Tabular Foundation Models for Discrete Choice Estimation

# 1 Introduction
Source: [https://arxiv.org/html/2607.13314](https://arxiv.org/html/2607.13314)
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Tabular Foundation Models for Discrete Choice Estimation

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Liu Liu, Dan Zhang\\AFFLeeds School of Business, University of Colorado Boulder \\EMAILliu\.liu\-1@colorado\.edu, dan\.zhang@colorado\.edu

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\\ABSTRACT

Tabular foundation models \(TFMs\) generate predictions on structured data via in\-context learning, without task\-specific estimation\. We ask whether TFMs can be effectively applied to discrete choice, a central demand estimation framework in marketing and operations, and find that directly applying TFMs yields limited performance\. The gap is structural: TFMs assume row\-independent observations, whereas discrete choice is inherently set\-valued and subject to persistent consumer preference heterogeneity\. We propose a reformulation that encodes both choice\-set dependence and individual heterogeneity within a row\-based learning framework\. Evaluated on a yogurt scanner panel, individual\-level heterogeneity encoding is the dominant driver of predictive accuracy\. The best reformulation outperforms hierarchical Bayesian estimation by 8% in holdout log\-likelihood and 3\.6% in hit rate, running 16 times faster, a practical advantage for large\-scale demand estimation\. The advantage is largest in the medium\-data regime \(10–40 purchase occasions per consumer\), where parametric Bayesian shrinkage most distorts estimates for atypical consumers\. Fine\-tuning on population choice data provides additional gains for consumers with shallow purchase histories, where in\-context learning has limited individual\-specific signal to condition on\. These results establish a principled approach for applying foundation models to consumer choice problems more broadly\.

Tabular foundation models \(TFMs\) have recently emerged as powerful pretrained architectures for structured, column\-based data\(Hollmannet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib1)\)\. Like large language models \(LLMs\) in text domains, TFMs are trained once on large collections of synthetic prediction tasks and subsequently applied to new datasets via*in\-context learning*: just as an LLM can perform a new task by conditioning on instructions or examples in the prompt at inference time rather than retraining, a TFM produces predictions on a new dataset by treating the labeled training examples as input context, with no task\-specific parameter estimation or gradient updates of any kind\. Because inference consists of a single forward pass through a fixed pretrained network, TFMs are computationally lightweight, and empirical evidence suggests they achieve competitive predictive accuracy across a wide range of tabular benchmarks with minimal tuning\. These properties make them an appealing tool for empirical applications in marketing and operations that involve structured, column\-formatted data\.

Discrete choice models \(DCMs\) constitute a central empirical framework in marketing, economics, and operations management for analyzing demand and consumer decision\-making among competing alternatives\. Under the random utility paradigm, each alternative in a choice set is associated with a latent utility, and observed choices arise from utility comparisons within the set\. Choice probabilities are therefore defined conditional on the composition of the choice set\. Canonical models, including multinomial logit, mixed logit, and latent class specifications, require explicit functional assumptions and are typically estimated via likelihood\-based procedures that may involve simulation and numerical optimization\(Train[2009](https://arxiv.org/html/2607.13314#bib.bib3)\)\.

A central challenge in discrete choice estimation is preference heterogeneity: consumers differ systematically in their tastes, and individual\-level preference parameters are typically not identified from the small number of observations available per person: often ten to twenty tasks in conjoint studies or a few dozen purchase occasions in scanner panel data\. The hierarchical Bayesian \(HB\) approach has become the dominant solution in marketing research and commercial practice\(Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55)\)\. HB models a two\-level structure: individual preference parameters𝜷i\\bm\{\\beta\}\_\{i\}are assumed drawn from a population distribution, and observed choices are generated by utility maximization at the individual level\. Posterior distributions over individual parameters are recovered via Markov chain Monte Carlo \(MCMC\), yielding estimates that are shrunk toward the population mean in proportion to the information available for each consumer\. The result is individual\-level preference estimates that support segmentation, targeting, and downstream optimization\.

These considerations raise a natural question: can TFMs be effectively applied to discrete choice problems, and can they match the predictive performance of HB without any dataset\-specific estimation? The answer is not obvious\. Two structural features of choice data sit in tension with the row\-wise inductive bias of tabular learning, and reconciling them is the central challenge we address\.

A first source of tension is the set\-valued nature of discrete choice\. TFMs are pretrained under a row\-wise supervised learning paradigm in which each observation is treated as an independent input\-output pair\. In contrast, discrete choice is inherently set\-valued: the probability of choosing an alternative is defined conditional on the composition of the choice set and depends on comparisons across alternatives within that set\. The alternatives in a given choice task therefore form a mutually dependent group, linked through substitution patterns and competitive interactions\. Moreover, the identity of the chosen alternative is invariant to the ordering of alternatives in the set, reflecting a permutation invariance at the choice set level that is not imposed by standard tabular representations\. Discrete choice prediction therefore cannot be expressed as a function of an isolated row without explicitly accounting for the relational structure of the choice set\.

A second source of tension is preference heterogeneity across consumers\. In most applications, multiple choice tasks are observed for each decision maker, and these choices are generated from a common underlying utility function\. Observations associated with the same individual are therefore not independent but reflect persistent heterogeneity in tastes and substitution patterns\. This feature contrasts with the pooled prediction setting implicit in row\-wise tabular learning, where all observations are treated as arising from a single homogeneous mapping from features to outcomes\. Taken together, discrete choice data are inherently relational and conditional, violating row independence both within choice sets and across observations belonging to the same individual\.

We find that directly applying TFMs to discrete choice data yields limited predictive performance, not from insufficient model capacity, but from this structural mismatch\. To address it, we propose a structured reformulation of discrete choice prediction for TFMs, in which choice data are represented as tabular prediction tasks that explicitly encode choice\-set dependence and individual heterogeneity\. We show that when these structural properties are appropriately encoded, pretrained TFMs applied purely via in\-context learning match or exceed the predictive performance of HB while being substantially faster to deploy\.

The reformulation has two main components\. First, a choice\-set\-to\-tabular representation that converts set\-valued competition into row\-level supervised inputs through set\-aware and pairwise constructions\. Second, an individual heterogeneity encoding that captures consumer\-specific preferences either through an explicit respondent identifier feature or through per\-respondent task construction for in\-context learning\. We further consider permutation\-based augmentation to approximate invariance with respect to alternative ordering\.

We evaluate these representations on a yogurt scanner panel, a revealed\-preference dataset in which consumers made repeated brand choices under real market conditions\. As the representative TFM, we use TabPFN\(Hollmannet al\.[2023](https://arxiv.org/html/2607.13314#bib.bib5),[2025](https://arxiv.org/html/2607.13314#bib.bib1)\), a pretrained transformer that achieves state\-of\-the\-art accuracy on small\-to\-medium tabular classification tasks while completing inference in seconds\. The analysis compares TabPFN against pooled MNL and hierarchical Bayesian benchmarks across a range of consumer purchase histories, isolating the contributions of choice\-set representation, consumer heterogeneity encoding, and choice data adaptation to predictive performance\. We also evaluate fine\-tuning as a route from the TFM’s learned prior toward a population choice\-aware prior, following the framework developed in Section[3](https://arxiv.org/html/2607.13314#S3)\.

This paper makes three contributions\. First, we identify a structural mismatch between discrete choice problems and TFMs, arising from the relational and conditional nature of choice data and the row\-wise inductive bias of tabular learning; we further develop a unifying shrinkage perspective that connects HB and TFM in\-context learning as instances of the same adaptive pooling principle, differing only in how the preference prior is acquired\. Second, we propose a structured reformulation of discrete choice prediction for TFMs, showing how appropriate data representation and task formulation can encode choice\-set structure and consumer heterogeneity within a row\-wise learning framework\. Third, we provide a systematic empirical evaluation establishing three findings: individual\-level heterogeneity, not choice\-set representation, is the dominant source of predictive gain; appropriately reformulated TFMs match or exceed HB in predictive accuracy at substantially lower computational cost, with the relative advantage greatest in the medium\-data regime; and fine\-tuning on population choice data provides additional gains for consumers with sparse purchase histories, where in\-context learning has limited individual\-specific signal to condition on\. Our analysis does not position TFMs as substitutes for structural DCMs; classical approaches remain essential for interpretation and counterfactual analysis\. Nor does it propose a task\-specific architecture in the spirit of ML\-for\-choice methods, which require estimating parameters on the focal dataset\. Rather, we find that a fixed pretrained model, when the representation pipeline reflects the structural features of choice data, serves as an effective and lightweight predictive alternative to HB\.

The remainder of the paper is organized as follows\. Section[2](https://arxiv.org/html/2607.13314#S2)reviews the related literature\. Section[3](https://arxiv.org/html/2607.13314#S3)introduces the prior\-data fitted network \(PFN\) framework underlying TabPFN, develops the connection to hierarchical Bayesian estimation through the shrinkage formula and a unified methodological progression, and distinguishes in\-context prediction from fine\-tuning\. Section[4](https://arxiv.org/html/2607.13314#S4)describes the two structural features of panel choice data \(set\-valued occasions and persistent consumer heterogeneity\) that motivate the reformulations\. Section[5](https://arxiv.org/html/2607.13314#S5)introduces the proposed reformulation and the associated data representations and task constructions\. Section[6](https://arxiv.org/html/2607.13314#S6)describes the dataset, experimental design, and empirical results\. Section[7](https://arxiv.org/html/2607.13314#S7)concludes\. Appendix[A](https://arxiv.org/html/2607.13314#A1)provides a self\-contained account of the HB framework\.

## 2Related Literature

This paper draws on and contributes to several streams of research\. We build on the discrete choice modeling tradition, including discrete choice and Bayesian methods for consumer heterogeneity, and we contribute to the literatures on machine learning for demand estimation and TFMs\. Across these streams, existing approaches either impose parametric structure \(as in hierarchical Bayes\) or require task\-specific estimation \(as in ML\-for\-choice methods\); no existing work has shown how to reformulate choice data so that a pretrained, estimation\-free model can respect its relational and heterogeneous structure\. We review each stream in turn and position our contribution relative to existing work\.

#### Discrete choice models\.

The discrete choice literature is grounded in the random utility framework, in which observed choices arise from utility maximization within a choice set\(McFadden[1974](https://arxiv.org/html/2607.13314#bib.bib20)\)\. Multinomial logit provides a tractable closed\-form expression for choice probabilities; mixed logit extends this by allowing coefficients to vary continuously across individuals, capturing unobserved preference heterogeneity\.Revelt and Train \([1998](https://arxiv.org/html/2607.13314#bib.bib22)\)demonstrate this framework for panels of repeated consumer choices, andMcFadden and Train \([2000](https://arxiv.org/html/2607.13314#bib.bib21)\)establish that the mixed logit family can approximate any random utility model arbitrarily well\.Train \([2009](https://arxiv.org/html/2607.13314#bib.bib3)\)provides a comprehensive treatment of these models and their simulation\-based estimation\. Latent class specifications offer a discrete alternative to continuous heterogeneity\(Kamakura and Russell[1989](https://arxiv.org/html/2607.13314#bib.bib50)\)\. In industrial organization,Berryet al\.\([1995](https://arxiv.org/html/2607.13314#bib.bib24)\)introduce a random\-coefficients demand system estimated from market\-level data\.Chintagunta and Nair \([2011](https://arxiv.org/html/2607.13314#bib.bib56)\)survey DCM applications in marketing;Berbegliaet al\.\([2022](https://arxiv.org/html/2607.13314#bib.bib2)\)provide a systematic empirical comparison across DCM families in retail settings\.

Conjoint analysis is the principal stated\-preference application of discrete choice, with the methodological tradition originating inGreen and Srinivasan \([1978](https://arxiv.org/html/2607.13314#bib.bib45)\)and formalized within the random utility framework byLouviereet al\.\([2000](https://arxiv.org/html/2607.13314#bib.bib49)\)\. The Bayesian heterogeneity estimation methods that dominate practice \(reviewed below\) were developed largely in the conjoint setting but apply equally to revealed\-preference panel data\. Our work addresses a complementary question applicable across both settings: whether pretrained, estimation\-free models can serve as predictive tools for choice data when representations are designed to reflect the relational and heterogeneous structure of discrete choice\.

