic reasoning, and end\-to\-end learning within a unified approach\.

### 2\.4Prior Work on Machine Learning in Portfolio Construction

Research on machine learning in portfolio construction broadly spans return prediction, alternative data, and tighter integration with optimization\. Studies such asGuet al\.[15](https://arxiv.org/html/2607.20533#bib.bib23)andChenet al\.[8](https://arxiv.org/html/2607.20533#bib.bib14)show that neural networks can outperform traditional models in forecasting cross\-sectional returns, but they generally provide point estimates without a principled treatment of uncertainty\. A parallel literature uses NLP to extract signals from news, analyst reports, earnings calls, and social media[38](https://arxiv.org/html/2607.20533#bib.bib60),[25](https://arxiv.org/html/2607.20533#bib.bib38),[18](https://arxiv.org/html/2607.20533#bib.bib28),[7](https://arxiv.org/html/2607.20533#bib.bib13), with recent extensions employing large language models[24](https://arxiv.org/html/2607.20533#bib.bib37); however, converting these signals into Black\-Litterman views remains largely ad hoc\. Other work combines machine learning with portfolio optimization through robust optimization, reinforcement learning, and deep hedging[1](https://arxiv.org/html/2607.20533#bib.bib2),[40](https://arxiv.org/html/2607.20533#bib.bib65),[6](https://arxiv.org/html/2607.20533#bib.bib10), while extensions of Black\-Litterman incorporate quantitative signals into the view vector𝐪\\mathbf\{q\}[17](https://arxiv.org/html/2607.20533#bib.bib26)\. Yet no existing approach provides a formal and interpretable mapping from structured, multi\-dimensional analysis to the full\(𝐏,𝐪,𝛀\)\(\\mathbf\{P\},\\mathbf\{q\},\\boldsymbol\{\\Omega\}\)representation, and the specification of𝛀\\boldsymbol\{\\Omega\}in particular remains weakly grounded, often reduced to a fixed multiple of the prior covariance[19](https://arxiv.org/html/2607.20533#bib.bib29)\.

## 3Neural Predicates and the Black\-Litterman Model

### 3\.1Symbolic Reasoning and Neural Predicates

Symbolic AI models intelligent behavior as rule\-governed manipulation of discrete, interpretable structures, its dominant formalism being*first\-order logic*\(FOL\)[36](https://arxiv.org/html/2607.20533#bib.bib55)\. A*predicate*PPof aritynnis a functionP:𝒟n→\{⊤,⊥\}P:\\mathcal\{D\}^\{n\}\\to\\\{\\top,\\bot\\\}over a domain𝒟\\mathcal\{D\}; an*atom*such as𝑏𝑢𝑙𝑙𝑖𝑠ℎ​\(x\)\\mathit\{bullish\}\(x\)is the smallest expression to which a truth value can be assigned, and a variable\-free atom obtained by substituting constants \(*grounding*\) is a*ground atom*—e\.g\.𝑏𝑢𝑙𝑙𝑖𝑠ℎ​\(𝑎𝑝𝑝𝑙𝑒\)\\mathit\{bullish\}\(\\mathit\{apple\}\), asking whether Apple is in a bullish stance\. Truth\-preserving*inference rules*, chief among them modus ponens \(fromφ→ψ\\varphi\\rightarrow\\psiandφ\\varphi, deriveψ\\psi\), then derive new facts, each conclusion carrying an explicit justification chain\. Systems built this way succeeded in well\-structured domains—theorem proving, expert systems, Prolog[21](https://arxiv.org/html/2607.20533#bib.bib32)—but cannot handle perceptual data, noise, or learning\.

The obstacle is representational\. A neural network computes a continuous functionf𝜽:ℝd→ℝkf\_\{\\boldsymbol\{\\theta\}\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{k\}over distributed, non\-interpretable activations, whereas symbolic systems operate on discrete structures in which a rule either fires or does not\. This creates the*symbol grounding problem*[16](https://arxiv.org/html/2607.20533#bib.bib25): a symbolic system cannot determine the truth of𝑢𝑛𝑑𝑒𝑟𝑣𝑎𝑙𝑢𝑒𝑑​\(x\)\\mathit\{undervalued\}\(x\)from raw data, so grounding must be stipulated externally by a human—the principal bottleneck as data grow\. Compounding it, discrete logical operations are step functions with zero gradient almost everywhere, so embedding symbolic reasoning in a training loop breaks backpropagation and precludes joint end\-to\-end learning\. Real decision\-making—including, we argue, generating investor views—requires both perception \(where neural networks excel\) and compositional reasoning \(where symbolic systems excel\)\.

A neural predicate[27](https://arxiv.org/html/2607.20533#bib.bib40)is a logical predicate whose extension is determined by a neural network rather than by enumeration\. Letqqhave possible output values𝐮=\(u1,…,um\)\\mathbf\{u\}=\(u\_\{1\},\\ldots,u\_\{m\}\)over input terms𝐭\\mathbf\{t\}\. A*neural annotated disjunction*\(nAD\) is

𝚗𝚗​\(mq,𝐭,u,𝐮\)::q​\(𝐭,u\),\\mathtt\{nn\}\(m\_\{q\},\\,\\mathbf\{t\},\\,u,\\,\\mathbf\{u\}\)\\mathrel\{::\}q\(\\mathbf\{t\},\\,u\),\(1\)
with semantics that the probability of the ground atomq​\(𝐭,ui\)q\(\\mathbf\{t\},u\_\{i\}\)equals theii\-th network output,

P​\(q​\(𝐭,ui\)=⊤\)=\[fmq​\(𝐭\)\]i,∑i=1m\[fmq​\(𝐭\)\]i=1,P\\bigl\(q\(\\mathbf\{t\},u\_\{i\}\)=\\top\\bigr\)=\\bigl\[f\_\{m\_\{q\}\}\(\\mathbf\{t\}\)\\bigr\]\_\{i\},\\quad\\sum\_\{i=1\}^\{m\}\\bigl\[f\_\{m\_\{q\}\}\(\\mathbf\{t\}\)\\bigr\]\_\{i\}=1,\(2\)
wherefmq:𝒳→Δm−1f\_\{m\_\{q\}\}:\\mathcal\{X\}\\to\\Delta^\{m\-1\}maps inputs to the probability simplex via a softmax layer\. For financial stance classification,𝚗𝚗\(f𝑠𝑡𝑎𝑛𝑐𝑒,x,s,\{𝑏𝑢𝑙𝑙𝑖𝑠ℎ,𝑏𝑒𝑎𝑟𝑖𝑠ℎ,𝑛𝑒𝑢𝑡𝑟𝑎𝑙\}\)::𝑠𝑡𝑎𝑛𝑐𝑒\(x,s\)\\mathtt\{nn\}\(f\_\{\\mathit\{stance\}\},x,s,\\\{\\mathit\{bullish\},\\mathit\{bearish\},\\mathit\{neutral\}\\\}\)::\\mathit\{stance\}\(x,s\), withxxencoding a company’s profile\. DeepProbLog[27](https://arxiv.org/html/2607.20533#bib.bib40)embeds nADs into ProbLog[9](https://arxiv.org/html/2607.20533#bib.bib15), computing query probabilities by*weighted model counting*\(WMC\); because network outputs appear as weights and WMC can be made differentiable, gradients back\-propagate into the network, enabling end\-to\-end learning\.

