arXiv subjects

David T. Frazier

Publications and source records attributed to David T. Frazier.

At least 19 recordsLinked to original sources

arXiv2026

Scalable likelihood-based inference for limited dependent variable models

Limited dependent variable models are central to empirical economics, but likelihood-based inference is infeasible when likelihoods involve high-dimensional integration over latent variables. This paper proposes Stochastically Estimated Gradient Ascent (SEGA), a scalable estimation approach for limited dependent variable models. Using Fisher's identity, SEGA replaces the intractable likelihood score with an unbiased augmented-data score evaluated at a single conditional draw of the latent variables, and embeds this score in a stochastic gradient ascent algorithm. With sufficiently many iterations, we show that SEGA is asymptotically equivalent to the infeasible maximum likelihood estimator. A variance estimator based on Fisher's and Louis' identities is proposed that allows inference to proceed in the usual manner. Applications to brand choice and household demand demonstrate the usefulness of SEGA for conducting inference in large-scale discrete-choice and censored-demand models.

econ.EM↗

arXiv2026

Variational predictive resampling

Bayesian inference provides principled uncertainty quantification, but accurate posterior sampling with MCMC can be computationally prohibitive for modern applications. Variational inference (VI) offers a scalable alternative and often yields accurate predictive distributions, but cheap variational families such as mean-field (MF) can produce over-concentrated approximations that miss posterior dependence. We propose variational predictive resampling (VPR), a scalable posterior sampling method that exploits VI's predictive strength within a predictive-resampling framework to better approximate the Bayesian posterior. Given a prior-likelihood pair, VPR repeatedly imputes future observations from the current variational predictive, updates the variational approximation after each imputation, and records the parameter value implied by the completed sample. We establish conditions under which the law of the parameter returned by VPR is well defined and show that its finite-horizon approximation converges to this limit. In a tractable Gaussian location model, we show that VPR with MF variational predictives converges to the exact Bayesian posterior, whereas the optimal MF-VI approximation retains a non-vanishing asymptotic gap. Experiments on linear regression, logistic regression, and hierarchical linear mixed-effects models demonstrate that VPR substantially improves posterior uncertainty quantification and recovers posterior dependence missed by MF-VI, while remaining computationally competitive with, and often more efficient than, MCMC.

stat.ME↗

arXiv2026

Concentration and Calibration in Predictive Bayesian Inference

Predictive Bayesian inference (PBI) represents a model-and prior-agnostic approach to standard Bayesian inference which allows users to quantify uncertainty for a functional of interest only by specifying a forward predictive model for future unobserved data. The flexibility and generality of this framework have led to a host of novel algorithms for implementing this approach, and many empirical applications, yet the reliability of the resulting inferences for the underlying statistical functional of interest remains unclear. Herein, we demonstrate that when using PBI for a population functional of interest, the resulting posterior concentrates onto a well-defined quantity that explicitly depends on the forward predictive model used to implement the predictive recursion underlying the method. Furthermore, the forward predictive model entirely determines the uncertainty quantification produced in PBI. Consequently, our results show that if the predictive model does not capture all relevant features of the data, and, even in very simple examples, the coverage of predictive Bayes credible sets for the population value of the functional of interest can be arbitrarily close to zero. We carefully explain why this occurs, and show that this behavior is directly tied to the inaccuracy of the forward predictive model used to produce future observations within the PBI framework. As a consequence, our results imply that in order for PBI to deliver calibrated posterior inferences, the resulting predictive engine used to generate posterior samples must contain, in a well-defined sense, the true DGP, else inferences generated under this framework will not be calibrated.

stat.ME↗

arXiv2026

Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

Copula models of multivariate data are popular because they allow separate specification of marginal distributions and the copula function. These components can be treated as inter-related modules in a modified Bayesian inference approach called ''cutting feedback'' that is robust to their misspecification. Recent work uses a two module approach, where all $d$ marginals form a single module, to robustify inference for the marginals against copula function misspecification, or vice versa. However, marginals can exhibit differing levels of misspecification, making it attractive to assign each its own module with an individual influence parameter controlling its contribution to a joint semi-modular inference (SMI) posterior. This generalizes existing two module SMI methods, which interpolate between cut and conventional posteriors using a single influence parameter. We develop a novel copula SMI method and select the influence parameters using Bayesian optimization. It provides an efficient continuous relaxation of the discrete optimization problem over $2^d$ cut/uncut configurations. We establish theoretical properties of the resulting semi-modular posterior and demonstrate the approach on simulated and real data. The real data application uses a skew-normal copula model of asymmetric dependence between equity volatility and bond yields, where robustifying copula estimation against marginal misspecification is strongly motivated.