#### Preference heterogeneity and Bayesian methods\.

Accounting for consumer heterogeneity is central to both predictive accuracy and managerial decision\-making in marketing\.Lenket al\.\([1996](https://arxiv.org/html/2607.13314#bib.bib52)\)established the hierarchical Bayes framework for conjoint analysis, showing that individual\-level partworth distributions can be recovered from designs too small to support individual estimation by pooling observations through a common population prior\.Rossiet al\.\([1996](https://arxiv.org/html/2607.13314#bib.bib53)\)demonstrate the value of this approach for scanner panel data, andAllenby and Rossi \([1998](https://arxiv.org/html/2607.13314#bib.bib54)\)develop and compare hierarchical Bayes and finite mixture specifications, providing the statistical framework that remains the benchmark for individual\-level preference recovery\.Rossiet al\.\([2005](https://arxiv.org/html/2607.13314#bib.bib55)\)codify this approach in a comprehensive treatment of Bayesian methods for marketing DCMs\.Dubéet al\.\([2010](https://arxiv.org/html/2607.13314#bib.bib57)\)use flexible heterogeneity models to disentangle structural state dependence from spurious persistence driven by preference variation\.Evgeniouet al\.\([2007](https://arxiv.org/html/2607.13314#bib.bib58)\)establish that the HB posterior mean is not uniquely Bayesian: it coincides with penalized regression toward a population mean, with the population prior precision playing the role of a regularization strength, placing HB within the broader class of shrinkage estimators\. Section[3](https://arxiv.org/html/2607.13314#S3)develops the connection between HB and TFMs\. Our heterogeneity encoding approaches, whether through a respondent identifier feature or per\-respondent in\-context learning, pursue the same goal as HB \(recovering individual\-level preference structure from observed choices\) but without parametric assumptions or iterative estimation\.

#### Demand estimation in operations management\.

DCMs are foundational to operations management, where choice probabilities underlie assortment selection, pricing, and revenue management\.Gallego and Wang \([2014](https://arxiv.org/html/2607.13314#bib.bib79)\)characterize optimal multi\-product pricing under the nested logit model, andFariaset al\.\([2013](https://arxiv.org/html/2607.13314#bib.bib63)\)show that nonparametric choice models estimated from transaction data yield substantial revenue improvements over parametric benchmarks in retail\. Estimation from operational data introduces additional challenges: sales records are censored by stockouts and assortment decisions, and per\-consumer purchase histories are short, with scanner panel data typically offering a few dozen observations per individual\. In this sparse\-data regime, parametric Bayesian shrinkage can distort estimates for atypical consumers as the population prior dominates the likelihood;Berbegliaet al\.\([2022](https://arxiv.org/html/2607.13314#bib.bib2)\)document that DCM selection has material consequences for operational decisions in retail\. Recent evidence suggests that the foundation model paradigm has the potential to ease these estimation burdens in real operational settings:Suiet al\.\([2026](https://arxiv.org/html/2607.13314#bib.bib62)\)evaluate time\-series foundation models against an incumbent deep\-learning production forecasting system at Alibaba and find that fine\-tuned Chronos\-2 improves demand forecasting accuracy by 3\.5–5\.4% across product categories, with consistent gains in both zero\-shot and fine\-tuned regimes\. Our results contribute a parallel finding on the choice modeling side: TFMs offer an effective alternative to HB, with the largest predictive advantage at 10–40 observations per respondent, precisely the depth typical of scanner panels and revenue management applications\.

#### Machine learning methods for discrete choice\.

A growing literature examines the intersection of machine learning and discrete choice modeling\.Athey and Imbens \([2019](https://arxiv.org/html/2607.13314#bib.bib67)\)survey prediction\-focused ML methods and their adaptation to economic settings, andvan Cranenburghet al\.\([2022](https://arxiv.org/html/2607.13314#bib.bib29)\)articulate the complementarity between predictive flexibility and structural interpretability\. One strand of this literature develops hybrid architectures that embed neural networks within utility\-theoretic structure:Sifringeret al\.\([2020](https://arxiv.org/html/2607.13314#bib.bib25)\)augment the multinomial logit utility function with a learned representation component, andWanget al\.\([2020](https://arxiv.org/html/2607.13314#bib.bib27)\)demonstrate that deep neural networks can recover economically meaningful quantities such as elasticities from choice data\.Aouad and Désir \([2025](https://arxiv.org/html/2607.13314#bib.bib26)\)establish a theoretical connection between neural networks and random utility models, proving that any RUM can be approximated arbitrarily well by a suitably constructed architecture\. A second strand forgoes structural constraints entirely:Singhet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib28)\)exploit the permutation\-invariant structure of choice probabilities to construct flexible nonparametric demand estimators, andGabel and Timoshenko \([2022](https://arxiv.org/html/2607.13314#bib.bib66)\)develop a scalable deep learning model for product choice with large assortments\.Compianiet al\.\([2025](https://arxiv.org/html/2607.13314#bib.bib80)\)incorporate embeddings from pretrained deep learning models into a mixed logit system, demonstrating that unstructured product representations can improve counterfactual demand predictions\. Transformer\-based architectures have also been adapted directly for choice modeling:Wanget al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib33)\)develop a transformer that treats assortment and purchase history as context, capturing cross\-item interactions without a prespecified utility structure\. Our work is distinguished from all of these in that the underlying model is held fixed at pretraining and applied purely via in\-context learning, with no task\-specific parameters estimated\. Rather than modifying the model, we focus on how data representation and task formulation enable a pretrained row\-wise architecture to capture the relational and heterogeneous structure of choice data\.

#### Large language models for preference elicitation\.

A nascent literature explores whether LLMs can serve as synthetic consumers\.Hortonet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib70)\)proposes treating LLMs as implicit computational models of human agents that can be queried about preferences\.Chenet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib71)\)demonstrate that GPT satisfies the generalized axiom of revealed preference at higher rates than human subjects\.Brandet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib69)\)find that querying LLMs in a conjoint\-like framework can recover willingness\-to\-pay estimates that are sometimes comparable to those from human surveys, though often inaccurate and occasionally wrong\-signed, andWanget al\.\([2024](https://arxiv.org/html/2607.13314#bib.bib75)\)propose combining small human conjoint samples with LLM\-generated data to reduce estimation error\.

Despite this progress, important methodological challenges remain\.Gui and Toubia \([2023](https://arxiv.org/html/2607.13314#bib.bib73)\)identify a fundamental identification problem: when LLMs are blinded to experimental design, treatment variations confound other variables, producing implausible demand estimates\.Ludwiget al\.\([2025](https://arxiv.org/html/2607.13314#bib.bib74)\)provide a formal econometric framework showing that prompt and model choices can yield dramatically different structural parameter estimates without a validation sample\. Our work is complementary but distinct: rather than using language models as synthetic consumers, we apply TFMs to actual consumer choice data\. This sidesteps the identification and calibration challenges that arise when LLMs substitute for human respondents, and focuses on the structural reformulation of real choice data as tabular prediction tasks\. Critically, we use in\-context learning in its technical sense \(providing labeled choice observations as context from which the foundation model predicts held\-out choices\), rather than eliciting preferences through natural language prompting\.

#### Tabular foundation models\.

The tabular machine learning literature has long been dominated by gradient\-boosted decision trees \(Chen and Guestrin \([2016](https://arxiv.org/html/2607.13314#bib.bib14)\);Keet al\.\([2017](https://arxiv.org/html/2607.13314#bib.bib15)\);Prokhorenkovaet al\.\([2018](https://arxiv.org/html/2607.13314#bib.bib16)\)\), which consistently achieve strong performance across benchmarks\(Grinsztajnet al\.[2022](https://arxiv.org/html/2607.13314#bib.bib17), Shwartz\-Ziv and Armon[2022](https://arxiv.org/html/2607.13314#bib.bib19), Ericksonet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib36)\)\. Deep learning approaches have been studied extensively\(Gorishniyet al\.[2021](https://arxiv.org/html/2607.13314#bib.bib18)\)but have not consistently surpassed tree\-based methods\. TFMs represent a qualitatively different approach: pretrained offline on large collections of synthetic tasks and deployed via in\-context learning without task\-specific parameter updates\.

The theoretical foundation is the prior\-data fitted network \(PFN\) framework ofMülleret al\.\([2022](https://arxiv.org/html/2607.13314#bib.bib6)\), which trains a transformer to approximate posterior predictive distributions by sampling data from a prior and learning to predict held\-out observations in context;Nagler \([2023](https://arxiv.org/html/2607.13314#bib.bib8)\)establishes its statistical consistency properties\.Hollmannet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib5)\)applies this framework to tabular classification, demonstrating that a single forward pass can match or exceed tuned AutoML pipelines on small classification benchmarks\.Hollmannet al\.\([2025](https://arxiv.org/html/2607.13314#bib.bib1)\)substantially extends this to datasets with up to 10,000 rows and 500 features, achieving state\-of\-the\-art accuracy across hundreds of benchmarks while completing inference in seconds\. Subsequent work has scaled this approach to larger datasets\(Prior Labs Team[2025](https://arxiv.org/html/2607.13314#bib.bib32), Quet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib7)\)and addressed the synthetic\-to\-real distribution gap\(Maet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib35)\)\. The ability of transformers to perform in\-context learning was established in the language domain byBrownet al\.\([2020](https://arxiv.org/html/2607.13314#bib.bib9)\);Akyüreket al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib11)\)andvon Oswaldet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib12)\)provide mechanistic accounts of the algorithms that self\-attention implements in context\.

A key limitation of standard TFMs, including those above, is the assumption of row independence: each observation is treated as an independent input–output pair during inference\.Cucumides and Geerts \([2026](https://arxiv.org/html/2607.13314#bib.bib30)\)formalize this limitation, proving that targets driven by relational patterns across rows are inaccessible to any row\-local predictor regardless of model capacity\. Discrete choice data are a specific and economically motivated instance of this failure: choice probabilities depend on comparisons across alternatives within a set, making them inaccessible to row\-local models without appropriate reformulation\. Broader relational deep learning methods address inter\-row structure through graph neural networks over multi\-table databases\(Feyet al\.[2024](https://arxiv.org/html/2607.13314#bib.bib37), Robinsonet al\.[2024](https://arxiv.org/html/2607.13314#bib.bib38)\), andEremeevet al\.\([2025](https://arxiv.org/html/2607.13314#bib.bib39)\)show that pretrained TFMs can be repurposed as graph foundation models by treating node neighborhoods as tabular context\. Our approach addresses the relational structure of discrete choice data not through graph construction but through structured data representations that encode choice\-set dependencies and individual heterogeneity within the existing row\-wise TFM framework\.

## 3Tabular Foundation Models and Their Connection to Hierarchical Bayes

Hierarchical Bayesian DCMs, the dominant approach for individual\-level preference estimation in marketing, address sparse per\-consumer data through a specific mechanism: shrink individual estimates toward a population mean, with the degree of pooling governed by how many choices each consumer has revealed\. This section introduces TFMs and develops their connection to HB\. The connection is more than analogy: both approaches solve the same sparse\-data problem through the same underlying principle \(shrinkage toward a prior\), differing only in how that prior is specified\. Understanding this connection clarifies what TFMs add to the researcher’s toolkit and what they require in exchange\.

Section[3\.1](https://arxiv.org/html/2607.13314#S3.SS1)introduces in\-context learning and the prior\-data fitted network \(PFN\) framework that underlies TabPFN\. Section[3\.2](https://arxiv.org/html/2607.13314#S3.SS2)develops the HB\-TFM connection formally, presenting the HB shrinkage formula and positioning both approaches within the broader methodological progression\. Section[3\.3](https://arxiv.org/html/2607.13314#S3.SS3)distinguishes in\-context prediction from fine\-tuning and explains the role of each in the empirical analysis\.

### 3\.1In\-Context Learning and the PFN Framework

#### The in\-context learning paradigm\.

The standard supervised\-learning workflow trains a separate model on each new dataset: specify a model family, optimize parameters on labeled observations, tune hyperparameters via cross\-validation, and evaluate on held\-out data\. This procedure is repeated from scratch whenever the dataset or prediction target changes\.*In\-context learning*\(ICL\) replaces per\-dataset parameter fitting with a conditioning operation\. A single networkqϕq\_\{\\phi\}is pretrained once across a large collection of tasks\. At prediction time,qϕq\_\{\\phi\}receives the labeled training set𝒟=\{\(xi,yi\)\}i=1n\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}as part of its input—the “context”—and produces a prediction for a new pointx∗x\_\{\*\}in a single forward pass, with the weightsϕ\\phiheld fixed:

y^∗=qϕ​\(x∗,𝒟\)\.\\hat\{y\}\_\{\*\}=q\_\{\\phi\}\(x\_\{\*\},\\mathcal\{D\}\)\.\(1\)The analogy to large language models is instructive: when a user prompts ChatGPT with several input\-output examples followed by a new input, the model produces a prediction without any weight update\(Brownet al\.[2020](https://arxiv.org/html/2607.13314#bib.bib9)\)\. TabPFN performs the same operation on rows of a table—the labeled training rows are the context, and the test row is the new input—exploiting the same underlying mechanism, self\-attention over context tokens, adapted to tabular data\.