### 3\.2Compositional Reasoning

The power of neural predicates lies in composing them through logical rules, which distinguishes such a system from a collection of independent classifiers\. Atomic predicates form the perceptual layer—for instance𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛​\(x,v\)\\mathit\{valuation\}\(x,v\),𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠​\(x,e\)\\mathit\{earnings\}\(x,e\), andℎ𝑒𝑎𝑙𝑡ℎ​\(x,h\)\\mathit\{health\}\(x,h\), each an nAD over its own categories—and rules compose them:

𝑠𝑡𝑎𝑛𝑐𝑒​\(x,𝑏𝑢𝑙𝑙𝑖𝑠ℎ\)\\displaystyle\\mathit\{stance\}\(x,\\mathit\{bullish\}\)←𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛​\(x,𝑢𝑛𝑑𝑒𝑟\),𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠​\(x,𝑠𝑡𝑟𝑜𝑛𝑔\),ℎ𝑒𝑎𝑙𝑡ℎ​\(x,ℎ𝑒𝑎𝑙𝑡ℎ𝑦\),\\displaystyle\\leftarrow\\mathit\{valuation\}\(x,\\mathit\{under\}\),\\,\\mathit\{earnings\}\(x,\\mathit\{strong\}\),\\,\\mathit\{health\}\(x,\\mathit\{healthy\}\),\(3\)𝑠𝑡𝑎𝑛𝑐𝑒​\(x,𝑏𝑒𝑎𝑟𝑖𝑠ℎ\)\\displaystyle\\mathit\{stance\}\(x,\\mathit\{bearish\}\)←𝑣𝑎𝑙𝑢𝑎𝑡𝑖𝑜𝑛​\(x,𝑜𝑣𝑒𝑟\),𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠​\(x,𝑤𝑒𝑎𝑘\)\.\\displaystyle\\leftarrow\\mathit\{valuation\}\(x,\\mathit\{over\}\),\\,\\mathit\{earnings\}\(x,\\mathit\{weak\}\)\.\(4\)
The derived𝑠𝑡𝑎𝑛𝑐𝑒​\(x,s\)\\mathit\{stance\}\(x,s\)is not a network but a logical consequence, its probability computed by the inference engine\. Treating each network output as the probability of the corresponding ground atom, under ProbLog’s independence assumption rule \([3](https://arxiv.org/html/2607.20533#S3.E3)\) fires with probability

P\(𝑏𝑢𝑙𝑙𝑖𝑠ℎvia\([3](https://arxiv.org/html/2607.20533#S3.E3)\)\)=P\(𝑣𝑎𝑙\.=𝑢𝑛𝑑𝑒𝑟\)⋅P\(𝑒𝑎𝑟𝑛\.=𝑠𝑡𝑟𝑜𝑛𝑔\)⋅P\(ℎ𝑒𝑎𝑙𝑡ℎ=ℎ𝑒𝑎𝑙𝑡ℎ𝑦\),P\(\\mathit\{bullish\}\\text\{ via \}\\eqref\{eq:bullish\_comp\}\)=P\(\\mathit\{val\.\}\{=\}\\mathit\{under\}\)\\cdot P\(\\mathit\{earn\.\}\{=\}\\mathit\{strong\}\)\\cdot P\(\\mathit\{health\}\{=\}\\mathit\{healthy\}\),\(5\)
and the total probability of each stance sums such terms over deriving rules, with corrections for overlap[9](https://arxiv.org/html/2607.20533#bib.bib15), yielding the output distribution

\(P​\(𝑏𝑢𝑙𝑙𝑖𝑠ℎ∣x\),P​\(𝑏𝑒𝑎𝑟𝑖𝑠ℎ∣x\),P​\(𝑛𝑒𝑢𝑡𝑟𝑎𝑙∣x\)\)\.\\bigl\(P\(\\mathit\{bullish\}\\mid x\),\\;P\(\\mathit\{bearish\}\\mid x\),\\;P\(\\mathit\{neutral\}\\mid x\)\\bigr\)\.\(6\)
This architecture is*interpretable by construction*: each probability in \([6](https://arxiv.org/html/2607.20533#S3.E6)\) traces back through explicit logical steps to the atomic predicate outputs, so the engine can report which rules contributed and with what weight\. Unlike post\-hoc attention or gradient attribution, the explanation is not a surrogate but the actual computational path—consequential where investment committees require justifiable reasoning and regulation may mandate explainability\.

### 3\.3Neural Predicates as View Generators

The central conceptual claim is that the output \([6](https://arxiv.org/html/2607.20533#S3.E6)\), denoted𝝅i∈Δ2\\boldsymbol\{\\pi\}\_\{i\}\\in\\Delta^\{2\}for companyii, is structurally isomorphic to a Black\-Litterman view triplet\(𝐩k,qk,Ωk\)\(\\mathbf\{p\}\_\{k\},q\_\{k\},\\Omega\_\{k\}\)\. The dominant stance determines the direction \(𝐩k\\mathbf\{p\}\_\{k\}\): bullish a positive absolute view, bearish a negative one, neutral no view\. A mapping from stance probabilities to excess returns, developed in Section[4](https://arxiv.org/html/2607.20533#S4), yields the view return \(qkq\_\{k\}\)\. And the Shannon entropy of the distribution,

H​\(𝝅i\)=−∑sP​\(s∣xi\)​log⁡P​\(s∣xi\),H\(\\boldsymbol\{\\pi\}\_\{i\}\)=\-\\sum\_\{s\}P\(s\\mid x\_\{i\}\)\\log P\(s\\mid x\_\{i\}\),\(7\)
measures its uncertainty—low for a concentrated \(confident\) distribution, high for a diffuse one—and maps directly to the view variance \(Ωk\\Omega\_\{k\}\), a data\-driven alternative to the ad hoc conventions detailed below\. The correspondence reflects a deeper alignment: both approaches represent beliefs as distributions, update them against evidence, and propagate uncertainty explicitly rather than collapsing it to a point\. The former generates the beliefs the latter incorporates\.

