Goodness-of-fit (GoF) tests are fundamental for assessing model adequacy. Score-based tests are appealing because they require fitting the model only once under the null. However, extending them to powerful nonparametric alternatives is difficult due to the lack of suitable score functions. Through a class of exponentially tilted models, we show that the resulting score-based GoF tests are equivalent to the tests based on integral probability metrics (IPMs) indexed by a function class. When the class is rich, the test is universally consistent. This simple yet insightful perspective enables reinterpretation of classical distance-based testing procedures-including those based on Kolmogorov-Smirnov distance, Wasserstein-1 distance, and maximum mean discrepancy-as arising from score-based constructions. Building on this insight, we propose a new nonparametric score-based GoF test through a special class of IPM induced by kernelized Stein's function class, called semiparametric kernelized Stein discrepancy (SKSD) test. Compared with other nonparametric score-based tests, the SKSD test is computationally efficient and accommodates general nuisance-parameter estimators, supported by a generic parametric bootstrap procedure. The SKSD test is universally consistent and attains Pitman efficiency. Moreover, SKSD test provides simple GoF tests for models with intractable likelihoods but tractable scores with the help of Stein's identity and we use two popular models, kernel exponential family and conditional Gaussian models, to illustrate the power of our method. Our method achieves power comparable to task-specific normality tests such as Anderson-Darling and Lilliefors, despite being designed for general nonparametric alternatives.

Unmeasured confounding can render identification strategies based on adjustment functionals invalid. We study the "Napkin graph", a causal structure that encapsulates patterns of M-bias, instrumental variables, and the classical back-door and front-door models within a single graphical framework, yet requires a nonstandard identification strategy: the average treatment effect is expressed as a ratio of two g-formulas. We develop novel estimators for this functional, including doubly robust one-step and targeted minimum loss-based estimators that remain asymptotically linear when nuisance functions are estimated at slower-than-parametric rates using machine learning. We also show how a generalized independence restriction encoded by the Napkin graph, known as a Verma constraint, can be exploited to improve efficiency, illustrating more generally how such constraints in hidden variable DAGs can inform semiparametric inference. The proposed methods are validated through simulations and applied to the Finnish Life Course study to estimate the effect of educational attainment on income. An accompanying R package, napkincausal, implements all proposed procedures.

This letter extends the exactly sparse Gaussian variational inference (ESGVI) algorithm for state estimation in two complementary directions. First, ESGVI is generalized to operate on matrix Lie groups, enabling the estimation of states with orientation components while respecting the underlying group structure. Second, factors are introduced to accommodate heavy-tailed and skewed noise distributions, as commonly encountered in ultra-wideband (UWB) localization due to non-line-of-sight (NLOS) and multipath effects. Both extensions are shown to integrate naturally within the ESGVI framework while preserving its sparse and derivative-free structure. The proposed approach is validated in a UWB localization experiment with NLOS-rich measurements, demonstrating improved accuracy and comparable consistency. Finally, a Python implementation within a factor-graph-based estimation framework is made open-source (https://github.com/decargroup/gvi_ws) to support broader research use.

Empirical Bayes (EB) improves the accuracy of simultaneous inference "by learning from the experience of others" (Efron, 2012). Classical EB theory focuses on latent variables that are iid draws from a fitted prior (Efron, 2019). Modern applications, however, feature complex structure, like arrays, spatial processes, or covariates. How can we apply EB ideas to these settings? We propose a generalized approach to empirical Bayes based on the notion of probabilistic symmetry. Our method pairs a simultaneous inference problem-with an unknown prior-to a symmetry assumption on the joint distribution of the latent variables. Each symmetry implies an ergodic decomposition, which we use to derive a corresponding empirical Bayes method. We call this methodBayesian empirical Bayes (BEB). We show how BEB recovers the classical methods of empirical Bayes, which implicitly assume exchangeability. We then use it to extend EB to other probabilistic symmetries: (i) EB matrix recovery for arrays and graphs; (ii) covariate-assisted EB for conditional data; (iii) EB spatial regression under shift invariance. We develop scalable algorithms based on variational inference and neural networks. In simulations, BEB outperforms existing approaches to denoising arrays and spatial data. On real data, we demonstrate BEB by denoising a cancer gene-expression matrix and analyzing spatial air-quality data from New York City.