#### The prior\-data fitted network framework\.

Mülleret al\.\([2022](https://arxiv.org/html/2607.13314#bib.bib6)\)provide a Bayesian interpretation of in\-context learning through the prior\-data fitted network \(PFN\) framework\. Given a training set𝒟\\mathcal\{D\}and a model family parameterized byθ\\theta, the Bayesian posterior predictive distribution for a test outcomey∗y\_\{\*\}at inputx∗x\_\{\*\}is

p​\(y∗∣x∗,𝒟\)=∫p​\(y∗∣x∗,θ\)​p​\(θ∣𝒟\)​𝑑θ\.p\(y\_\{\*\}\\mid x\_\{\*\},\\mathcal\{D\}\)=\\int p\(y\_\{\*\}\\mid x\_\{\*\},\\theta\)\\,p\(\\theta\\mid\\mathcal\{D\}\)\\,d\\theta\.\(2\)Equation \([2](https://arxiv.org/html/2607.13314#S3.E2)\) averages the likelihood over all parameter values weighted by their posterior given the training data, simultaneously accounting for model fit and parameter uncertainty\. For most model families this integral is analytically intractable and requires MCMC, which is expensive to run anew for each new dataset\.

The PFN insight is to amortize this computation\. Rather than evaluating \([2](https://arxiv.org/html/2607.13314#S3.E2)\) at prediction time, the networkqϕq\_\{\\phi\}is trained once to approximate it\. Pretraining proceeds by repeatedly \(i\) sampling a complete synthetic dataset from a priorp​\(𝒟\)p\(\\mathcal\{D\}\)over data\-generating processes, \(ii\) splitting it into a context𝒟\\mathcal\{D\}and held\-out test pairs\{\(x∗\(k\),y∗\(k\)\)\}\\\{\(x\_\{\*\}^\{\(k\)\},y\_\{\*\}^\{\(k\)\}\)\\\}, and \(iii\) updatingϕ\\phito minimize the negative log\-predictive loss on the held\-out pairs given the context\. After pretraining,qϕ​\(y∗∣x∗,𝒟\)≈p​\(y∗∣x∗,𝒟\)q\_\{\\phi\}\(y\_\{\*\}\\mid x\_\{\*\},\\mathcal\{D\}\)\\approx p\(y\_\{\*\}\\mid x\_\{\*\},\\mathcal\{D\}\)for any dataset drawn from the prior, in a single forward pass\. The weightsϕ\\phiparameterize a learned prediction algorithm; they do not represent an explicit posterior overθ\\theta, but rather a general prediction procedure trained to behave like a Bayesian predictor across the full distribution of pretraining tasks\.

#### The pretraining prior\.

The priorp​\(𝒟\)p\(\\mathcal\{D\}\)determines which prediction tasks the network handles well\. TabPFN’s prior combines two families of synthetic data\-generating processes: structural causal models, which generate datasets with realistic correlation structure and causal directionality, and Bayesian neural networks with randomly drawn weights, which produce a rich family of smooth and nonsmooth response surfaces\. Simpler data\-generating processes are up\-weighted, encoding an Occam’s razor preference for regular relationships\. We refer to the statistical regularities thatϕ\\phiencodes through pretraining on this prior as the TFM’s*learned prior*: the general prediction heuristics internalized from millions of synthetic tasks\. Before fine\-tuning, this learned prior is general\-purpose—not estimated from the focal choice panel, not calibrated to the target product category, and not adapted to the specific data under study\.

#### Statistical properties\.

Hollmannet al\.\([2023](https://arxiv.org/html/2607.13314#bib.bib5)\)demonstrated that this framework is practically viable, showing that a single forward pass matches or exceeds tuned AutoML pipelines on small classification benchmarks\.Hollmannet al\.\([2025](https://arxiv.org/html/2607.13314#bib.bib1)\)scaled the approach to datasets with up to 10,000 training rows and 500 features, achieving state\-of\-the\-art accuracy while completing inference in seconds\.Nagler \([2023](https://arxiv.org/html/2607.13314#bib.bib8)\)provides the first frequentist analysis of these predictors, and two properties bear on our argument\. The prediction variance declines as the context sizenngrows, at a rate that depends on how strongly the network responds to any single context example rather than at a universal1/n1/nrate\. Low variance does not, however, guarantee accuracy: the predictor is consistent only when the pretraining priorp​\(𝒟\)p\(\\mathcal\{D\}\)places sufficient mass on the true data\-generating process, a condition that the architecture controlling variance does not itself ensure\. A model pretrained on generic synthetic data may be biased on choice data whose structure lies outside the pretraining distribution; this observation motivates the reformulation developed in Sections[4](https://arxiv.org/html/2607.13314#S4)and[5](https://arxiv.org/html/2607.13314#S5)\.

### 3\.2HB and TFMs: A Unified Shrinkage View

#### The HB model\.

The HB approach resolves the sparse\-data problem by treating individual preference parameters as draws from a common population distribution\(Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55), Allenby and Rossi[1998](https://arxiv.org/html/2607.13314#bib.bib54)\)\. Let𝜷i∈ℝK\\bm\{\\beta\}\_\{i\}\\in\\mathbb\{R\}^\{K\}denote consumerii’s vector of taste parameters\. At the population level,𝜷i∼𝒩​\(𝝁,𝚺\)\\bm\{\\beta\}\_\{i\}\\sim\\mathcal\{N\}\(\\bm\{\\mu\},\\bm\{\\Sigma\}\), where𝝁∈ℝK\\bm\{\\mu\}\\in\\mathbb\{R\}^\{K\}is the population mean taste vector and𝚺\\bm\{\\Sigma\}is the population covariance matrix, both estimated jointly from the full panel\. At the individual level, consumerii’s choices are governed by a likelihoodp​\(yi​t∣𝐱i​t,𝜷i\)p\(y\_\{it\}\\mid\\mathbf\{x\}\_\{it\},\\bm\{\\beta\}\_\{i\}\), where𝐱i​t\\mathbf\{x\}\_\{it\}collects observed alternative attributes on choice occasiontt\. Appendix[A](https://arxiv.org/html/2607.13314#A1)develops the full MNL specification used in the empirical analysis\.

#### The HB shrinkage formula\.

The key shrinkage property is most transparent under the conjugate specification\. Let𝐗i\\mathbf\{X\}\_\{i\}denote theTi×KT\_\{i\}\\times Kmatrix stacking consumerii’s attribute vectors acrossTiT\_\{i\}choice occasions\. Under the Gaussian individual likelihoodyi​t=𝐱i​t′​𝜷i\+εi​ty\_\{it\}=\\mathbf\{x\}\_\{it\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\+\\varepsilon\_\{it\}withεi​t∼iid𝒩​\(0,σ2\)\\varepsilon\_\{it\}\\stackrel\{\{\\scriptstyle\\mathrm\{iid\}\}\}\{\{\\sim\}\}\\mathcal\{N\}\(0,\\sigma^\{2\}\), the model is conjugate and the posterior mean of𝜷i\\bm\{\\beta\}\_\{i\}has the closed form\(Lindley and Smith[1972](https://arxiv.org/html/2607.13314#bib.bib51), Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55)\):

𝜷¯i=\(𝐈−𝐁i\)​𝜷^iMLE\+𝐁i​𝝁,\\bar\{\\bm\{\\beta\}\}\_\{i\}\\;=\\;\(\\mathbf\{I\}\-\\mathbf\{B\}\_\{i\}\)\\,\\hat\{\\bm\{\\beta\}\}\_\{i\}^\{\\,\\mathrm\{MLE\}\}\\;\+\\;\\mathbf\{B\}\_\{i\}\\,\\bm\{\\mu\},\(3\)where𝐁i=\(𝐈\+𝚺​𝐗i′​𝐗i/σ2\)−1\\mathbf\{B\}\_\{i\}=\\bigl\(\\mathbf\{I\}\+\\bm\{\\Sigma\}\\mathbf\{X\}\_\{i\}^\{\\prime\}\\mathbf\{X\}\_\{i\}/\\sigma^\{2\}\\bigr\)^\{\-1\}is the shrinkage matrix and𝜷^iMLE\\hat\{\\bm\{\\beta\}\}\_\{i\}^\{\\,\\mathrm\{MLE\}\}is consumerii’s maximum likelihood estimate computed fromii’s data alone\. For scanner panel data and choice\-based conjoint, the logit likelihood replaces the Gaussian, breaking conjugacy; equation \([3](https://arxiv.org/html/2607.13314#S3.E3)\) holds approximately, and the qualitative behavior is the same\. The posterior mean is a matrix\-weighted average of the consumer’s own data and the population mean𝝁\\bm\{\\mu\}, governed by the shrinkage matrix𝐁i\\mathbf\{B\}\_\{i\}\. When consumeriihas many observations,𝐗i′​𝐗i\\mathbf\{X\}\_\{i\}^\{\\prime\}\\mathbf\{X\}\_\{i\}is large,𝐁i→𝟎\\mathbf\{B\}\_\{i\}\\to\\mathbf\{0\}, and𝜷¯i\\bar\{\\bm\{\\beta\}\}\_\{i\}approaches the individual MLE\. When observations are sparse,𝐁i→𝐈\\mathbf\{B\}\_\{i\}\\to\\mathbf\{I\}and𝜷¯i\\bar\{\\bm\{\\beta\}\}\_\{i\}collapses to𝝁\\bm\{\\mu\}\. This adaptive pooling is the mechanism that makes HB produce stable individual\-level estimates even from short purchase histories\.

#### Shrinkage and regularization\.

Evgeniouet al\.\([2007](https://arxiv.org/html/2607.13314#bib.bib58)\)show that the HB posterior mean is not uniquely Bayesian: the shrinkage formula \([3](https://arxiv.org/html/2607.13314#S3.E3)\) is equivalent to penalized regression toward a population mean, with𝚺−1\\bm\{\\Sigma\}^\{\-1\}playing the role of a regularization strength\. HB is therefore an instance of a broader principle: shrinkage toward a shared representation, which also underlies the methods reviewed below\.

#### The unified progression\.

Every method for individual\-level preference estimation from sparse panel data is a shrinkage estimator: it pools information across consumers toward a shared representation, with the degree of pooling governed by the available data\. Methods differ in how that shared prior is specified\. Table[1](https://arxiv.org/html/2607.13314#S3.T1)organizes this progression\.

Table 1:Methods for individual\-level preference recovery as shrinkage estimators, organized by how the shared prior is specified\.TFMs occupy the rightmost position in this progression\. In HB, the prior𝒩​\(𝝁,𝚺\)\\mathcal\{N\}\(\\bm\{\\mu\},\\bm\{\\Sigma\}\)is a parametric family specified by the analyst and estimated from the focal sample\. In a TFM, the prior is encoded in the pretrained weightsϕ\\phi, learned from millions of synthetic datasets outside the focal sample\. At prediction time, each consumer’s in\-context examples condition the pretrained prior on that consumer’s observed choices: the functional analog of the HB posterior update in \([3](https://arxiv.org/html/2607.13314#S3.E3)\), but computed in a single forward pass without MCMC\.

Three concrete differences follow\. First, computational speed: a single forward pass replaces MCMC, reducing inference from minutes to seconds\. Second, distributional flexibility: the learned prior is not restricted to the multivariate normal assumed by standard HB and can encode irregular distributions, nonlinear feature interactions, and response patterns that the normal prior would smooth over\. Third, cold\-start performance: because the prior is estimated from synthetic tasks rather than the focal sample, a TFM can produce calibrated predictions even from a single choice observation per consumer\.

In exchange, the learned prior is not directly interpretable as a distribution over preferences: the practitioner cannot inspect𝝁\\bm\{\\mu\}and𝚺\\bm\{\\Sigma\}or assess the direction of shrinkage for a given consumer\. There are no posterior draws, so formal uncertainty quantification requires different tools than the standard HB output\. And the learned prior may be biased if the true data\-generating process falls outside the pretraining distribution, the risk thatNagler \([2023](https://arxiv.org/html/2607.13314#bib.bib8)\)formalizes and that the reformulations in Sections[4](https://arxiv.org/html/2607.13314#S4)and[5](https://arxiv.org/html/2607.13314#S5)are designed to address\.

### 3\.3In\-Context Prediction and Fine\-Tuning

Given the learned prior encoded in the pretrained weights, there are two ways to apply a TFM to a specific choice panel: in\-context prediction, which keeps the weights fixed, and fine\-tuning, which adapts those weights before prediction\.

In\-context prediction keeps the pretrained weightsϕ\\phifixed\. Given a context𝒟\\mathcal\{D\}and test covariatesx∗x\_\{\*\}, prediction is formed asqϕ​\(x∗,𝒟\)q\_\{\\phi\}\(x\_\{\*\},\\mathcal\{D\}\)\. No gradient updates are performed on the focal dataset\. Adaptation occurs only through the examples supplied in context and their representation\. For panel choice data, population information can therefore enter only through the organization of the context: by pooling consumers in a shared table, including consumer identity as a feature, or constructing separate per\-consumer contexts\.