### 3\.4The Black\-Litterman Model

The Black\-Litterman model[4](https://arxiv.org/html/2607.20533#bib.bib5),[5](https://arxiv.org/html/2607.20533#bib.bib6)combines the information in market equilibrium with subjective views to form expected return estimates\. Its central innovation is the prior\. Rather than estimating returns directly—which amplifies estimation error and destabilizes weights \(Section[2](https://arxiv.org/html/2607.20533#S2)\)—it begins from market equilibrium\. Under the CAPM[37](https://arxiv.org/html/2607.20533#bib.bib57),[23](https://arxiv.org/html/2607.20533#bib.bib36), the market portfolio with weights𝐰𝑚𝑘𝑡\\mathbf\{w\}\_\{\\mathit\{mkt\}\}and return covariance𝚺\\boldsymbol\{\\Sigma\}is mean\-variance efficient, so the*implied equilibrium excess returns*follow by reverse optimization,

𝚷=δ​𝚺​𝐰𝑚𝑘𝑡,\\boldsymbol\{\\Pi\}=\\delta\\boldsymbol\{\\Sigma\}\\mathbf\{w\}\_\{\\mathit\{mkt\}\},\(8\)
with market risk\-aversionδ=\(𝔼​\[rm\]−rf\)/σm2\\delta=\(\\mathbb\{E\}\[r\_\{m\}\]\-r\_\{f\}\)/\\sigma\_\{m\}^\{2\}\. The model treats𝚷\\boldsymbol\{\\Pi\}as the prior mean over expected returns𝝁\\boldsymbol\{\\mu\}:

𝝁∼𝒩​\(𝚷,τ​𝚺\),\\boldsymbol\{\\mu\}\\sim\\mathcal\{N\}\\bigl\(\\boldsymbol\{\\Pi\},\\,\\tau\\boldsymbol\{\\Sigma\}\\bigr\),\(9\)
whereτ\>0\\tau\>0scales the prior’s uncertainty \(typicallyτ∈\[0\.01,0\.05\]\\tau\\in\[0\.01,0\.05\];[5](https://arxiv.org/html/2607.20533#bib.bib6)\)\. This prior is grounded in market prices rather than finite\-sample noise and, absent views, recommends the well\-diversified market portfolio—a default historical mean\-variance optimization almost never produces[17](https://arxiv.org/html/2607.20533#bib.bib26)\. Departures are driven entirely by views\.

### 3\.5Views and the Posterior

The model supportsKKviews, encoded in the pick matrix𝐏∈ℝK×N\\mathbf\{P\}\\in\\mathbb\{R\}^\{K\\times N\}, the view return vector𝐪∈ℝK\\mathbf\{q\}\\in\\mathbb\{R\}^\{K\}, and the diagonal uncertainty matrix𝛀∈ℝK×K\\boldsymbol\{\\Omega\}\\in\\mathbb\{R\}^\{K\\times K\}\. Rowkkof𝐏\\mathbf\{P\}specifies the portfolio to which viewkkapplies: an*absolute*view has a single entry of11; a*relative*view has positive entries summing to11and negative entries summing to−1\-1\. Eachqkq\_\{k\}gives the expected return on that portfolio, and eachωk\>0\\omega\_\{k\}\>0its variance \(small for high confidence\)\. The views are noisy observations of the prior,

𝐪=𝐏​𝝁\+𝜺,𝜺∼𝒩​\(𝟎,𝛀\),\\mathbf\{q\}=\\mathbf\{P\}\\boldsymbol\{\\mu\}\+\\boldsymbol\{\\varepsilon\},\\qquad\\boldsymbol\{\\varepsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\boldsymbol\{\\Omega\}\),\(10\)
with uncorrelated errors\. Bayes’ theorem then yields a Gaussian posterior with closed\-form mean[17](https://arxiv.org/html/2607.20533#bib.bib26)

𝝁𝐵𝐿=𝚷\+τ​𝚺​𝐏⊤​\(𝐏​τ​𝚺​𝐏⊤\+𝛀\)−1​\(𝐪−𝐏​𝚷\),\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}=\\boldsymbol\{\\Pi\}\+\\tau\\boldsymbol\{\\Sigma\}\\mathbf\{P\}^\{\\top\}\\bigl\(\\mathbf\{P\}\\tau\\boldsymbol\{\\Sigma\}\\mathbf\{P\}^\{\\top\}\+\\boldsymbol\{\\Omega\}\\bigr\)^\{\-1\}\\bigl\(\\mathbf\{q\}\-\\mathbf\{P\}\\boldsymbol\{\\Pi\}\\bigr\),\(11\)
the equilibrium return plus a correction proportional to the*view surprise*𝐪−𝐏​𝚷\\mathbf\{q\}\-\\mathbf\{P\}\\boldsymbol\{\\Pi\}; assets covered by no view retain their equilibrium returns\. The scalarτ\\tausets the prior’s weight against the views and is difficult to calibrate[19](https://arxiv.org/html/2607.20533#bib.bib29); we treat it as given\. With posterior covariance𝐌=\[\(τ​𝚺\)−1\+𝐏⊤​𝛀−1​𝐏\]−1\\mathbf\{M\}=\[\(\\tau\\boldsymbol\{\\Sigma\}\)^\{\-1\}\+\\mathbf\{P\}^\{\\top\}\\boldsymbol\{\\Omega\}^\{\-1\}\\mathbf\{P\}\]^\{\-1\}, the total return covariance𝚺\+𝐌\\boldsymbol\{\\Sigma\}\+\\mathbf\{M\}feeds a mean\-variance optimization

𝐰∗=arg​max𝐰⁡\{𝐰⊤​𝝁𝐵𝐿−δ2​𝐰⊤​\(𝚺\+𝐌\)​𝐰\},\\mathbf\{w\}^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{\\mathbf\{w\}\}\\Bigl\\\{\\mathbf\{w\}^\{\\top\}\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}\-\\tfrac\{\\delta\}\{2\}\\,\\mathbf\{w\}^\{\\top\}\(\\boldsymbol\{\\Sigma\}\+\\mathbf\{M\}\)\\mathbf\{w\}\\Bigr\\\},\(12\)
with unconstrained solution

𝐰∗=1δ​\(𝚺\+𝐌\)−1​𝝁𝐵𝐿\.\\mathbf\{w\}^\{\*\}=\\tfrac\{1\}\{\\delta\}\(\\boldsymbol\{\\Sigma\}\+\\mathbf\{M\}\)^\{\-1\}\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}\.\(13\)
Absent views,𝝁𝐵𝐿=𝚷\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}=\\boldsymbol\{\\Pi\}gives𝐰∗=𝐰𝑚𝑘𝑡\\mathbf\{w\}^\{\*\}=\\mathbf\{w\}\_\{\\mathit\{mkt\}\}, so every deviation is attributable to a specific view; and because𝝁𝐵𝐿\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}is a precision\-weighted average, the equilibrium prior acts as a regularizer, mitigating the error\-amplification pathology of Section[2](https://arxiv.org/html/2607.20533#S2)\.

### 3\.6The View Specification Problem

The model incorporates views rigorously but offers no method for*generating*them or*quantifying*their uncertainty—its principal practical limitation\. Views originate from informal judgment[5](https://arxiv.org/html/2607.20533#bib.bib6),[17](https://arxiv.org/html/2607.20533#bib.bib26): the multi\-dimensional content of an analysis collapses into a scalarqkq\_\{k\}by an undocumented, irreproducible process on which two analysts may disagree\. The uncertainty𝛀\\boldsymbol\{\\Omega\}is more acute still, demanding calibrated beliefs about the reliability of one’s own judgment\. The most widely used approach[19](https://arxiv.org/html/2607.20533#bib.bib29)sets

𝛀=α⋅𝐏​\(τ​𝚺\)​𝐏⊤,\\boldsymbol\{\\Omega\}=\\alpha\\cdot\\mathbf\{P\}\(\\tau\\boldsymbol\{\\Sigma\}\)\\mathbf\{P\}^\{\\top\},\(14\)
with an ad hoc scalarα\\alphathat reflects only the analyst’s willingness to assert a number and imposes the generally unjustified assumption that view uncertainty is proportional to return volatility\. Manual elicitation of a triplet\(𝐩k,qk,ωk\)\(\\mathbf\{p\}\_\{k\},q\_\{k\},\\omega\_\{k\}\)per asset also does not scale, and systematic strategies address scale only by sacrificing view structure \(Section[2](https://arxiv.org/html/2607.20533#S2)\)\. The sophistication of the posterior \([11](https://arxiv.org/html/2607.20533#S3.E11)\) thus rests on inputs of uncertain quality—precisely what Section[4](https://arxiv.org/html/2607.20533#S4)addresses, deriving directions from the dominant stance, magnitudes from a principled stance\-to\-return mapping, and uncertainties from the output distribution’s entropy \([7](https://arxiv.org/html/2607.20533#S3.E7)\), each explicit, reproducible, and grounded in the analysis\.