stat.ME↗

arXiv2026

Anomaly detection using surprisals

Anomaly detection methods are widely used but often rely on ad hoc rules or strong assumptions, and they often focus on tail events, missing ``inlier'' anomalies that occur in low-density gaps between modes. We propose a unified framework that defines an anomaly as an observation with unusually low probability under a (possibly misspecified) model. For each observation we compute its surprisal (the negative log generalized density) and define an anomaly score as the probability of a surprisal at least as large as that observed. This reduces anomaly detection for complex univariate or multivariate data to estimating the upper tail of a univariate surprisal distribution. We develop two model-robust estimators of these tail probabilities: an empirical estimator based on the observed surprisal distribution and an extreme-value estimator that fits a Generalized Pareto Distribution above a high threshold. For the empirical method we give conditions under which tail ordering is preserved and derive finite-sample confidence guarantees via the Dvoretzky--Kiefer--Wolfowitz inequality. For the GPD method we establish broad tail conditions ensuring classical extreme-value behavior. Simulations and applications to French mortality and Test-cricket data show the approach remains effective under substantial model misspecification.

stat.ME↗

arXiv2026

Preconditioned Robust Neural Posterior Estimation for Misspecified Simulators

Simulation-based inference (SBI) enables parameter estimation for complex stochastic models with intractable likelihoods when model simulation is feasible. Neural posterior estimation (NPE) is a popular SBI approach that often achieves accurate inference with far fewer simulations than classical approaches. But in practice, neural approaches can be unreliable for two reasons: incompatible data summaries arising from model misspecification yield unreliable posteriors due to extrapolation, and prior-predictive draws can produce extreme summaries that lead to difficulties in obtaining an accurate posterior for the observed data of interest. Existing preconditioning schemes target well-specified settings, and their behaviour under misspecification remains unexplored. We study preconditioning under misspecification and propose preconditioned robust neural posterior estimation, which computes data-dependent weights that focus training near the observed summaries and fits a robust neural posterior approximation. We also introduce a forest-proximity preconditioning approach that uses tree-based proximity scores to down-weight outlying simulations and concentrate computation around the observed dataset. Across two synthetic examples and one real example with incompatible summaries and extreme prior-predictive behaviour, we demonstrate that preconditioning combined with robust NPE increases stability and improves accuracy, calibration, and posterior-predictive fit over standard baseline methods.

stat.ME↗

arXiv2025

TabMGP: Martingale Posterior with TabPFN

Bayesian inference provides principled uncertainty quantification but is often limited by the challenges of prior and likelihood elicitation. The martingale posterior (MGP) (Fong et al., 2023) offers an alternative by replacing these requirements with a predictive rule. In addition, the MGP focuses inference on parameters defined through a loss function. This framework is especially resonant in the era of foundation transformers; practitioners increasingly leverage models like TabPFN for their state-of-the-art capabilities, yet often require epistemic uncertainty for a scientific estimand $\theta$ that need not parameterise the implicit latent model. The MGP provides a mechanism to recover these posterior distributions. We introduce TabMGP, an MGP built on TabPFN for tabular data. TabMGP produces credible sets with near-nominal coverage and often outperforms both handcrafted MGP constructions and standard Bayesian baselines.

stat.ME↗

arXiv2025

Predictively Oriented Posteriors

We advocate for a new statistical principle that combines the most desirable aspects of both parameter inference and density estimation. This leads us to the predictively oriented (PrO) posterior, which expresses uncertainty as a consequence of predictive ability. We show that these posteriors converge to the predictively optimal model average and predictively dominate both classical and generalised Bayes posterior predictive distributions. Further, PrO posteriors adapt to the level of model misspecification: while they concentrate around the true model in the same way as classical and generalised Bayesian strategies if the model can recover the data-generating distribution, they do not concentrate around a single model in the presence of non-trivial forms of model misspecification. Instead, they stabilise towards a non-degenerate predictively optimal posterior distribution that represents a form of irreducible uncertainty due to model misspecification. We put forward a sampling algorithm for PrO posteriors based on mean field Langevin dynamics, and verify the practical significance of our theoretical developments on a number of numerical examples.