Fine\-tuning changes the model weights before in\-context prediction\. Starting from the pretrained weightsϕ\\phi, the model is further trained on pooled choice observations from the focal population:ϕ→fine\-tune on​𝒟popϕchoice\\phi\\xrightarrow\{\\text\{fine\-tune on \}\\mathcal\{D\}\_\{\\mathrm\{pop\}\}\}\\phi\_\{\\mathrm\{choice\}\}\. Operationally, this is gradient\-based adaptation of neural network weights\. It is not Bayesian updating: the procedure does not analytically condition a prior or produce posterior draws over preference parameters\. Conceptually, it shifts the learned prior from the off\-the\-shelf model, which encodes general tabular regularities, toward one adapted to the empirical regularities of the focal choice population\. After fine\-tuning, predictions use the adapted weightsqϕchoice​\(x∗,𝒟i\)q\_\{\\phi\_\{\\mathrm\{choice\}\}\}\(x\_\{\*\},\\mathcal\{D\}\_\{i\}\), where𝒟i\\mathcal\{D\}\_\{i\}denotes the consumer\-specific calibration context supplied at prediction time\. Fine\-tuning does not replace in\-context prediction; it changes the pretrained mapping through which the context is interpreted\.

## 4From Row\-Level Prediction to Panel Choice Data

Section[3](https://arxiv.org/html/2607.13314#S3)described TabPFN as a pretrained model for row\-and\-column supervised prediction, and developed its connection to hierarchical Bayesian estimation through the shrinkage formula and unified methodological progression\. Panel choice data have additional structure\. A purchase occasion is not a single labeled row, but a choice among alternatives in a set; and repeated choices by the same consumer are linked by persistent preferences\. This section examines these two features and, in Section[4\.3](https://arxiv.org/html/2607.13314#S4.SS3), draws out their implications for applying a learned prior to panel choice prediction, motivating the reformulations in Section[5](https://arxiv.org/html/2607.13314#S5)\. The goal is to identify what information must be made visible to the model\.

### 4\.1Set\-Valued Choice Occasions

DCMs describe consumer decisions through random utility maximization: a consumer selects the alternative yielding the greatest utility, decomposed into a component explained by observed attributes and an unobserved stochastic term\. Consumeriiderives utility

ui​j​t=𝐱i​j​t′​𝜷i\+εi​j​tu\_\{ijt\}=\\mathbf\{x\}\_\{ijt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\+\\varepsilon\_\{ijt\}\(4\)from alternativejjin choice setStS\_\{t\}on occasiontt, where𝐱i​j​t\\mathbf\{x\}\_\{ijt\}are observed alternative attributes,𝜷i\\bm\{\\beta\}\_\{i\}is a consumer\-specific vector of taste parameters, andεi​j​t\\varepsilon\_\{ijt\}is an idiosyncratic random shock\. Under i\.i\.d\. Type I extreme\-value errors, the probability that consumeriichooses alternativejjfrom choice setStS\_\{t\}on occasiontttakes the multinomial logit form

Pi​j​t=exp⁡\(𝐱i​j​t′​𝜷i\)∑k∈Stexp⁡\(𝐱i​k​t′​𝜷i\)\.P\_\{ijt\}=\\frac\{\\exp\(\\mathbf\{x\}\_\{ijt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\)\}\{\\sum\_\{k\\in S\_\{t\}\}\\exp\(\\mathbf\{x\}\_\{ikt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\)\}\.\(5\)The specification \([5](https://arxiv.org/html/2607.13314#S4.E5)\) has two features\. First, the prediction object is a distribution over a set: the probability assigned to one alternative is defined relative to the other alternatives available on the same occasion\. Second, the alternatives inStS\_\{t\}are unordered\. A model’s prediction should not depend on whether the chosen alternative is listed first, second, or fourth in the data representation\.

These features differ from the native input\-output structure of standard tabular prediction\. In a row\-level supervised task, each observation is a feature vector paired with a label\. In a choice task, the observed outcome is attached to a group of rows: exactly one alternative in the set is chosen and the remaining alternatives are not\. If each alternative is converted into an independent row, the model may see the attributes of the chosen and unchosen alternatives, but the mutually exclusive nature of the choice set is no longer represented explicitly\. The set\-valued structure therefore motivates reformulations that make comparison within a choice occasion more visible, such as wide representations, pairwise comparisons, within\-set rank features, or post\-processing that normalizes predicted probabilities within each set\.

At the same time, set structure is only one part of the problem\. The empirical question is not merely whether a TFM can recognize that alternatives compete within a set, but also whether it can use repeated consumer histories to predict future choices\. This second structure is the focus of the next subsection\.

### 4\.2Repeated Choices and Consumer Heterogeneity

Panel choice data contain repeated observations from the same decision maker\. In the random utility notation above, the same𝜷i\\bm\{\\beta\}\_\{i\}governs all observed choices by consumerii\. Thus, a consumer’s earlier purchases are informative about their later purchases because they reveal persistent features of that consumer’s preferences\. This cross\-observation dependence is the reason that heterogeneity has played such a central role in marketing DCMs: hierarchical Bayesian, finite\-mixture, and latent\-class specifications all aim to recover consumer\-level preference differences while borrowing strength across consumers\(Allenby and Rossi[1998](https://arxiv.org/html/2607.13314#bib.bib54), Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55), Train[2009](https://arxiv.org/html/2607.13314#bib.bib3)\)\.

A TFM does not automatically represent this structure\. TabPFN can attend to all examples supplied in context, but the model must be given a representation that tells it which observations belong to the same consumer, or it must be organized so that a consumer’s own history forms the relevant context for prediction\. Without such information, repeated choices from the same consumer are simply pooled examples in a supervised prediction task\. The model may still learn average relationships between features and choices, but it has no direct way to personalize predictions using the consumer’s own purchase history\.

This distinction maps directly onto the unified shrinkage view developed in Section[3](https://arxiv.org/html/2607.13314#S3)\. In HB\-MNL, population information enters through an estimated distribution over consumer taste parameters, and individual\-level predictions shrink toward that distribution when consumer histories are sparse\. In TabPFN, population and consumer information enter differently\. With fixed pretrained weights, the learned prior is unchanged; the model can use population or consumer structure only to the extent that the context organization exposes it\. For example, including consumer identity as a categorical feature keeps all consumers in a shared context while marking repeated observations from the same consumer\. Constructing one context per consumer instead emphasizes personalization from that consumer’s own history but reduces direct borrowing across consumers within the in\-context task\.

This is why we treat consumer heterogeneity as a central design dimension rather than a minor feature\-engineering choice\. The set\-valued nature of choice occasions motivates alternative representations of a choice set, but the repeated\-observation structure determines whether the model can use the core information that panel data provide: past choices by the same consumer and regularities across consumers\.

### 4\.3Implications for TabPFN Reformulation

The preceding discussion suggests two axes for adapting TabPFN to panel choice data\. The first is the*choice\-representation axis*: how to convert a set\-valued choice occasion into inputs suitable for a row\-and\-column prediction model\. A long representation treats each alternative as a row; a wide representation stores the entire choice set in one row; a pairwise representation converts each observed choice into comparisons between alternatives; and rank or set\-summary features make relative position within the choice set more explicit\.

The second is the*heterogeneity axis*: how to make repeated consumer histories available to the model\. A pooled representation suppresses consumer identity and asks the model to learn average choice patterns\. A respondent\-categorical representation includes consumer identity as a feature, allowing the model to condition on repeated observations from the same consumer within a shared population context\. A per\-respondent in\-context representation treats each consumer as its own prediction task, using that consumer’s calibration choices as the context for its future choices\. Fine\-tuning, studied separately in Section[6\.3](https://arxiv.org/html/2607.13314#S6.SS3), adds another route by updating the pretrained model on pooled choice data before prediction\.

These two axes are conceptually distinct\. Choice representation determines how the model expresses the within\-occasion comparison; heterogeneity encoding determines how the model can use information across occasions and across consumers\. Section[5](https://arxiv.org/html/2607.13314#S5)operationalizes these axes as concrete TabPFN inputs\. Section[6](https://arxiv.org/html/2607.13314#S6)then evaluates them empirically, distinguishing gains that come from representing the choice set differently from gains that come from making consumer heterogeneity visible to the model\.

## 5Reformulating Panel Choice Data for TabPFN

Section[4](https://arxiv.org/html/2607.13314#S4)identified two design axes for adapting TabPFN to panel choice data\. The first is how to represent a set\-valued choice occasion as tabular inputs\. The second is how to expose consumer heterogeneity to the model\. This section describes the concrete reformulations we evaluate\. We organize the discussion around these two axes rather than around individual specifications, because the empirical analysis asks which source of information accounts for predictive gains: choice\-set representation or consumer heterogeneity\. Sections[5\.1](https://arxiv.org/html/2607.13314#S5.SS1)and[5\.2](https://arxiv.org/html/2607.13314#S5.SS2)develop the representation axis and Section[5\.3](https://arxiv.org/html/2607.13314#S5.SS3)the heterogeneity axis; Section[5\.4](https://arxiv.org/html/2607.13314#S5.SS4)then contrasts in\-context with fine\-tuned prediction, a further dimension along which these reformulations can be applied\.

### 5\.1Choice\-Set Representations

LetSt=\{1,…,Jt\}S\_\{t\}=\\\{1,\\ldots,J\_\{t\}\\\}denote the alternatives available on choice occasiontt, and letyi​j​t=1y\_\{ijt\}=1if consumeriichooses alternativejjon that occasion\. Each alternative has attributes𝐱i​j​t\\mathbf\{x\}\_\{ijt\}\. A TabPFN classifier requires tabular rows and labels, so the first step is to convert the choice occasion into a supervised prediction problem\. We consider three main representations: long, wide, and pairwise\.

#### Long representation\.

The long representation treats each alternative in a choice set as one row\. For consumerii, occasiontt, and alternativejj, the input row is𝐱i​j​t\\mathbf\{x\}\_\{ijt\}, and the label is

yi​j​t=\{1,if alternative​j​is chosen on occasion​t,0,otherwise\.y\_\{ijt\}=\\begin\{cases\}1,&\\text\{if alternative \}j\\text\{ is chosen on occasion \}t,\\\\ 0,&\\text\{otherwise\.\}\\end\{cases\}Thus, a four\-alternative choice occasion produces four binary rows\. This representation is closest to the row\-level classification format for which TabPFN is designed\. It also preserves alternative\-level attributes directly\. However, the choice\-set constraint is not automatic: the model produces a binary score for each alternative row, and we convert these scores into choice probabilities by normalizing them within the choice set\.

Specifically, lets^i​j​t\\hat\{s\}\_\{ijt\}denote the TabPFN score or predicted probability assigned to alternativejj\. We form choice probabilities as

P^i​j​t=s^i​j​t∑k∈Sts^i​k​t\.\\hat\{P\}\_\{ijt\}=\\frac\{\\hat\{s\}\_\{ijt\}\}\{\\sum\_\{k\\in S\_\{t\}\}\\hat\{s\}\_\{ikt\}\}\.This post\-processing step restores the requirement that predicted probabilities sum to one within each choice set\.

#### Wide representation\.

The wide representation stores the entire choice set in a single row\. The attributes of all alternatives inStS\_\{t\}are concatenated into one feature vector,

𝐳i​t=\(𝐱i​1​t,𝐱i​2​t,…,𝐱i​Jt​t\),\\mathbf\{z\}\_\{it\}=\\left\(\\mathbf\{x\}\_\{i1t\},\\mathbf\{x\}\_\{i2t\},\\ldots,\\mathbf\{x\}\_\{iJ\_\{t\}t\}\\right\),and the label is the identity of the chosen alternative\. Therefore, the wide representation becomes a multi\-class classification problem\.

The wide representation makes the mutually exclusive nature of the choice occasion explicit: one row corresponds to one choice, and the classifier directly predicts the chosen alternative\. Its primary limitation is that it does not encode the repeated\-alternative structure of the problem: the TFM observes separate columns for each alternative’s attributes but is not told that corresponding columns across alternatives represent the same underlying feature\. Relationships such as price comparisons across brands must therefore be inferred from the data rather than supplied by the representation\. The wide representation also requires a fixed alternative ordering; we consider permutation augmentation to reduce dependence on arbitrary column positions\.