## 4Incorporating Neural Predicates into Black\-Litterman

### 4\.1Overview of the Proposed Approach

The approach interposes a neural predicate system between raw company\-level data and the Black\-Litterman view interface, proceeding through four stages \(Figure[1](https://arxiv.org/html/2607.20533#S4.F1)\)\.

CompanyAnalytical Data𝐱i\\mathbf\{x\}\_\{i\}NeuralPredicate System𝝅i\\boldsymbol\{\\pi\}\_\{i\}View Matrices𝐏,𝐪,𝛀\\mathbf\{P\},\\,\\mathbf\{q\},\\,\\boldsymbol\{\\Omega\}BL Posterior𝝁𝐵𝐿\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}PortfolioWeights𝐰∗\\mathbf\{w\}^\{\*\}

Figure 1:End\-to\-end architecture of the proposed approach\. Structured analytical data for each asset is processed by a compositional neural predicate system to produce a probability distribution over market stances\. This distribution is mapped to the Black\-Litterman view triplet\(𝐏,𝐪,𝛀\)\(\\mathbf\{P\},\\mathbf\{q\},\\boldsymbol\{\\Omega\}\), which is combined with the equilibrium prior via the Bayesian update of Section[3\.5](https://arxiv.org/html/2607.20533#S3.SS5)to produce the posterior expected return vector𝝁𝐵𝐿\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}, from which optimal portfolio weights are derived\.#### Stage 1: Analytical data\.

For each assetii, a structured profile𝐱i∈𝒳\\mathbf\{x\}\_\{i\}\\in\\mathcal\{X\}encodes the multi\-dimensional outputs of financial analysis—valuation, earnings quality, balance\-sheet diagnostics, and any other relevant structured quantities\. Crucially,𝐱i\\mathbf\{x\}\_\{i\}retains the full structure of the analysis rather than collapsing it to a scalar; translating that structure into a probabilistic judgment is the predicate system’s task\.

#### Stage 2: Neural predicate evaluation\.

The profile is passed through a compositional neural predicate system \(Section[3](https://arxiv.org/html/2607.20533#S3)\), which evaluates a hierarchy of atomic predicates and composes their outputs into a stance distribution

𝝅i=\(P​\(𝑏𝑢𝑙𝑙𝑖𝑠ℎ∣𝐱i\),P​\(𝑏𝑒𝑎𝑟𝑖𝑠ℎ∣𝐱i\),P​\(𝑛𝑒𝑢𝑡𝑟𝑎𝑙∣𝐱i\)\)∈Δ2\.\\boldsymbol\{\\pi\}\_\{i\}=\\bigl\(P\(\\mathit\{bullish\}\\mid\\mathbf\{x\}\_\{i\}\),\\;P\(\\mathit\{bearish\}\\mid\\mathbf\{x\}\_\{i\}\),\\;P\(\\mathit\{neutral\}\\mid\\mathbf\{x\}\_\{i\}\)\\bigr\)\\in\\Delta^\{2\}\.\(15\)
This vector is the sufficient statistic for view generation: all of𝐱i\\mathbf\{x\}\_\{i\}relevant to the update is captured in these three dimensions\.

#### Stage 3: View matrix construction\.

The distribution𝝅i\\boldsymbol\{\\pi\}\_\{i\}is mapped to the view triplet, developed in Subsections[4\.2](https://arxiv.org/html/2607.20533#S4.SS2)–[4\.5](https://arxiv.org/html/2607.20533#S4.SS5), yieldingK≤NK\\leq Nviews encoded in𝐏∈ℝK×N\\mathbf\{P\}\\in\\mathbb\{R\}^\{K\\times N\},𝐪∈ℝK\\mathbf\{q\}\\in\\mathbb\{R\}^\{K\}, and𝛀∈ℝK×K\\boldsymbol\{\\Omega\}\\in\\mathbb\{R\}^\{K\\times K\}\.

#### Stage 4: Bayesian update and portfolio construction\.

The triplet enters the Black\-Litterman update \([11](https://arxiv.org/html/2607.20533#S3.E11)\) to produce𝝁𝐵𝐿\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}, then the mean\-variance optimization \([12](https://arxiv.org/html/2607.20533#S3.E12)\) to produce weights𝐰∗\\mathbf\{w\}^\{\*\}\. This stage is entirely standard; the approach’s contribution lies in Stage 3\.

### 4\.2Mapping Neural Predicate Outputs to View Returns

The view returnqiq\_\{i\}is a scalar expected excess return, which we now derive from𝝅i\\boldsymbol\{\\pi\}\_\{i\}\.

#### Stance\-return correspondence\.

We introduce three stance return parameters,

r𝑏𝑢𝑙𝑙𝑖𝑠ℎ\>0,r𝑏𝑒𝑎𝑟𝑖𝑠ℎ<0,r𝑛𝑒𝑢𝑡𝑟𝑎𝑙≈0,r\_\{\\mathit\{bullish\}\}\>0,\\quad r\_\{\\mathit\{bearish\}\}<0,\\quad r\_\{\\mathit\{neutral\}\}\\approx 0,\(16\)
encoding the investment hypothesis: a bullish assessment implies positive excess returnr𝑏𝑢𝑙𝑙𝑖𝑠ℎr\_\{\\mathit\{bullish\}\}, a bearish one negative return\|r𝑏𝑒𝑎𝑟𝑖𝑠ℎ\|\|r\_\{\\mathit\{bearish\}\}\|, and a neutral one near zero\. The natural choice forr𝑛𝑒𝑢𝑡𝑟𝑎𝑙r\_\{\\mathit\{neutral\}\}is the equilibrium returnΠi\\Pi\_\{i\}, reflecting that a neutral assessment adds nothing beyond what the market has priced; a predicate assigning equal probability to all stances then gravitates toward the prior, and the posterior return sits nearΠi\\Pi\_\{i\}\.