stat.ME↗

arXiv2025

Optimization-centric cutting feedback for semiparametric models

Complex statistical models are often built by combining multiple submodels, called modules. Here we consider modular inference where the modules contain both parametric and nonparametric components. In such cases, standard Bayesian inference can be highly sensitive to misspecification in any module, and influential prior specifications for the nonparametric components can compromise inference for the parametric components, and vice versa. We propose a novel "optimization-centric" approach to cutting feedback for semiparametric modular inference, which can address misspecification and prior-data conflicts. The proposed cut posteriors are defined via a variational optimization problem like other generalized posteriors, but regularization is based on R\'enyi divergence, instead of Kullback-Leibler divergence (KLD). We show empirically that defining the cut posterior using R\'enyi divergence delivers more robust inference than KLD, and R\'enyi divergence reduces the tendency to underestimate uncertainty when the variational approximations impose strong parametric or independence assumptions. Novel posterior concentration results that accommodate the R\'enyi divergence and allow for semiparametric components are derived, extending existing results for cut posteriors that only apply to KLD and parametric models. These new methods are demonstrated in a benchmark example and two real examples: Gaussian process adjustments for confounding in causal inference and misspecified copula models with nonparametric marginals.

stat.ME↗

arXiv2025

Sequential Scoring Rule Evaluation for Forecast Method Selection

This paper shows that sequential statistical analysis techniques can be generalised to the problem of selecting between alternative forecasting methods using scoring rules. A return to basic principles is necessary in order to show that ideas and concepts from sequential statistical methods can be adapted and applied to sequential scoring rule evaluation (SSRE). One key technical contribution of this paper is the development of a large deviations type result for SSRE schemes using a change of measure that parallels a traditional exponential tilting form. Further, we also show that SSRE will terminate in finite time with probability one, and that the moments of the SSRE stopping time exist. A second key contribution is to show that the exponential tilting form underlying our large deviations result allows us to cast SSRE within the framework of generalised e-values. Relying on this formulation, we devise sequential testing approaches that are both powerful and maintain control on error probabilities underlying the analysis. Through several simulated examples, we demonstrate that our e-values based SSRE approach delivers reliable results that are more powerful than more commonly applied testing methods precisely in the situations where these commonly applied methods can be expected to fail.

math.ST↗

arXiv2025

Robustifying Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is one of the most popular "likelihood-free" methods. These methods have been applied in a wide range of fields by providing solutions to intractable likelihood problems in which exact Bayesian approaches are either infeasible or computationally costly. However, the performance of ABC can be unreliable when dealing with model misspecification. To circumvent the poor behavior of ABC in these settings, we propose a novel ABC approach that is robust to model misspecification. This new method can deliver more accurate statistical inference under model misspecification than alternatives and also enables the detection of summary statistics that are incompatible with the assumed data-generating process. We demonstrate the effectiveness of our approach through several simulated examples, where it delivers more accurate point estimates and uncertainty quantification over standard ABC approaches when the model is misspecified. Additionally, we apply our approach to an empirical example, further showcasing its advantages over alternative methods.

stat.ME↗

arXiv2025

Simulation-based Bayesian inference under model misspecification

Simulation-based Bayesian inference (SBI) methods are widely used for parameter estimation in complex models where evaluating the likelihood is challenging but generating simulations is relatively straightforward. However, these methods commonly assume that the simulation model accurately reflects the true data-generating process, an assumption that is frequently violated in realistic scenarios. In this paper, we focus on the challenges faced by SBI methods under model misspecification. We consolidate recent research aimed at mitigating the effects of misspecification, highlighting three key strategies: i) robust summary statistics, ii) generalised Bayesian inference, and iii) error modelling and adjustment parameters. To illustrate both the vulnerabilities of popular SBI methods and the effectiveness of misspecification-robust alternatives, we present empirical results on an illustrative example.

stat.ME↗

arXiv2024

Probabilistic Predictions of Option Prices with Modular Approximate Bayesian Inference