#### Pairwise representation\.

The pairwise representation converts a multinomial choice into binary comparisons\. For each observed choice of alternativejjover another available alternativekk, we construct a row describing the comparison between the two alternatives\. In the difference version, the input is𝐱i​j​t−𝐱i​k​t\\mathbf\{x\}\_\{ijt\}\-\\mathbf\{x\}\_\{ikt\}, and the label indicates thatjjis preferred tokk\. In the ordered version, the input concatenates the two alternatives,\[𝐱i​j​t,𝐱i​k​t\]\[\\mathbf\{x\}\_\{ijt\},\\mathbf\{x\}\_\{ikt\}\], and the label indicates whether the first alternative is the one chosen\.

Pairwise representations are attractive because they express choice as a set of revealed preference inequalities\. However, they also change the prediction problem: the model learns binary comparisons rather than direct choice probabilities over the full set\. To evaluate a choice occasion, we aggregate pairwise predictions back to alternative\-level scores and then normalize those scores within the choice set\. We consider several aggregation rules in the empirical sweep, including Borda\-style aggregation and logit\-sum aggregation\.

### 5\.2Relative and Set\-Level Features

The long representation is row\-natural but does not by itself tell the model how an alternative compares with the other alternatives available on the same occasion\. We therefore consider simple features that make within\-set relative position explicit\.

The first set of features are rank features\. For each attribute that varies within a choice set, we compute the rank of an alternative relative to the other alternatives in that set\. In the yogurt application considered in our numerical study, price varies by brand and occasion, so a price\-rank feature indicates whether a brand is relatively cheap or expensive within the current choice set\. Rank features are attractive because they are scale\-free and directly encode comparison, while keeping the long representation in a row\-level format\.

The second set of features are set\-summary features\. These features describe the choice set around an alternative, such as within\-set means, minima, maxima, or deviations from the set average\. The goal is to give the model information about the competitive environment faced by the focal alternative\. Such features provide the row\-level classifier with information that would otherwise be implicit in the full choice set\.

These features are useful for separating two questions empirically\. If relative or set\-summary features improve prediction substantially, then making within\-set comparison more explicit is important\. If their effect is small relative to the effect of consumer heterogeneity, then the main challenge lies less in representing the choice set and more in representing persistent consumer preferences\.

### 5\.3Encoding Consumer Heterogeneity

The second design axis concerns consumer heterogeneity\. We compare three ways of organizing the same panel data: pooled prediction, respondent\-categorical prediction, and per\-respondent in\-context prediction\.

#### Pooled prediction\.

The pooled specification suppresses consumer identity\. Training rows from all consumers are combined into a single context, but the model is not told which rows belong to the same consumer\. This specification is the closest TabPFN analogue to a homogeneous DCM\. It allows the model to learn average relationships between alternative attributes and choices, but it does not provide a direct channel for consumer\-specific preferences\.

#### Respondent\-categorical prediction\.

The respondent\-categorical specification includes consumer identity as a categorical feature\. All consumers remain in a shared context, but repeated observations from the same consumer share the same respondent identifier\. This construction gives the model two kinds of information at once\. First, because all consumers are pooled into one context, the model can condition on population\-level regularities in the data\. Second, because consumer identity is included as a feature, the model can use a consumer’s own past choices to personalize predictions for that consumer\.

This construction is not equivalent to estimating a random coefficient vector for each consumer\. The respondent identifier is a feature supplied to the pretrained model, and the model decides how to use it through its learned in\-context prediction mechanism\. Nevertheless, it is the most direct way to make repeated consumer histories visible while retaining a shared population context\.

#### Per\-respondent in\-context prediction\.

The per\-respondent specification creates a separate ICL problem for each consumer\. For consumerii, the context consists only of that consumer’s calibration choices, and the test rows are that same consumer’s holdout choices\. This construction emphasizes personalization: the model predicts future choices for a consumer using only that consumer’s own observed history\.

The advantage of this approach is that it aligns naturally with the panel prediction problem for an existing consumer\. The limitation is that it removes direct cross\-consumer pooling from the in\-context task\. When a consumer has few calibration choices, the context may be too small for the model to infer stable consumer\-specific patterns\. This limitation is especially important in shallow panels and motivates the fine\-tuning analysis in Section[6\.3](https://arxiv.org/html/2607.13314#S6.SS3), where population information can enter through weight updates before consumer\-level prediction\.

### 5\.4In\-Context Prediction and Fine\-Tuned Prediction

All reformulations can be used with TabPFN in\-context prediction\. In this case, the pretrained weights remain fixed, and the model adapts to the choice task only through the context supplied at prediction time\. This is the setting evaluated in Section[6\.2](https://arxiv.org/html/2607.13314#S6.SS2)\. It asks how much the TFM’s learned prior can do when the choice data are represented in different ways and when consumer heterogeneity is or is not made visible to the model\.

We also evaluate fine\-tuned TabPFN for a subset of representative specifications\. Fine\-tuning starts from pretrained TabPFN weights and updates them using the pooled training observations from the focal choice population\. After fine\-tuning, the model is evaluated using the same train\-holdout split and the same prediction metrics as in the in\-context analysis\. As discussed in Section[3\.3](https://arxiv.org/html/2607.13314#S3.SS3), we interpret fine\-tuning as moving the model from the TFM’s learned prior toward a population choice\-aware prior\. The fine\-tuning analysis is therefore not a separate method family, but a second way of using TabPFN: instead of only conditioning a fixed prior on context, the model first adapts its weights to the empirical regularities of the target choice population\.

## 6Empirical Application

We evaluate the proposed reformulations using a yogurt scanner panel, a revealed\-preference dataset in which consumers make repeated brand choices across grocery shopping occasions\. The empirical analysis proceeds in three steps\. First, we describe the data, evaluation protocol, and benchmark DCMs, including both homogeneous MNL and hierarchical Bayesian models that allow for consumer\-level heterogeneity\. Second, we evaluate TabPFN with fixed pretrained weights, asking how much can be gained by conditioning the TFM’s learned prior on calibration choices and by organizing those choices to reveal consumer heterogeneity\. Third, we evaluate fine\-tuned TabPFN, asking whether updating the pretrained model with population choice data improves prediction for consumers with sparse choice histories\. Together, these analyses show when a tabular foundation model can compete with hierarchical DCMs, and which parts of the gain come from representation, heterogeneity, and population\-specific prior updating\.

### 6\.1Data and Benchmarks

We use the yogurt scanner panel originally analyzed byJainet al\.\([1994](https://arxiv.org/html/2607.13314#bib.bib78)\)and distributed in thelogitrpackage in R\. The data contain repeated grocery purchase decisions from 98 consumers \(those with at least five purchase occasions in the raw data\), each choosing among four yogurt brands: Dannon, Hiland, Weight Watchers, and Yoplait\. There is no outside option, so every observed occasion is a purchase of one of the four brands\. Each brand corresponds to one alternative in the four\-element choice set and is described by five features: three brand indicators, a price, and a binary indicator for whether the brand was featured in a newspaper advertisement\. This setting is well suited to our analysis because it is a revealed\-preference panel: consumers make actual purchase decisions under prices and promotional conditions that are not designed by the researcher, and the same consumers are observed repeatedly over time\.

The number of purchase occasions varies substantially across consumers\. This variation is central to our empirical design because it allows us to study how each method performs as the amount of consumer\-level choice history changes\. We therefore group consumers into four purchase\-depth bins, shown in Table[2](https://arxiv.org/html/2607.13314#S6.T2)\. The shallow bin contains consumers for which individual preferences are weakly identified without pooling, whereas the deep bin contains consumers with longer histories that can support more reliable individual\-level inference\. Thus, consumer depth is not merely a feature of the data, but an empirical axis for studying when different forms of shrinkage and heterogeneity modeling are most useful\.

Table 2:Consumer distribution by purchase depth\.For each consumer, we reserve the three most recent purchase occasions as holdout observations and use all earlier occasions for estimation or in\-context prediction\. This protocol is held fixed across all methods, so every model is evaluated on the same future choice occasions\. The training observations serve as estimation data for the benchmark choice models\. For TabPFN, they serve as in\-context examples: in the respondent\-categorical construction, observations from all consumers are pooled into a shared context with consumer identity included as a categorical feature; in the per\-respondent construction, each consumer’s training observations form a separate context for predicting that consumer’s holdout choices\. This design matches the panel\-prediction problem faced by firms that use past purchase histories to predict subsequent choices by existing customers\.

We evaluate predictive performance using two metrics\. The primary metric is holdout log\-likelihood \(LL\), defined as the average per\-occasion log predicted probability assigned to the chosen brand on held\-out occasions\. Higher values indicate better probabilistic choice prediction\. We also report hit rate, defined as the fraction of held\-out occasions for which the model assigns the highest predicted probability to the brand actually chosen\. Log\-likelihood evaluates the full predicted choice distribution, while hit rate summarizes argmax accuracy\.

We compare TabPFN against four discrete\-choice benchmarks\. The first is pooled MNL, which imposes homogeneous preferences across consumers and serves as the classical non\-hierarchical baseline\. The remaining three are hierarchical Bayesian mixed\-logit models estimated usingbayesm\(Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55)\)\. The first HB specification assumes that consumer\-level taste parameters are drawn from a single multivariate normal population distribution\. The second allows the population distribution to be a finite mixture of normals, which can capture discrete segments or multimodal heterogeneity\. The third uses a Dirichlet process prior, allowing the number and composition of latent preference groups to be learned more flexibly from the data\. Together, these benchmarks span the standard homogeneous model, the canonical HB specification widely used in marketing practice, and more flexible heterogeneous specifications\. For the Bayesian benchmarks, holdout choice probabilities are computed by averaging choice probabilities over retained posterior draws, rather than by evaluating the choice model only at posterior mean parameters\.

We use TabPFN\(Hollmannet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib1)\)as the representative TFM\. The ICL results use TabPFN v2\.6, the strongest released version available for off\-the\-shelf prediction in our experiments\. The fine\-tuning results use TabPFN v2\.5, because classifier fine\-tuning is currently supported for that version in the released library\. We keep this version difference explicit when reporting fine\-tuning results, so that improvements from population\-specific fine\-tuning can be separated from improvements due to the newer off\-the\-shelf model\.

### 6\.2TabPFN In\-Context Learning Results

We first evaluate TabPFN in its ICL form, keeping the pretrained model weights fixed\. In this analysis, TabPFN brings to the yogurt task a learned prior from pretraining, but that prior is not updated using the yogurt choice data\. Any adaptation to the focal market therefore occurs through the calibration observations supplied in context and through the way those observations are represented\. This setting allows us to separate two sources of predictive improvement: gains from reformulating the choice problem for a TFM, and gains from exposing consumer\-level heterogeneity to the model\.

We begin with the homogeneous setting, where all consumers are pooled and the model is not given any consumer identifier or consumer\-specific context\. Table[3](https://arxiv.org/html/2607.13314#S6.T3)compares pooled MNL with selected homogeneous TabPFN reformulations\. The full sweep contains many additional representation variants; here we report representative specifications that span the main representation families\.

Table 3:Homogeneous in\-context prediction results\.The homogeneous results show that changing the representation alone yields only modest improvements\. The best homogeneous TabPFN specification, the long representation, improves predicted probability by approximately3\.9%3\.9\\%over pooled MNL, but the gain is small relative to the improvements reported below once consumer heterogeneity is introduced\. Other representations are close to pooled MNL, and some improve one metric while worsening the other\. This pattern suggests that the learned prior, by itself, does not solve the main statistical problem in the panel\. The dominant source of predictive difficulty is not simply the row\-wise format of the data, but the fact that repeated observations from the same consumer reflect persistent consumer\-level preferences\.

We next introduce consumer heterogeneity into the TabPFN context\. We consider two ways of doing so\. In the respondent\-categorical construction, all consumers remain in a shared context and consumer identity is included as a categorical feature\. In the per\-respondent construction, each consumer forms a separate ICL problem, using that consumer’s calibration choices to predict its holdout choices\. These two constructions differ in how population information can enter the prediction problem, but they share the same purpose: both expose consumer\-level heterogeneity to the model\.

Table[4](https://arxiv.org/html/2607.13314#S6.T4)compares the heterogeneous TabPFN specifications with the benchmark DCMs\. We evaluated 37 TabPFN variants across the long, wide, and pairwise reformulation families; the table reports a representative selection spanning these families\.