#### The view return mapping\.

We define the view return as the expected stance return under the predicate distribution:

qi=P​\(𝑏𝑢𝑙𝑙𝑖𝑠ℎ∣𝐱i\)⋅r𝑏𝑢𝑙𝑙\+P​\(𝑏𝑒𝑎𝑟𝑖𝑠ℎ∣𝐱i\)⋅r𝑏𝑒𝑎𝑟\+P​\(𝑛𝑒𝑢𝑡𝑟𝑎𝑙∣𝐱i\)⋅Πi\.q\_\{i\}=P\(\\mathit\{bullish\}\\mid\\mathbf\{x\}\_\{i\}\)\\cdot r\_\{\\mathit\{bull\}\}\+P\(\\mathit\{bearish\}\\mid\\mathbf\{x\}\_\{i\}\)\\cdot r\_\{\\mathit\{bear\}\}\+P\(\\mathit\{neutral\}\\mid\\mathbf\{x\}\_\{i\}\)\\cdot\\Pi\_\{i\}\.\(17\)
This mapping is linear in the predicate probabilities, soqiq\_\{i\}varies smoothly as the distribution shifts; it recoversr𝑏𝑢𝑙𝑙𝑖𝑠ℎr\_\{\\mathit\{bullish\}\},r𝑏𝑒𝑎𝑟𝑖𝑠ℎr\_\{\\mathit\{bearish\}\}, orΠi\\Pi\_\{i\}under certainty; and a uniform distribution yieldsqi=\(r𝑏𝑢𝑙𝑙\+r𝑏𝑒𝑎𝑟\+Πi\)/3q\_\{i\}=\(r\_\{\\mathit\{bull\}\}\+r\_\{\\mathit\{bear\}\}\+\\Pi\_\{i\}\)/3, a moderate return the uncertainty mechanism of Subsection[4\.4](https://arxiv.org/html/2607.20533#S4.SS4)will downweight\.

#### Calibration of stance return parameters\.

The parametersr𝑏𝑢𝑙𝑙𝑖𝑠ℎr\_\{\\mathit\{bullish\}\}andr𝑏𝑒𝑎𝑟𝑖𝑠ℎr\_\{\\mathit\{bearish\}\}represent the average excess return of a correct bullish or bearish assessment and can be estimated by regressing realized excess returns on past stance probabilities\. Their estimation is left to the implementation; here we treat them as given\.

#### Thresholding and view inclusion\.

Not every output should generate a view: a near\-uniform distribution carries little directional information and would add overhead and potential numerical instability\. We introduce a*view inclusion threshold*γ∈\(0,13\)\\gamma\\in\(0,\\frac\{1\}\{3\}\), including a view for assetiionly if

maxs⁡P​\(s∣𝐱i\)≥13\+γ,\\max\_\{s\}P\(s\\mid\\mathbf\{x\}\_\{i\}\)\\geq\\frac\{1\}\{3\}\+\\gamma,\(18\)
i\.e\. only if some stance is materially above the uniform baseline\. Largerγ\\gammayields fewer, higher\-confidence views; smallerγ\\gammayields more views at the cost of noisier signals\.

### 4\.3Constructing the Pick Matrix from Predicate Structure

The compositional structure of the predicate system provides a natural basis for𝐏\\mathbf\{P\}, and both absolute and relative views arise from predicate outputs\.

#### Absolute views from single\-asset predicates\.

When a predicate applied to assetiialone satisfies the threshold \([18](https://arxiv.org/html/2607.20533#S4.E18)\), it generates an*absolute view*asserting returnqiq\_\{i\}for that asset, with

\[𝐏\]k​i=1,\[𝐏\]k​j=0​for all​j≠i\.\[\\mathbf\{P\}\]\_\{ki\}=1,\\quad\[\\mathbf\{P\}\]\_\{kj\}=0\\text\{ for all \}j\\neq i\.\(19\)
This is the approach’s most direct use: each company’s profile generates one absolute view\.

#### Relative views from comparative predicates\.

Given assetsiiandjjwith outputs𝝅i\\boldsymbol\{\\pi\}\_\{i\}and𝝅j\\boldsymbol\{\\pi\}\_\{j\}, a*relative view*asserts the differential

qi​j=qi−qj,q\_\{ij\}=q\_\{i\}\-q\_\{j\},\(20\)
with pick vector\[𝐏\]k,i=1\[\\mathbf\{P\}\]\_\{k,i\}=1,\[𝐏\]k,j=−1\[\\mathbf\{P\}\]\_\{k,j\}=\-1, others zero—useful in long\-short or market\-neutral strategies\. More generally, given a bullish setℒ\\mathcal\{L\}and bearish set𝒮\\mathcal\{S\}, a group\-relative view sets\[𝐏\]k,i=1/\|ℒ\|\[\\mathbf\{P\}\]\_\{k,i\}=1/\|\\mathcal\{L\}\|fori∈ℒi\\in\\mathcal\{L\}and\[𝐏\]k,j=−1/\|𝒮\|\[\\mathbf\{P\}\]\_\{k,j\}=\-1/\|\\mathcal\{S\}\|forj∈𝒮j\\in\\mathcal\{S\}, expressing that the average bullish asset outperforms the average bearish one byqℒ−q𝒮q\_\{\\mathcal\{L\}\}\-q\_\{\\mathcal\{S\}\}\.

#### Predicate structure and pick matrix sparsity\.

A naive implementation generates one view per asset, making𝐏\\mathbf\{P\}the identity—a valid special case\. But the compositional structure permits richer, semantically meaningful multi\-asset views: a macro predicate firing on a whole sector can generate a single view on the equal\-weighted sector portfolio, leaving within\-sector allocation to asset\-level predicates\. This hierarchical structure—macro predicates for broad views, micro predicates for idiosyncratic ones—mirrors the multi\-level reasoning institutional investors employ\.

### 4\.4Deriving View Uncertainty from Predicate Confidence

The uncertainty matrix𝛀\\boldsymbol\{\\Omega\}is the component existing practice handles least satisfactorily; as established in Subsection[3\.6](https://arxiv.org/html/2607.20533#S3.SS6), setting it proportional to the prior covariance[19](https://arxiv.org/html/2607.20533#bib.bib29)has no connection to analytical confidence\. This subsection is the paper’s primary theoretical contribution: a principled, data\-driven mapping from the dispersion of the predicate output to the view uncertaintyωi\\omega\_\{i\}\.

#### Entropy as a measure of predicate uncertainty\.

The Shannon entropy of𝝅i\\boldsymbol\{\\pi\}\_\{i\}is

H​\(𝝅i\)=−∑s∈𝒮P​\(s∣𝐱i\)​log⁡P​\(s∣𝐱i\),H\(\\boldsymbol\{\\pi\}\_\{i\}\)=\-\\sum\_\{s\\in\\mathcal\{S\}\}P\(s\\mid\\mathbf\{x\}\_\{i\}\)\\log P\(s\\mid\\mathbf\{x\}\_\{i\}\),\(21\)
with𝒮=\{𝑏𝑢𝑙𝑙𝑖𝑠ℎ,𝑏𝑒𝑎𝑟𝑖𝑠ℎ,𝑛𝑒𝑢𝑡𝑟𝑎𝑙\}\\mathcal\{S\}=\\\{\\mathit\{bullish\},\\mathit\{bearish\},\\mathit\{neutral\}\\\}and natural logarithms\. It is zero at maximum certainty \(probability one on a single stance\) and maximal atlog⁡3≈1\.099\\log 3\\approx 1\.099under the uniform distribution, making it a natural measure of the predicate’s epistemic confidence\.