A new approximate Bayesian inferential framework is proposed that exploits multiple information sources -- daily spot returns, high-frequency spot data and option prices -- and enables fast calculation of probabilistic predictions of future option prices. This approach operates directly from the theoretical option pricing model, and does not require an explicit statistical model, or likelihood, for the observed option prices. We demonstrate that our approach produces accurate probabilistic option-price predictions in realistic scenarios and, despite not explicitly modelling option-pricing errors via a statistical model, the method is shown to be robust to the presence of such errors. Predictive accuracy based on the Heston option pricing model is illustrated empirically for short-maturity options, with the rapidity of real-time updates of the predictive distributions highlighted.

q-fin.ST↗

arXiv2024

The Statistical Accuracy of Neural Posterior and Likelihood Estimation

Neural posterior estimation (NPE) and neural likelihood estimation (NLE) are machine learning approaches that provide accurate posterior, and likelihood, approximations in complex modeling scenarios, and in situations where conducting amortized inference is a necessity. While such methods have shown significant promise across a range of diverse scientific applications, the statistical accuracy of these methods is so far unexplored. In this manuscript, we give, for the first time, an in-depth exploration on the statistical behavior of NPE and NLE. We prove that these methods have similar theoretical guarantees to common statistical methods like approximate Bayesian computation (ABC) and Bayesian synthetic likelihood (BSL). While NPE and NLE methods are just as accurate as ABC and BSL, we prove that this accuracy can often be achieved at a vastly reduced computational cost, and will therefore deliver more attractive approximations than ABC and BSL in certain problems. We verify our results theoretically and in several examples from the literature.

stat.ML↗

arXiv2024

Predictive performance of power posteriors

We analyse the impact of using tempered likelihoods in the production of posterior predictions. While the choice of temperature has an impact on predictive performance in small samples, we formally show that in moderate-to-large samples, tempering does not impact posterior predictions.

math.ST↗

arXiv2024

Exact Sampling of Gibbs Measures with Estimated Losses

A popular strategy for ameliorating some of the shortcomings of Bayesian posterior inference is to instead target a Gibbs measure based on losses that connect a parameter of interest to observed data, and which are known to be robust to misspecification. Existing theory for these procedures treats these losses as being analytically available, but in many situations these losses must be stochastically estimated using pseudo-observations. In these settings, and even under strong assumptions, we show that posteriors based on standard Markov chain Monte Carlo (MCMC) algorithms exhibit strong dependence on the number of these pseudo-observations, and require utilizing a diverging number of pseudo-observations to ensure posterior concentration. To remedy this issue, we introduce a modified piecewise deterministic Markov process (PDMP) sampler, and formally show that its posterior draws have no dependence on the number of pseudo-observations used to estimate the loss within a Gibbs measure. We verify the practical utility of this approach on five examples spanning intractable likelihoods and intractable losses.

math.ST↗

arXiv2024

Misspecification-robust Sequential Neural Likelihood for Simulation-based Inference

Simulation-based inference techniques are indispensable for parameter estimation of mechanistic and simulable models with intractable likelihoods. While traditional statistical approaches like approximate Bayesian computation and Bayesian synthetic likelihood have been studied under well-specified and misspecified settings, they often suffer from inefficiencies due to wasted model simulations. Neural approaches, such as sequential neural likelihood (SNL) avoid this wastage by utilising all model simulations to train a neural surrogate for the likelihood function. However, the performance of SNL under model misspecification is unreliable and can result in overconfident posteriors centred around an inaccurate parameter estimate. In this paper, we propose a novel SNL method, which through the incorporation of additional adjustment parameters, is robust to model misspecification and capable of identifying features of the data that the model is not able to recover. We demonstrate the efficacy of our approach through several illustrative examples, where our method gives more accurate point estimates and uncertainty quantification than SNL.

stat.ME↗

arXiv2023

Calibrated Generalized Bayesian Inference

We propose a simple approach that provides accurate uncertainty quantification for Bayesian inference in misspecified or approximate models, and for generalized (Gibbs) posteriors. While existing solutions in this context are based on explicit Gaussian approximations or post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by substituting the usual posterior with an intuitively appealing alternative that conveys the same information. This solution applies to both likelihood-based and loss-based posteriors, and is formally demonstrated to reliably quantify uncertainty. This new approach is demonstrated through a range of examples, including generalized linear models, and doubly intractable models.

stat.ME↗