Table 4:Heterogeneous in\-context prediction results\.The heterogeneous results change the picture sharply\. Moving from homogeneous TabPFN to heterogeneity\-aware TabPFN produces a much larger improvement than moving across representation families within the homogeneous setting\. The best homogeneous TabPFN specification has LL=−1\.115=\-1\.115, whereas the best heterogeneous specification, long TabPFN with rank features and respondent\-categorical encoding, reaches LL=−0\.648=\-0\.648and a hit rate of0\.7720\.772\. This improvement is also large relative to the hierarchical benchmarks\. The best HB specification, the single\-normal population model, achieves LL=−0\.725=\-0\.725and hit rate0\.7450\.745; the more flexible finite\-mixture and Dirichlet process specifications do not improve on it in this dataset\. Thus, the best in\-context TabPFN specification improves on the strongest HB benchmark by approximately8\.0%8\.0\\%in predicted probability and3\.6%3\.6\\%in hit rate\. The best TabPFN configuration also runs in approximately4\.44\.4seconds, against70\.470\.4seconds for HB\-MNL with the single\-normal population, a sixteenfold reduction in wall\-clock time; the finite\-mixture and Dirichlet process specifications take comparable time to the single\-normal specification\. This computational efficiency is itself a contribution of the in\-context approach: the cost of inference has been amortized into pretraining, and no further MCMC is required at deployment\.

These comparisons clarify what is, and is not, driving the gains\. The long, wide, and pairwise representations differ in performance, but these differences are second\-order relative to the effect of introducing consumer heterogeneity\. In the pooled setting, the best TabPFN reformulation improves over pooled MNL by only about3\.9%3\.9\\%in predicted probability\. By contrast, adding consumer heterogeneity to the long\-rank representation improves predicted probability by approximately60%60\\%over the pooled long\-rank baseline\. This decomposition supports the interpretation developed in Section[4](https://arxiv.org/html/2607.13314#S4): the primary challenge in adapting off\-the\-shelf tabular prediction to panel choice data is not only the set\-valued nature of each choice occasion, but the cross\-observation dependence induced by persistent consumer preferences\.

The two heterogeneity constructions should therefore be read primarily as alternative ways of making consumer\-level preference variation available to the model\. Their common pattern is more important than their difference: once consumer heterogeneity is introduced, predictive performance improves substantially across representation families\. Still, the two constructions are not identical\. Respondent\-categorical encoding keeps all consumers in a shared context, allowing the model to use population\-level regularities while learning consumer\-specific differences through the categorical feature\. Per\-respondent ICL isolates each consumer and therefore relies more heavily on the amount of calibration history available for that consumer\. This distinction helps explain why per\-respondent ICL performs well for some long\-format specifications but becomes unstable for the wide representation, where the median consumer provides too few context rows to support a multiclass prediction problem\. The wide representation is also the most sensitive to representation choice within the heterogeneous setting: combining wide with the respondent\-categorical encoding and permutation augmentation simultaneously produces results well below the pooled MNL baseline, and we report these entries for completeness as evidence that some refinements of the wide representation interact poorly with one another\.

The aggregate comparison masks important variation across consumer depth\. Table[2](https://arxiv.org/html/2607.13314#footnote2)compares the strongest HB benchmark against in\-context TabPFN within each purchase\-depth bin under two specifications of TabPFN\. The first holds the specification fixed across all bins at the aggregate winner \(long with rank features and respondent\-categorical encoding\) so that a single deployable model is evaluated everywhere\. The second reports the best TabPFN variant within each bin, where “best” is taken as the variant with the highest holdout hit rate, with log\-likelihood as a tie\-breaker among methods that reach the bin’s hit\-rate ceiling\. The two columns together let the reader distinguish what the fixed aggregate winner gives up at a given depth from how much remains on the table for variant\-specific gains\.

Table 5:In\-context TabPFN versus HB by consumer purchase depth\.222The best TabPFN variants per bin are: pairwise\-difference with antisymmetric\-Borda aggregation under per\-respondent ICL in B1; long with rank features and respondent\-categorical encoding in B2 and B3 \(the same as the fixed specification\); and long with rank features under per\-respondent ICL in B4\. Several other heterogeneity\-aware TabPFN variants also reach the0\.94870\.9487hit\-rate ceiling in B4, matching HB on hit; among them, long with rank features under per\-respondent ICL has the highest log\-likelihood\. The aggregate row is omitted for the best\-per\-bin column because no single TabPFN model achieves all bin\-specific bests simultaneously\.Two broader patterns in the depth distribution are nonetheless clear\. First, both HB and TabPFN benefit from deeper consumer histories\. HB improves monotonically across depth bins, from LL=−1\.010=\-1\.010in the shallow bin to LL=−0\.279=\-0\.279in the deepest bin, with hit rate rising from0\.6240\.624to0\.9490\.949\. TabPFN shows the same broad pattern from shallow to medium\-deep consumers: its LL improves from−1\.028\-1\.028in B1 to−0\.625\-0\.625in B2 and−0\.306\-0\.306in B3, while hit rate rises from0\.5700\.570to0\.7980\.798and then0\.9370\.937\. In the deepest bin, however, TabPFN no longer improves, whereas HB continues to benefit from the longer consumer histories\. This suggests that the two approaches use additional individual\-level observations differently: HB’s parametric consumer\-level posterior continues to sharpen with depth, while the fixed\-prior TabPFN specification appears most useful in the middle of the depth distribution\.

Second, TabPFN’s aggregate advantage over HB is concentrated in the medium\-depth bins\. In B2, TabPFN improves on HB by approximately21%21\\%in predicted probability and16%16\\%in hit rate\. In B3, TabPFN improves on HB by approximately14%14\\%in predicted probability and5%5\\%in hit rate\. By contrast, HB outperforms TabPFN in the shallowest bin and in the deepest bin on both metrics\. Thus, the overall TabPFN gain is not a uniform dominance result\. Rather, in\-context TabPFN is most competitive when consumers have enough history for the model to infer consumer\-specific patterns, but not so much history that the hierarchical Bayesian model can estimate individual preferences very precisely\.

This depth pattern is consistent with the unified shrinkage view\. With very shallow histories, the TFM’s learned prior has limited consumer\-specific information to condition on, and HB’s population pooling remains valuable\. With medium\-depth histories, TabPFN can exploit the consumer identifier and rank\-based representation to extract useful individual\-level signal, producing its clearest advantage over HB\. With deep histories, HB catches up or surpasses TabPFN because the individual\-level posterior is well informed by each consumer’s own data\.

Overall, the in\-context results show that the TFM’s learned prior can be useful for discrete choice prediction, but only when the choice data are organized so that consumer heterogeneity is visible to the model\. Representation matters, especially the long representation with simple within\-set rank features, but the main empirical result is the large jump from homogeneous to heterogeneity\-aware prediction\. The depth\-bin analysis further shows that this advantage is concentrated in the middle of the consumer\-depth distribution\. This pattern motivates the fine\-tuning analysis in the next subsection: if the shallow bin is precisely where the learned prior has too little consumer\-specific information to condition on, then updating the prior using population choice data should be most valuable there\.

### 6\.3TabPFN Fine\-Tuning Results

The preceding subsection evaluates TabPFN with fixed pretrained weights\. We now ask whether performance improves when the learned prior is updated using the focal choice population\. Conceptually, fine\-tuning differs from ICL because population information enters the model weights before prediction\. This makes fine\-tuning the closest TabPFN analogue to the population\-learning step in HB, where information from all consumers is used to estimate the population distribution that anchors individual\-level predictions\.

A practical constraint is that classifier fine\-tuning is currently supported for TabPFN v2\.5, whereas the strongest in\-context results in the previous subsection use TabPFN v2\.6\. We therefore report three quantities: v2\.5 ICL, v2\.5 fine\-tuning, and v2\.6 ICL\. The comparison between v2\.5 ICL and v2\.5 fine\-tuning isolates the effect of fine\-tuning holding model version fixed\. The comparison with v2\.6 ICL shows how much of the performance gap to the newer off\-the\-shelf model is closed by fine\-tuning\. Table[6](https://arxiv.org/html/2607.13314#S6.T6)reports these comparisons for four representative specifications\.

Table 6:Aggregate fine\-tuning results\.The aggregate results show that fine\-tuning helps, but does not uniformly dominate the newer off\-the\-shelf model\. Holding the model version fixed at v2\.5, fine\-tuning improves log\-likelihood for all four specifications, with gains in predicted probability ranging from0\.5%0\.5\\%to3\.8%3\.8\\%\. The largest gain occurs for the long\-rank respondent\-categorical specification, where fine\-tuning improves predicted probability by3\.8%3\.8\\%and hit rate by5\.6%5\.6\\%\. However, v2\.6 ICL remains the strongest aggregate specification for the main winner, with LL=−0\.648=\-0\.648and hit rate=0\.772=0\.772\. For this method, the v2\.6 upgrade improves predicted probability by7\.5%7\.5\\%relative to v2\.5 ICL, while fine\-tuning v2\.5 improves it by3\.8%3\.8\\%, closing roughly half of the version gap\.

The pattern is consistent with the prior\-updating interpretation\. Fine\-tuning is most useful when the model has a channel through which population\-level choice regularities can be expressed\. The respondent\-categorical specifications allow the model to learn how consumer identity relates to choice behavior within a shared population context\. By contrast, fine\-tuning provides only modest calibration gains for the pooled and per\-respondent specifications, and its effects on hit rate are mixed\. This suggests that fine\-tuning is not simply adding generic flexibility; it is most valuable when the representation gives the updated prior a way to organize population\-level heterogeneity\.

The aggregate comparison, however, does not fully reveal where fine\-tuning is useful\. The depth\-bin analysis in the previous subsection suggests that the shallow bin is the most prior\-dependent regime: consumers in B1 have too little calibration history for consumer\-specific patterns to be reliably inferred from context alone\. If fine\-tuning moves TabPFN toward a population choice\-aware prior, its effect should therefore be largest in B1\. Table[7](https://arxiv.org/html/2607.13314#S6.T7)tests this implication for the strongest overall TabPFN specification, long with rank features and respondent\-categorical encoding\.

Table 7:Fine\-tuning results by consumer depth for long\-rank respondent\-categorical TabPFN\.The depth\-bin results sharpen the interpretation\. In the shallow bin, fine\-tuned v2\.5 is the best model on both metrics, beating HB, v2\.5 ICL, and v2\.6 ICL\. This is exactly the regime in which consumer\-level context is thinnest and predictions rely most heavily on the prior\. Fine\-tuning improves B1 hit rate by approximately22%22\\%relative to v2\.5 ICL and by13%13\\%relative to v2\.6 ICL\. It also improves B1 predicted probability relative to both fixed\-prior TabPFN variants and HB\. The B1 advantage is attributable to fine\-tuning rather than to a confound with the newer off\-the\-shelf model: in this bin, the version upgrade from v2\.5 ICL to v2\.6 ICL contributes only about0\.5%0\.5\\%in predicted probability, whereas fine\-tuning within v2\.5 contributes about4\.1%4\.1\\%\. This pattern suggests that updating the pretrained prior with population choice data is especially valuable when consumer\-specific histories are too sparse for ICL alone\.

The advantage of fine\-tuning is less pronounced in deeper bins\. In B2, v2\.6 ICL dominates both HB and fine\-tuned v2\.5\. In B3, all TabPFN variants perform well relative to HB, with fine\-tuned v2\.5 slightly ahead on LL and v2\.6 ICL ahead on hit rate\. In B4, HB regains the advantage in LL and ties or leads on hit rate, consistent with the depth analysis above: when each consumer has a long history, the hierarchical model can estimate individual preferences precisely\. Thus, fine\-tuning does not uniformly dominate either HB or off\-the\-shelf TabPFN\. Its contribution is concentrated where the theory predicts it should be: among consumers with sparse histories, where population\-level regularization is most valuable\.

Taken together, the fine\-tuning results qualify and extend the in\-context findings\. Off\-the\-shelf TabPFN v2\.6 remains the best aggregate model in this application, but fine\-tuning v2\.5 substantially improves over v2\.5 ICL and is especially effective in the shallowest consumer bin\. This supports the interpretation of fine\-tuning as moving TabPFN from a learned prior toward a population choice\-aware prior\. ICL conditions on the observed calibration choices; fine\-tuning changes the prior that the model brings to those choices\. The two operations therefore play different roles in adapting TFMs to panel choice data\.

## 7Conclusion

This paper studies the application of TFMs to discrete choice problems and identifies a fundamental structural mismatch between the two\. Discrete choice is inherently set\-valued and heterogeneous, with outcomes defined by comparisons within choice sets and linked across observations through individual\-level preferences\. In contrast, TFMs are built on a row\-wise learning paradigm that imposes no explicit relational or set\-valued structure\. This mismatch is not a matter of model capacity, but of representation: the economic structure of choice is not directly aligned with the inductive bias of tabular learning\. As a result, direct applications of TFMs yield performance comparable to pooled models that do not capture heterogeneity or within\-set competition\.