#### The uncertainty mapping\.

We propose

ωi=ωmin\+\(ωmax−ωmin\)⋅H​\(𝝅i\)log⁡3,\\omega\_\{i\}=\\omega\_\{\\min\}\+\(\\omega\_\{\\max\}\-\\omega\_\{\\min\}\)\\cdot\\frac\{H\(\\boldsymbol\{\\pi\}\_\{i\}\)\}\{\\log 3\},\(22\)
with bounds0<ωmin<ωmax0<\\omega\_\{\\min\}<\\omega\_\{\\max\}\. The normalized entropyH​\(𝝅i\)/log⁡3∈\[0,1\]H\(\\boldsymbol\{\\pi\}\_\{i\}\)/\\log 3\\in\[0,1\]interpolates linearly: a concentrated predicate \(H=0H=0\) givesωi=ωmin\\omega\_\{i\}=\\omega\_\{\\min\}, a diffuse one \(H=log⁡3H=\\log 3\) givesωmax\\omega\_\{\\max\}\. Sinceωi\\omega\_\{i\}is monotone increasing in entropy, the posterior weights confident assessments more and uncertain ones less—exactly the Bayesian principle that evidence quality governs evidence weight\.

#### Relation to the prior covariance\.

To keepωi\\omega\_\{i\}commensurate with the prior covarianceτ​𝚺\\tau\\boldsymbol\{\\Sigma\}, we calibrate the bounds against its diagonal:

ωmin=αmin⋅τ​\[𝚺\]i​i,ωmax=αmax⋅τ​\[𝚺\]i​i,\\omega\_\{\\min\}=\\alpha\_\{\\min\}\\cdot\\tau\[\\boldsymbol\{\\Sigma\}\]\_\{ii\},\\qquad\\omega\_\{\\max\}=\\alpha\_\{\\max\}\\cdot\\tau\[\\boldsymbol\{\\Sigma\}\]\_\{ii\},\(23\)
with0<αmin<αmax0<\\alpha\_\{\\min\}<\\alpha\_\{\\max\}dimensionless\. This preserves the scale\-invariance of the update—more volatile assets get larger view uncertainties, consistent with the intuition that precise views are harder to form on volatile assets\. Substituting into \([22](https://arxiv.org/html/2607.20533#S4.E22)\),

ωi=τ​\[𝚺\]i​i​\[αmin\+\(αmax−αmin\)⋅H​\(𝝅i\)log⁡3\],\\omega\_\{i\}=\\tau\[\\boldsymbol\{\\Sigma\}\]\_\{ii\}\\left\[\\alpha\_\{\\min\}\+\(\\alpha\_\{\\max\}\-\\alpha\_\{\\min\}\)\\cdot\\frac\{H\(\\boldsymbol\{\\pi\}\_\{i\}\)\}\{\\log 3\}\\right\],\(24\)
which nests the Idzorek convention \([14](https://arxiv.org/html/2607.20533#S3.E14)\) as the special caseH​\(𝝅i\)=0H\(\\boldsymbol\{\\pi\}\_\{i\}\)=0withαmin=α\\alpha\_\{\\min\}=\\alpha\. The mapping thus generalizes existing practice by making the confidence parameter a function of the predicate output rather than an analyst\-specified constant\.

#### Alternative dispersion measures\.

Other measures may suit specific contexts\. The*margin*of the dominant stance,

mi=maxs⁡P​\(s∣𝐱i\)−13,m\_\{i\}=\\max\_\{s\}P\(s\\mid\\mathbf\{x\}\_\{i\}\)\-\\frac\{1\}\{3\},\(25\)
may be more interpretable, while the*Gini impurity*1−∑sP​\(s∣𝐱i\)21\-\\sum\_\{s\}P\(s\\mid\\mathbf\{x\}\_\{i\}\)^\{2\}is computationally simpler\. Each induces a different functional form forωi\\omega\_\{i\}but shares the property of mapping concentrated distributions to low uncertainty and diffuse ones to high; the choice is an empirical question left to the implementation\.

### 4\.5Multi\-Predicate Composition and View Aggregation

A richer architecture deploys multiple specialized predicates and aggregates their outputs into one view per asset\.

#### Multiple predicate outputs\.

SupposeLLpredicates, indexed byℓ\\ell, are applied to assetii, each producing𝝅i\(ℓ\)∈Δ2\\boldsymbol\{\\pi\}\_\{i\}^\{\(\\ell\)\}\\in\\Delta^\{2\}and addressing a distinct dimension \(valuation, financial health, and so on\), with individual view returns and uncertainties

qi\(ℓ\)=P\(ℓ\)​\(𝑏𝑢𝑙𝑙𝑖𝑠ℎ∣𝐱i\)​r𝑏𝑢𝑙𝑙𝑖𝑠ℎ\+P\(ℓ\)​\(𝑏𝑒𝑎𝑟𝑖𝑠ℎ∣𝐱i\)​r𝑏𝑒𝑎𝑟𝑖𝑠ℎ\+P\(ℓ\)​\(𝑛𝑒𝑢𝑡𝑟𝑎𝑙∣𝐱i\)​Πi,q\_\{i\}^\{\(\\ell\)\}=P^\{\(\\ell\)\}\(\\mathit\{bullish\}\\mid\\mathbf\{x\}\_\{i\}\)r\_\{\\mathit\{bullish\}\}\+P^\{\(\\ell\)\}\(\\mathit\{bearish\}\\mid\\mathbf\{x\}\_\{i\}\)r\_\{\\mathit\{bearish\}\}\+P^\{\(\\ell\)\}\(\\mathit\{neutral\}\\mid\\mathbf\{x\}\_\{i\}\)\\Pi\_\{i\},\(26\)
ωi\(ℓ\)=τ​\[𝚺\]i​i​\[αmin\+\(αmax−αmin\)⋅H​\(𝝅i\(ℓ\)\)log⁡3\]\.\\omega\_\{i\}^\{\(\\ell\)\}=\\tau\[\\boldsymbol\{\\Sigma\}\]\_\{ii\}\\left\[\\alpha\_\{\\min\}\+\(\\alpha\_\{\\max\}\-\\alpha\_\{\\min\}\)\\cdot\\frac\{H\(\\boldsymbol\{\\pi\}\_\{i\}^\{\(\\ell\)\}\)\}\{\\log 3\}\\right\]\.\(27\)
The problem is to combine\{\(qi\(ℓ\),ωi\(ℓ\)\)\}ℓ=1L\\\{\(q\_\{i\}^\{\(\\ell\)\},\\omega\_\{i\}^\{\(\\ell\)\}\)\\\}\_\{\\ell=1\}^\{L\}into a single\(qi,ωi\)\(q\_\{i\},\\omega\_\{i\}\)\.