Multivariate zero-inflated count data arise in a wide range of areas such as economics, social sciences, and biology. To infer causal relationships in zero-inflated count data, we propose a new zero-inflated Poisson Bayesian network (ZIPBN) model. We show that the proposed ZIPBN is identifiable with cross-sectional data. The proof is based on the well-known characterization of Markov equivalence class which is applicable to other distribution families. For causal structural learning, we introduce a fully Bayesian inference approach which exploits the parallel tempering Markov chain Monte Carlo algorithm to efficiently explore the multi-modal network space. We demonstrate the utility of the proposed ZIPBN in causal discoveries for zero-inflated count data by simulation studies with comparison to alternative Bayesian network methods. Additionally, real single-cell RNA-sequencing data with known causal relationships will be used to assess the capability of ZIPBN for discovering causal relationships in real-world problems.

We study the problem of inferring time-varying Gaussian Markov random fields, where the underlying graphical model is both sparse and changes {sparsely} over time. Most of the existing methods for the inference of time-varying Markov random fields (MRFs) rely on the \textit{regularized maximum likelihood estimation} (MLE), that typically suffer from weak statistical guarantees and high computational time. Instead, we introduce a new class of constrained optimization problems for the inference of sparsely-changing Gaussian MRFs (GMRFs). The proposed optimization problem is formulated based on the exact $\ell_0$ regularization, and can be solved in near-linear time and memory. Moreover, we show that the proposed estimator enjoys a provably small estimation error. We derive sharp statistical guarantees in the high-dimensional regime, showing that such problems can be learned with as few as one sample per time period. Our proposed method is extremely efficient in practice: it can accurately estimate sparsely-changing GMRFs with more than 500 million variables in less than one hour.

Virtually any model we use in machine learning to make predictions does not perfectly represent reality. So, most of the learning happens under model misspecification. In this work, we present a novel analysis of the generalization performance of Bayesian model averaging under model misspecification and i.i.d.

The key distinguishing property of a Bayesian approach is marginalization, rather than using a single setting of weights. Bayesian marginalization can particularly improve the accuracy and calibration of modern deep neural networks, which are typically underspecified by the data, and can represent many compelling but different solutions. We show that deep ensembles provide an effective mechanism for approximate Bayesian marginalization, and propose a related approach that further improves the predictive distribution by marginalizing within basins of attraction, without significant overhead. We also investigate the prior over functions implied by a vague distribution over neural network weights, explaining the generalization properties of such models from a probabilistic perspective. From this perspective, we explain results that have been presented as mysterious and distinct to neural network generalization, such as the ability to fit images with random labels, and show that these results can be reproduced with Gaussian processes. We also show that Bayesian model averaging alleviates double descent, resulting in monotonic performance improvements with increased flexibility.

Understanding the consequences of mutation for molecular fitness and function is a fundamental problem in biology. Recently, generative probabilistic models have emerged as a powerful tool for estimating fitness from evolutionary sequence data, with accuracy sufficient to predict both laboratory measurements of function and disease risk in humans, and to design novel functional proteins. Existing techniques rest on an assumed relationship between density estimation and fitness estimation, a relationship that we interrogate in this article. We prove that fitness is not identifiable from observational sequence data alone, placing fundamental limits on our ability to disentangle fitness landscapes from phylogenetic history. We show on real datasets that perfect density estimation in the limit of infinite data would, with high confidence, result in poor fitness estimation; current models perform accurate fitness estimation because of, not despite, misspecification. Our results challenge the conventional wisdom that bigger models trained on bigger datasets will inevitably lead to better fitness estimation, and suggest novel estimation strategies going forward.

Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming relaxations have long been a theoretically powerful tool for analyzing properties of probabilistic inference, but have not been practical owing to the high computational cost of typical solvers for solving the resulting SDPs. In this paper, we propose an efficient method for computing the partition function or MAP estimate in a pairwise MRF by instead exploiting a recently proposed coordinate-descent-based fast semidefinite solver. We also extend semidefinite relaxations from the typical binary MRF to the full multi-class setting, and develop a compact semidefinite relaxation that can again be solved efficiently using the solver. We show that the method substantially outperforms (both in terms of solution quality and speed) the existing state of the art in approximate inference, on benchmark problems drawn from previous work. We also show that our approach can scale to large MRF domains such as fully-connected pairwise CRF models used in computer vision.