To address this, we propose a structured reformulation of discrete choice prediction that explicitly aligns data representation and task construction with the underlying decision structure\. The central insight is that representation serves as the interface between model inductive bias and economic structure\. By encoding both choice\-set dependence and individual\-level heterogeneity within a row\-based framework, the reformulation enables pretrained TFMs to approximate the original set\-conditioned prediction problem\. Under this alignment, TFMs match or exceed the predictive performance of established heterogeneity\-aware DCMs, with the strongest aggregate results achieved via ICL and additional gains available through fine\-tuning on population choice data, particularly for consumers with sparse purchase histories\.

A key mechanism underlying these gains is the encoding of individual\-level heterogeneity within the TFM framework\. Heterogeneity encoding is the dominant driver of predictive improvement, with choice\-set representation providing a secondary but consistent contribution\. By conditioning predictions on respondent identity \(either through an explicit indicator feature or through per\-respondent in\-context learning\), TFMs recover personalized patterns without explicit parameter estimation or MCMC\. The appropriate form of this encoding depends on data depth: respondent\-indicator approaches are more effective when per\-consumer histories are moderately rich, while per\-respondent ICL and fine\-tuning on population choice data are most effective in sparse settings, with fine\-tuning providing the largest gains where consumer\-specific histories are too thin for ICL to reliably infer individual preferences\. These two regimes reflect complementary mechanisms: ICL extracts the most individual\-specific signal when moderate histories are available, while fine\-tuning compensates in the sparse regime by incorporating population\-level information into the prior itself\. Empirically, TFMs offer the greatest aggregate advantage over HB in the medium\-data regime \(roughly 10–40 observations per consumer\), where HB’s parametric normal prior is most likely to bind for atypical consumers; the two methods converge in both sparse and data\-rich settings\. This highlights that the success of TFMs in discrete choice depends not only on representation, but also on the structure and depth of available individual data\.

These findings suggest that the effectiveness of TFMs in discrete choice does not arise from model flexibility alone, but from aligning the prediction problem with the model’s inductive bias\. The unified shrinkage view developed in Section[3](https://arxiv.org/html/2607.13314#S3)provides the theoretical grounding for this alignment: TFMs and HB estimation occupy adjacent positions in a common methodological progression from pooled to individual\-level inference, differing primarily in how the prior over consumer preferences is acquired \(learned from synthetic data during pretraining\) and updated \(conditioned in\-context or via fine\-tuning, rather than via MCMC\)\. Rather than viewing TFMs as direct replacements for existing approaches, they are better understood as prediction\-oriented tools whose performance depends on how the problem is formulated\. When appropriately aligned, they offer a computationally lightweight alternative that avoids task\-specific estimation while retaining competitive accuracy\.

More broadly, this paper highlights a general principle for applying foundation models in structured domains\. In problems where outcomes are defined over sets and linked across observations, structure cannot be left for the model to infer implicitly; it must be encoded in the representation\. Discrete choice provides a clear example of this principle, but the same tension is likely to arise in other decision\-making settings with relational and hierarchical structure\.

Several directions emerge directly from this work\. Within the choice modeling context, the dominant role of heterogeneity encoding identifies richer preference representation as the highest\-leverage near\-term extension: latent class and mixed\-membership specifications within an ICL framework would capture the discrete segmentation structure that many marketing applications require\. Looking beyond prediction, the most important extension is toward counterfactual demand analysis: price elasticities, market share responses to new products, and welfare implications of assortment decisions\. Adapting PFN architectures to causal settings, where models would be pretrained on causal data\-generating processes to enable zero\-shot estimation of treatment effects and interventional demand distributions without specifying a parametric demand system, is a natural next step; integrating such approaches with the discrete choice reformulation developed here would substantially broaden the range of empirical questions TFMs can address in marketing and operations management\. At the architectural level, models that natively represent the set\-valued structure of choice, treating the choice set as an unordered collection of alternatives rather than a fixed class enumeration, would reduce reliance on the external reformulation the present work provides\.

Beyond extensions of the present framework, the rapid development of the TFM literature itself opens a broad research agenda for business scholars\. Recent architectures scale ICL to hundreds of thousands of observations\(Prior Labs Team[2025](https://arxiv.org/html/2607.13314#bib.bib32), Quet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib7)\), making TFMs applicable to the large transaction logs that characterize commercial demand estimation and revenue management\. Multi\-table relational extensions, which attend across linked tables of transactions, products, and consumers, are beginning to emerge\(Feyet al\.[2024](https://arxiv.org/html/2607.13314#bib.bib37), Eremeevet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib39)\)and would allow TFMs to leverage the richer database structures that firms already maintain, an important step toward deploying these models in realistic settings\. Domain\-specific pretraining on actual business data, rather than generic synthetic priors, represents a further frontier: models pretrained on curated real\-world tables have shown substantial improvements over models trained only on synthetic priors\(Maet al\.[2025](https://arxiv.org/html/2607.13314#bib.bib35)\), and pretraining on a retailer’s transaction history could yield a demand\-specialized TFM that transfers across product categories without dataset\-specific estimation\. Finally, interpretability remains an open challenge for practical adoption in business settings: extracting price elasticities and attribute importances from TFM predictions, without imposing parametric structure on the demand function, would make the approach actionable for the pricing, assortment, and targeting decisions that motivate choice modeling in the first place\.

## References

- What learning algorithm is in\-context learning? Investigations with linear models\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1)\.
- G\. M\. Allenby and P\. E\. Rossi \(1998\)Marketing models of consumer heterogeneity\.Journal of Econometrics89\(1–2\),pp\. 57–78\.Cited by:[Appendix A](https://arxiv.org/html/2607.13314#A1.SSx2.p1.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1),[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px1.p1.9),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.2.3.1.1),[§4\.2](https://arxiv.org/html/2607.13314#S4.SS2.p1.2)\.
- A\. Aouad and A\. Désir \(2025\)Representing random utility choice models with neural networks\.Management Science\.Note:Articles in AdvanceExternal Links:[Document](https://dx.doi.org/10.1287/mnsc.2023.02189)Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- S\. Athey and G\. W\. Imbens \(2019\)Machine learning methods that economists should know about\.Annual Review of Economics11,pp\. 685–725\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- G\. Berbeglia, A\. Garassino, and G\. Vulcano \(2022\)A comparative empirical study of discrete choice models in retail operations\.Management Science68\(6\),pp\. 4005–4023\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px3.p1.1)\.
- S\. Berry, J\. Levinsohn, and A\. Pakes \(1995\)Automobile prices in market equilibrium\.Econometrica63\(4\),pp\. 841–890\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1)\.
- J\. Brand, A\. Israeli, and D\. Ngwe \(2023\)Using LLMs for market research\.Working PaperTechnical Report23\-062,Harvard Business School\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p1.1)\.
- T\. B\. Brown, B\. Mann, N\. Ryder, M\. Subbiah, J\. Kaplan, P\. Dhariwal, A\. Neelakantan, P\. Shyam, G\. Sastry, A\. Askell,et al\.\(2020\)Language models are few\-shot learners\.Advances in Neural Information Processing Systems33,pp\. 1877–1901\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§3\.1](https://arxiv.org/html/2607.13314#S3.SS1.SSS0.Px1.p1.6)\.
- T\. Chen and C\. Guestrin \(2016\)XGBoost: a scalable tree boosting system\.InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,pp\. 785–794\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- Y\. Chen, T\. X\. Liu, Y\. Shan, and S\. Zhong \(2023\)The emergence of economic rationality of GPT\.Proceedings of the National Academy of Sciences120\(51\),pp\. e2316205120\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p1.1)\.
- P\. K\. Chintagunta and H\. S\. Nair \(2011\)Discrete\-choice models of consumer demand in marketing\.Marketing Science30\(6\),pp\. 977–996\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1)\.
- G\. Compiani, I\. Morozov, and S\. Seiler \(2025\)Demand estimation with text and image data\.External Links:2503\.20711Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- T\. Cucumides and F\. Geerts \(2026\)Grables: tabular learning beyond independent rows\.External Links:2602\.03945Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p3.1)\.
- J\. Dubé, G\. J\. Hitsch, and P\. E\. Rossi \(2010\)State dependence and alternative explanations for consumer inertia\.RAND Journal of Economics41\(3\),pp\. 417–445\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1)\.
- D\. Eremeev, G\. Bazhenov, O\. Platonov, A\. Babenko, and L\. Prokhorenkova \(2025\)Turning tabular foundation models into graph foundation models\.External Links:2508\.20906Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p3.1),[§7](https://arxiv.org/html/2607.13314#S7.p7.1)\.
- N\. Erickson, L\. Purucker, A\. Tschalzev, D\. Holzmüller, P\. Mutalik Desai, D\. Salinas, and F\. Hutter \(2025\)TabArena: a living benchmark for machine learning on tabular data\.InAdvances in Neural Information Processing Systems,External Links:2506\.16791Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- T\. Evgeniou, M\. Pontil, and O\. Toubia \(2007\)A convex optimization approach to modeling consumer heterogeneity in conjoint estimation\.Marketing Science26\(6\),pp\. 805–818\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1),[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px3.p1.1),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.5.2.3.1.1)\.
- V\. F\. Farias, S\. Jagabathula, and D\. Shah \(2013\)A nonparametric approach to modeling choice with limited data\.Management Science59\(2\),pp\. 305–322\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px3.p1.1)\.
- M\. Fey, W\. Hu, K\. Huang, J\. E\. Lenssen, R\. Ranjan, J\. Robinson, R\. Ying, J\. You, and J\. Leskovec \(2024\)Position: relational deep learning—Graph representation learning on relational databases\.InProceedings of the 41st International Conference on Machine Learning,Proceedings of Machine Learning Research\.External Links:2312\.04615Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p3.1),[§7](https://arxiv.org/html/2607.13314#S7.p7.1)\.
- S\. Gabel and A\. Timoshenko \(2022\)Product choice with large assortments: a scalable deep\-learning model\.Management Science68\(3\),pp\. 1808–1827\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- G\. Gallego and R\. Wang \(2014\)Multiproduct price optimization and competition under the nested logit model with product\-differentiated price sensitivities\.Operations Research62\(2\),pp\. 450–461\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px3.p1.1)\.
- Y\. Gorishniy, I\. Rubachev, V\. Khrulkov, and A\. Babenko \(2021\)Revisiting deep learning models for tabular data\.InAdvances in Neural Information Processing Systems,Vol\.34,pp\. 18932–18943\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- P\. E\. Green and V\. Srinivasan \(1978\)Conjoint analysis in consumer research: issues and outlook\.Journal of Consumer Research5\(2\),pp\. 103–123\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.4.1.3.1.1)\.
- L\. Grinsztajn, E\. Oyallon, and G\. Varoquaux \(2022\)Why do tree\-based models still outperform deep learning on typical tabular data?\.InAdvances in Neural Information Processing Systems,Vol\.35,pp\. 507–520\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- G\. Gui and O\. Toubia \(2023\)The challenge of using LLMs to simulate human behavior: a causal inference perspective\.External Links:2312\.15524Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p2.1)\.
- N\. Hollmann, S\. Müller, K\. Eggensperger, and F\. Hutter \(2023\)TabPFN: a transformer that solves small tabular classification problems in a second\.InInternational Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2607.13314#S1.p9.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§3\.1](https://arxiv.org/html/2607.13314#S3.SS1.SSS0.Px4.p1.3)\.
- N\. Hollmann, S\. Müller, L\. Purucker, A\. Krishnakumar, M\. Körfer, S\. B\. Hoo, R\. T\. Schirrmeister, and F\. Hutter \(2025\)Accurate predictions on small data with a tabular foundation model\.Nature637\(8045\),pp\. 319–326\.Cited by:[§1](https://arxiv.org/html/2607.13314#S1.p1.1),[§1](https://arxiv.org/html/2607.13314#S1.p9.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§3\.1](https://arxiv.org/html/2607.13314#S3.SS1.SSS0.Px4.p1.3),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.6.3.3.1.1),[§6\.1](https://arxiv.org/html/2607.13314#S6.SS1.p6.1)\.
- J\. J\. Horton, A\. Filippas, and B\. S\. Manning \(2023\)Large language models as simulated economic agents: what can we learn from Homo Silicus?\.Working PaperTechnical Report31122,National Bureau of Economic Research\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p1.1)\.
- D\. C\. Jain, N\. J\. Vilcassim, and P\. K\. Chintagunta \(1994\)A random\-coefficients logit brand\-choice model applied to panel data\.Journal of Business & Economic Statistics12\(3\),pp\. 317–328\.Cited by:[§6\.1](https://arxiv.org/html/2607.13314#S6.SS1.p1.1)\.
- W\. A\. Kamakura and G\. J\. Russell \(1989\)A probabilistic choice model for market segmentation and elasticity structure\.Journal of Marketing Research26\(4\),pp\. 379–390\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.1.1.3.1.1)\.
- G\. Ke, Q\. Meng, T\. Finley, T\. Wang, W\. Chen, W\. Ma, Q\. Ye, and T\. Liu \(2017\)LightGBM: a highly efficient gradient boosting decision tree\.InAdvances in Neural Information Processing Systems,Vol\.30\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- P\. J\. Lenk, W\. S\. DeSarbo, P\. E\. Green, and M\. R\. Young \(1996\)Hierarchical Bayes conjoint analysis: recovery of partworth heterogeneity from reduced experimental designs\.Marketing Science15\(2\),pp\. 173–191\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.2.3.1.1)\.
- D\. V\. Lindley and A\. F\. M\. Smith \(1972\)Bayes estimates for the linear model\.Journal of the Royal Statistical Society: Series B34\(1\),pp\. 1–41\.Cited by:[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px2.p1.7)\.
- J\. J\. Louviere, D\. A\. Hensher, and J\. D\. Swait \(2000\)Stated choice methods: analysis and applications\.Cambridge University Press,Cambridge\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p2.1)\.
- J\. Ludwig, S\. Mullainathan, and A\. Rambachan \(2025\)Large language models: an applied econometric framework\.Working PaperTechnical Report33344,National Bureau of Economic Research\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p2.1)\.
- J\. Ma, V\. Thomas, R\. Hosseinzadeh, A\. Labach, H\. Kamkari, J\. C\. Cresswell, K\. Golestan, G\. Yu, A\. L\. Caterini, and M\. Volkovs \(2025\)TabDPT: scaling tabular foundation models on real data\.InAdvances in Neural Information Processing Systems,External Links:2410\.18164Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§7](https://arxiv.org/html/2607.13314#S7.p7.1)\.
- D\. McFadden and K\. Train \(2000\)Mixed MNL models for discrete response\.Journal of Applied Econometrics15\(5\),pp\. 447–470\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1)\.
- D\. McFadden \(1974\)Conditional logit analysis of qualitative choice behavior\.InFrontiers in Econometrics,P\. Zarembka \(Ed\.\),pp\. 105–142\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1)\.
- S\. Müller, N\. Hollmann, S\. P\. Arango, J\. Grabocka, and F\. Hutter \(2022\)Transformers can do Bayesian inference\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§3\.1](https://arxiv.org/html/2607.13314#S3.SS1.SSS0.Px2.p1.4),[Table 1](https://arxiv.org/html/2607.13314#S3.T1.2.6.3.3.1.1)\.
- T\. Nagler \(2023\)Statistical foundations of prior\-data fitted networks\.InInternational Conference on Machine Learning,pp\. 25660–25676\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§3\.1](https://arxiv.org/html/2607.13314#S3.SS1.SSS0.Px4.p1.3),[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px4.p4.2)\.
- Prior Labs Team \(2025\)TabPFN\-2\.5: advancing the state of the art in tabular foundation models\.Note:Technical report\.[https://priorlabs\.ai/](https://priorlabs.ai/)External Links:2511\.08667Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§7](https://arxiv.org/html/2607.13314#S7.p7.1)\.
- L\. Prokhorenkova, G\. Gusev, A\. Vorobev, A\. V\. Dorogush, and A\. Gulin \(2018\)CatBoost: unbiased boosting with categorical features\.InAdvances in Neural Information Processing Systems,Vol\.31\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- J\. Qu, D\. Holzmüller, G\. Varoquaux, and M\. Le Morvan \(2025\)TabICL: a tabular foundation model for in\-context learning on large data\.InProceedings of the 42nd International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.267,pp\. 50817–50847\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1),[§7](https://arxiv.org/html/2607.13314#S7.p7.1)\.
- D\. Revelt and K\. Train \(1998\)Mixed logit with repeated choices: households’ choices of appliance efficiency level\.Review of Economics and Statistics80\(4\),pp\. 647–657\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1)\.
- J\. Robinson, R\. Ranjan, W\. Hu, K\. Huang, J\. Han, A\. Dobles, M\. Fey, J\. E\. Lenssen, Y\. Yuan, Z\. Zhang, X\. He, and J\. Leskovec \(2024\)RelBench: a benchmark for deep learning on relational databases\.InAdvances in Neural Information Processing Systems,External Links:2407\.20060Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p3.1)\.
- P\. E\. Rossi, G\. M\. Allenby, and R\. McCulloch \(2005\)Bayesian statistics and marketing\.Wiley,Chichester\.Cited by:[Appendix A](https://arxiv.org/html/2607.13314#A1.SSx2.p1.1),[§1](https://arxiv.org/html/2607.13314#S1.p3.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1),[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px1.p1.9),[§3\.2](https://arxiv.org/html/2607.13314#S3.SS2.SSS0.Px2.p1.7),[§4\.2](https://arxiv.org/html/2607.13314#S4.SS2.p1.2),[§6\.1](https://arxiv.org/html/2607.13314#S6.SS1.p5.1)\.
- P\. E\. Rossi, R\. E\. McCulloch, and G\. M\. Allenby \(1996\)The value of purchase history data in target marketing\.Marketing Science15\(4\),pp\. 321–340\.Cited by:[Appendix A](https://arxiv.org/html/2607.13314#A1.SSx3.p2.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px2.p1.1)\.
- R\. Shwartz\-Ziv and A\. Armon \(2022\)Tabular data: deep learning is not all you need\.Information Fusion81,pp\. 84–90\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p1.1)\.
- B\. Sifringer, V\. Lurkin, and A\. Alahi \(2020\)Enhancing discrete choice models with representation learning\.Transportation Research Part B: Methodological140,pp\. 236–261\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- A\. Singh, Y\. Liu, and H\. Yoganarasimhan \(2023\)Choice models and permutation invariance: demand estimation in differentiated products markets\.arXiv preprint arXiv:2307\.07090\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- Y\. Sui, C\. Xiao, L\. Xin, D\. Huang, and L\. Cao \(2026\)A bitter lesson for retail demand forecasting: Evidence from Fine\-Tuning Foundation Models\.Working PaperTechnical Report6788658,SSRN\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px3.p1.1)\.
- K\. E\. Train \(2009\)Discrete choice methods with simulation\.Cambridge university press\.Cited by:[Appendix A](https://arxiv.org/html/2607.13314#A1.SSx3.p2.1),[§1](https://arxiv.org/html/2607.13314#S1.p2.1),[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px1.p1.1),[§4\.2](https://arxiv.org/html/2607.13314#S4.SS2.p1.2)\.
- S\. van Cranenburgh, S\. Wang, A\. Vij, F\. Pereira, and J\. Walker \(2022\)Choice modelling in the age of machine learning – discussion paper\.Journal of Choice Modelling42,pp\. 100340\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- J\. von Oswald, E\. Niklasson, E\. Randazzo, J\. Sacramento, A\. Mordvintsev, A\. Zhmoginov, and M\. Vladymyrov \(2023\)Transformers learn in\-context by gradient descent\.InInternational Conference on Machine Learning,pp\. 35151–35174\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px6.p2.1)\.
- H\. Wang, X\. Li, and K\. Talluri \(2023\)Transformer choice net: a transformer neural network for choice prediction\.External Links:2310\.08716Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.
- M\. Wang, D\. J\. Zhang, and H\. Zhang \(2024\)Large language models for market research: a data\-augmentation approach\.External Links:2412\.19363Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px5.p1.1)\.
- S\. Wang, Q\. Wang, and J\. Zhao \(2020\)Deep neural networks for choice analysis: extracting complete economic information for interpretation\.Transportation Research Part C: Emerging Technologies118,pp\. 102701\.Cited by:[§2](https://arxiv.org/html/2607.13314#S2.SS0.SSS0.Px4.p1.1)\.