#### Precision\-weighted aggregation\.

The natural probabilistic rule weights each view return by its inverse uncertainty:

qi=∑ℓ=1L\(ωi\(ℓ\)\)−1​qi\(ℓ\)∑ℓ=1L\(ωi\(ℓ\)\)−1,q\_\{i\}=\\frac\{\\sum\_\{\\ell=1\}^\{L\}\(\\omega\_\{i\}^\{\(\\ell\)\}\)^\{\-1\}q\_\{i\}^\{\(\\ell\)\}\}\{\\sum\_\{\\ell=1\}^\{L\}\(\\omega\_\{i\}^\{\(\\ell\)\}\)^\{\-1\}\},\(28\)
with aggregate uncertainty, under independent errors,

ωi=\(∑ℓ=1L\(ωi\(ℓ\)\)−1\)−1\.\\omega\_\{i\}=\\left\(\\sum\_\{\\ell=1\}^\{L\}\(\\omega\_\{i\}^\{\(\\ell\)\}\)^\{\-1\}\\right\)^\{\-1\}\.\(29\)
Equation \([29](https://arxiv.org/html/2607.20533#S4.E29)\) is the standard combination of independent Gaussian estimates: concordant predicates yield a more confident aggregate than any single one\. If predicates disagree in direction, their contributions partially cancel in \([28](https://arxiv.org/html/2607.20533#S4.E28)\), pullingqiq\_\{i\}toward neutrality—but \([29](https://arxiv.org/html/2607.20533#S4.E29)\) does not increase under disagreement, since it aggregates precisions regardless of conflict\. We address this next\.

#### Disagreement\-adjusted uncertainty\.

Directional disagreement is itself informative about assessment difficulty and should raise uncertainty\. We augment \([29](https://arxiv.org/html/2607.20533#S4.E29)\) with a penalty:

ωi𝑎𝑑𝑗=ωi⋅\(1\+λ⋅𝕍​\[qi\(ℓ\)\]\),\\omega\_\{i\}^\{\\mathit\{adj\}\}=\\omega\_\{i\}\\cdot\\left\(1\+\\lambda\\cdot\\mathbb\{V\}\\\!\\left\[q\_\{i\}^\{\(\\ell\)\}\\right\]\\right\),\(30\)
where𝕍​\[qi\(ℓ\)\]=1L​∑ℓ\(qi\(ℓ\)−qi\)2\\mathbb\{V\}\[q\_\{i\}^\{\(\\ell\)\}\]=\\frac\{1\}\{L\}\\sum\_\{\\ell\}\(q\_\{i\}^\{\(\\ell\)\}\-q\_\{i\}\)^\{2\}is the empirical variance of the individual view returns andλ≥0\\lambda\\geq 0scales the penalty\. Under agreement \(𝕍=0\\mathbb\{V\}=0\) the adjusted uncertainty equals \([29](https://arxiv.org/html/2607.20533#S4.E29)\); under strong disagreement it inflates, letting the update discount the view accordingly\.

#### Hierarchical predicate composition\.

Alternatively, multi\-predicate composition can be folded into the logical structure itself \(Section[3\.2](https://arxiv.org/html/2607.20533#S3.SS2)\): atomic predicates compose through rules into a derived stance predicate whose distribution𝝅i\\boldsymbol\{\\pi\}\_\{i\}, computed by WMC, feeds \([17](https://arxiv.org/html/2607.20533#S4.E17)\) and \([24](https://arxiv.org/html/2607.20533#S4.E24)\) directly, without a separate aggregation step\. This has the advantage that the composition rules are explicit and interpretable, but the limitation that they must be specified in advance and may miss relevant interactions\. The numerical aggregation of \([28](https://arxiv.org/html/2607.20533#S4.E28)\)–\([30](https://arxiv.org/html/2607.20533#S4.E30)\) is more flexible but less transparent; in practice a hybrid—logical composition for well\-understood relationships, numerical aggregation elsewhere—may work best\.

### 4\.6Theoretical Properties of the Proposed Approach

Consistency, calibration, interpretability, and modularity follow directly from the approach’s structure\. Consistency arises because the same predicates, rules, and stance parameters \([16](https://arxiv.org/html/2607.20533#S4.E16)\) apply to every asset, ensuring that view returns and uncertainties are comparable across the universe\. Calibration is achieved by derivingωi\\omega\_\{i\}from entropy \([21](https://arxiv.org/html/2607.20533#S4.E21)\): uncertain predicates generate largerωi\\omega\_\{i\}and therefore weaker updates, while confident predicates produce smallerωi\\omega\_\{i\}, a property that follows from the mapping \([24](https://arxiv.org/html/2607.20533#S4.E24)\) itself\. Interpretability is inherent rather than post hoc, since every portfolio weight can be traced from𝐱i\\mathbf\{x\}\_\{i\}through atomic predicate outputs, inference rules,\(qi,ωi\)\(q\_\{i\},\\omega\_\{i\}\), and ultimately to𝝁𝐵𝐿\\boldsymbol\{\\mu\}\_\{\\mathit\{BL\}\}and the optimal allocation, unlike approximation\-based explanations such as saliency methods[26](https://arxiv.org/html/2607.20533#bib.bib39)\. Finally, the approach is modular: predicate systems, Black\-Litterman parameters, and stance\-return calibrations can each be updated independently, enabling maintainability, validation, and deployment without retraining the entire system\.

## 5Numerical Example

To illustrate the approach of Section[4](https://arxiv.org/html/2607.20533#S4), we consider a toy implementation on a two\-asset universe using a DeepProbLog inference engine, a large language model as the neural predicate component, and a standard Black\-Litterman optimizer\. The example is purely demonstrative: the firms are hypothetical, the analytical profiles are stipulated, and no claims about predictive performance are made\. Its purpose is simply to show that the proposed mapping is computationally feasible and yields economically interpretable outcomes\. The universe contains two companies,AcmeandGlobex, with parameters reported in Table[1](https://arxiv.org/html/2607.20533#S5.T1)\. The return correlation is fixed atρ=0\.25\\rho=0\.25, the market risk\-aversion coefficient atδ=2\.5\\delta=2\.5, and the stance returns atr𝑏𝑢𝑙𝑙=0\.20r\_\{\\mathit\{bull\}\}=0\.20,r𝑛𝑒𝑢𝑡=0\.00r\_\{\\mathit\{neut\}\}=0\.00, andr𝑏𝑒𝑎𝑟=−0\.15r\_\{\\mathit\{bear\}\}=\-0\.15\. These values are chosen for expository simplicity; their empirical calibration is left to the implementation\.

Table 1:Asset\-level parameters for the illustrative example\. Market capitalization weights are used to derive the equilibrium implied returns𝚷\\boldsymbol\{\\Pi\}via equation \([8](https://arxiv.org/html/2607.20533#S3.E8)\)\.### Neural Predicate Evaluation

Each company’s analytical profile is passed to a neural predicate implemented as a call to GPT\-4o[34](https://arxiv.org/html/2607.20533#bib.bib50)via the OpenAI API, structured according to the LangChain wrapper detailed in the public repository\. The model is prompted with the company’s structured financial profile and constrained, via Pydantic schema validation and structured output enforcement, to return a probability distribution over the three stance categories summing to exactly 100\. Temperature is set to zero to ensure deterministic outputs, consistent with the predicate consistency requirement discussed in Section[3](https://arxiv.org/html/2607.20533#S3)\.