## Appendix AThe Hierarchical Bayesian Framework for Discrete Choice

This appendix provides a self\-contained account of the hierarchical Bayesian \(HB\) framework for discrete choice estimation\. HB is the dominant method in marketing practice for recovering individual\-level preference parameters from panel choice data, and it serves as the primary benchmark in the empirical analysis of the main text\.

### The Challenge: Limited Individual\-Level Data

Estimating individual\-level consumer preferences is a central challenge in marketing research\. In conjoint studies, a typical respondent completes 10 to 20 choice tasks; in scanner panel data, a household may have a few dozen purchase occasions\. Either way, the number of observations per person is far too small to estimate individual preference parameters reliably without borrowing strength from other consumers\. A model fit to one person’s data alone would be highly unstable, overfitting idiosyncratic noise rather than recovering genuine preferences\.

### The Two\-Level Model

The HB approach resolves this by modeling individual parameters as draws from a common population distribution\(Rossiet al\.[2005](https://arxiv.org/html/2607.13314#bib.bib55), Allenby and Rossi[1998](https://arxiv.org/html/2607.13314#bib.bib54)\)\. The model has two levels\.

At the*individual level*, consumeriimakes choices according to a utility maximization rule\. On choice occasiontt, alternativejjis chosen from setStS\_\{t\}if

ui​j​t=𝐱i​j​t′​𝜷i\+εi​j​t\>ui​k​tfor all​k≠j​in​St,u\_\{ijt\}=\\mathbf\{x\}\_\{ijt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\+\\varepsilon\_\{ijt\}\>u\_\{ikt\}\\quad\\text\{for all \}k\\neq j\\text\{ in \}S\_\{t\},\(6\)where𝐱i​j​t\\mathbf\{x\}\_\{ijt\}is a vector of observed alternative attributes,𝜷i\\bm\{\\beta\}\_\{i\}is an individual\-specific coefficient vector \(part\-worths\), andεi​j​t\\varepsilon\_\{ijt\}is an idiosyncratic error term\. Assuming i\.i\.d\. Type I extreme value errors, the probability that consumeriichooses alternativejjfromStS\_\{t\}takes the multinomial logit form:

P​\(j∣St;𝜷i\)=exp⁡\(𝐱i​j​t′​𝜷i\)∑k∈Stexp⁡\(𝐱i​k​t′​𝜷i\)\.P\(j\\mid S\_\{t\};\\,\\bm\{\\beta\}\_\{i\}\)=\\frac\{\\exp\(\\mathbf\{x\}\_\{ijt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\)\}\{\\sum\_\{k\\in S\_\{t\}\}\\exp\(\\mathbf\{x\}\_\{ikt\}^\{\\prime\}\\bm\{\\beta\}\_\{i\}\)\}\.\(7\)
At the*population level*, individual parameters are assumed to follow a multivariate normal distribution,

𝜷i∼𝒩​\(𝝁,𝚺\),\\bm\{\\beta\}\_\{i\}\\sim\\mathcal\{N\}\(\\bm\{\\mu\},\\bm\{\\Sigma\}\),\(8\)with population mean𝝁\\bm\{\\mu\}and covariance𝚺\\bm\{\\Sigma\}estimated jointly with all individual parameters from the full panel\. The posterior distribution over𝜷i\\bm\{\\beta\}\_\{i\}is obtained by combining the individual choice likelihood implied by \([7](https://arxiv.org/html/2607.13314#A1.E7)\) with the population prior in \([8](https://arxiv.org/html/2607.13314#A1.E8)\)\.

### Shrinkage and Estimation

The key property of the posterior mean estimate is*shrinkage*: the estimate for consumeriican be written approximately as a weighted average of the individual’s own maximum likelihood estimate and the population mean𝝁\\bm\{\\mu\}, with weights determined by how many observations that person contributed\. A consumer with few observations receives a posterior pulled strongly toward𝝁\\bm\{\\mu\}; a consumer with many observations receives one close to their own choice history\. Information flows in both directions: each individual’s choices update the population distribution, and the population distribution regularizes each individual’s estimate\. This adaptive pooling is what allows HB to produce stable individual\-level estimates even from short purchase histories\.

Estimation proceeds via Markov chain Monte Carlo \(MCMC\), which yields full posterior distributions and individual\-level parameter draws\. These draws support downstream tasks such as consumer segmentation, targeting, and assortment optimization\. HB has been widely adopted in academic marketing research and commercial software\(Rossiet al\.[1996](https://arxiv.org/html/2607.13314#bib.bib53), Train[2009](https://arxiv.org/html/2607.13314#bib.bib3)\), and represents the state of practice for conjoint and scanner panel applications\.

### Relationship to Tabular Foundation Models

The TFM reformulation developed in the main text pursues the same goal as HB: recovering reliable individual\-level choice predictions from limited per\-consumer data\. The mechanisms differ\. In HB, pooling is explicit and parametric: the researcher specifies the prior, MCMC recovers the full posterior, and the amount of shrinkage each consumer receives is transparent\. In a TFM, pooling is implicit and amortized: a prior over data\-generating processes is learned during pretraining on synthetic data, and individual\-level adaptation occurs through the examples supplied in context at prediction time, without any gradient updates\. The cost is computational speed \(a single forward pass replaces MCMC\); the cost is that the learned prior is not directly interpretable\.