The DeepProbLog inference engine, initialized with theExactEnginefor exact probabilistic inference, compiles the logical program and evaluates six ground queries — one per stance per company — in6\.506\.50seconds of wall\-clock time, of which4\.274\.27seconds are attributable to theAcmeLLM call and2\.212\.21seconds to theGlobexLLM call\. The resulting stance distributions are reported in Table[2](https://arxiv.org/html/2607.20533#S5.T2)\.

Table 2:Neural predicate output distributions𝝅i\\boldsymbol\{\\pi\}\_\{i\}for each company, as returned by the GPT\-4o classifier\. Probabilities sum to 100 by construction\.The distributions exhibit clear directional differentiation: the predicate assigns a dominant bullish stance toAcmeand a dominant bearish stance toGlobex, with moderate residual probability on the remaining categories in each case\. The neutral probability of0\.250\.25is identical for both companies, reflecting a symmetric residual uncertainty in the underlying analytical profiles as interpreted by the language model\.

### View Confidence and Uncertainty

View confidence for each company is computed as one minus the normalized Shannon entropy of the stance distribution,

ci=1−H​\(𝝅i\)log⁡3,c\_\{i\}=1\-\\frac\{H\(\\boldsymbol\{\\pi\}\_\{i\}\)\}\{\\log 3\},\(31\)
whereH​\(𝝅i\)H\(\\boldsymbol\{\\pi\}\_\{i\}\)is defined in \([21](https://arxiv.org/html/2607.20533#S4.E21)\) andlog⁡3\\log 3is the maximum possible entropy over three equiprobable stances\. A confidence ofci=1c\_\{i\}=1corresponds to a degenerate distribution \(certainty about a single stance\); a confidence ofci=0c\_\{i\}=0corresponds to a uniform distribution \(maximal uncertainty\)\. The computed confidences are reported in Table[3](https://arxiv.org/html/2607.20533#S5.T3)\.

Table 3:Black\-Litterman view parameters derived from the neural predicate outputs\. View returnsqiq\_\{i\}are computed via equation \([17](https://arxiv.org/html/2607.20533#S4.E17)\) using the stance return parametersr𝑏𝑢𝑙𝑙=0\.20r\_\{\\mathit\{bull\}\}=0\.20,r𝑛𝑒𝑢𝑡=0\.00r\_\{\\mathit\{neut\}\}=0\.00,r𝑏𝑒𝑎𝑟=−0\.15r\_\{\\mathit\{bear\}\}=\-0\.15\. View confidencecic\_\{i\}is computed via equation \([31](https://arxiv.org/html/2607.20533#S5.E31)\)\. The view uncertaintyωi\\omega\_\{i\}is derived fromcic\_\{i\}via the Idzorek method[19](https://arxiv.org/html/2607.20533#bib.bib29)applied to the normalized entropy\.The view returns are obtained directly from the stance probabilities through \([17](https://arxiv.org/html/2607.20533#S4.E17)\): forAcme,qAcme=0\.60×0\.20\+0\.15×\(−0\.15\)\+0\.25×0\.00=0\.120−0\.023=0\.097q\_\{\\textsc\{Acme\}\}=0\.60\\times 0\.20\+0\.15\\times\(\-0\.15\)\+0\.25\\times 0\.00=0\.120\-0\.023=0\.097\. The confidence values,0\.1470\.147and0\.0630\.063, are modest because neither distribution is strongly concentrated on a single stance, which in turn keeps the Black\-Litterman posterior close to the market\-implied prior\.

### Black\-Litterman Update and Portfolio Allocation

The view triplet\(𝐏,𝐪,𝛀\)\(\\mathbf\{P\},\\mathbf\{q\},\\boldsymbol\{\\Omega\}\)is constructed from the predicate outputs using the mappings of Section[4](https://arxiv.org/html/2607.20533#S4)and passed to the Black\-Litterman update \([11](https://arxiv.org/html/2607.20533#S3.E11)\)\. The scalarτ\\tauis set following the standard convention[5](https://arxiv.org/html/2607.20533#bib.bib6), and the uncertainty matrix𝛀\\boldsymbol\{\\Omega\}is derived from the view confidencecic\_\{i\}via the Idzorek method[19](https://arxiv.org/html/2607.20533#bib.bib29)\. The resulting expected returns and portfolio weights are reported in Table[4](https://arxiv.org/html/2607.20533#S5.T4)\.

Table 4:Expected returns and portfolio weights\. The base portfolio reflects market\-capitalization weights, equivalent to the Black\-Litterman model with no investor views\. The adjusted portfolio reflects the max\-Sharpe tangency portfolio under the Black\-Litterman posterior incorporating the neural predicate views\. The delta reports the absolute difference in allocation between the two portfolios\.The results are economically coherent: the bullish view onAcmeraises its posterior return relative to Globex’s and increases its portfolio weight by3\.743\.74percentage points, while the bearish view onGlobexproduces an equal offsetting reduction under the full\-investment constraint\. The adjustments remain moderate because the view confidences in Table[3](https://arxiv.org/html/2607.20533#S5.T3)are modest\. More concentrated stance distributions would lower entropy, reduceωi\\omega\_\{i\}, and generate stronger portfolio tilts, whereas near\-uniform distributions would yield little or no deviation from the prior\. This graduated response to confidence is a structural feature of the approach rather than a calibrated parameter\.

### Implementation Note

The complete source code for this example — including the DeepProbLog program, the LangChain wrapper for the GPT\-4o neural predicate, and the Black\-Litterman optimizer — is publicly available at[https://github\.com/themarcosf/deepproblog\-black\-litterman\-demo](https://github.com/themarcosf/deepproblog-black-litterman-demo)\. The implementation uses the DeepProbLogExactEnginefor probabilistic inference, which performs exact weighted model counting and is appropriate for programs of this scale\. For larger investment universes with more complex predicate compositions, approximate inference engines may be required; this is one of the scalability considerations left to future implementation work\. Because the framework incorporates generative AI components, individual outputs and downstream portfolio allocations may exhibit non\-deterministic behavior across runs, even under equivalent inputs and configurations\.

## 6Conclusion

The Black\-Litterman model combines market equilibrium with investor judgment but leaves the generation of the view triplet\(𝐏,𝐪,𝛀\)\(\\mathbf\{P\},\\mathbf\{q\},\\boldsymbol\{\\Omega\}\)largely informal and difficult to reproduce\. We address this limitation through neural predicates[27](https://arxiv.org/html/2607.20533#bib.bib40), whose probability distributions over discrete stances naturally map to Black\-Litterman views: stance direction determines𝐏\\mathbf\{P\}, expected stance returns determine𝐪\\mathbf\{q\}, and entropy determines𝛀\\boldsymbol\{\\Omega\}\. The Bayesian update, equilibrium prior, and optimization procedure remain unchanged; the contribution lies entirely in a principled and interpretable method for producing their inputs\. By construction, the approach provides consistency, calibration, interpretability, and modularity\. It is, however, a theoretical proposal: implementation, calibration, and empirical validation are left to future work\. More broadly, the work highlights the potential of neuro\-symbolic AI to integrate modern machine learning with established financial theory in a manner that is scalable, transparent, and theoretically grounded\.

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