Amortized Bayesian inference (ABI) offers fast, scalable approximations to posterior densities by training neural surrogates on data simulated from the statistical model. However, ABI methods are highly sensitive to model misspecification: when observed data fall outside the training distribution (generative scope of the statistical models), neural surrogates can behave unpredictably. This makes it a challenge in a model comparison setting, where multiple statistical models are considered, of which at least some are misspecified. Recent work on self-consistency (SC) provides a promising remedy to this issue, accessible even for empirical data (without ground-truth labels). In this work, we investigate how SC can improve amortized model comparison conceptualized in four different ways. Across two synthetic and two real-world case studies, we find that approaches for model comparison that estimate marginal likelihoods through approximate parameter posteriors consistently outperform methods that directly approximate model evidence or posterior model probabilities. SC training improves robustness when the likelihood is available, even under severe model misspecification. The benefits of SC for methods without access of analytic likelihoods are more limited and inconsistent. Our results suggest practical guidance for reliable amortized Bayesian model comparison: prefer parameter posterior-based methods and augment them with SC training on empirical datasets to mitigate extrapolation bias under model misspecification.

As regulatory agencies increasingly recognise real-world evidence as a complement to traditional clinical trial data, interest has grown in applying Bayesian methods across both interventional and observational research (Boulanger and Carlin (2021). A central objective in many clinical investigations is the delineation of patient subgroups that exhibit comparable disease-related characteristics (He, Belouali, Patricoski, Lehmann, Ball, Anagnostou, Kreimeyer, and Botsis (2023)). Electronic Health Records (EHR) have become an important resource for such phenotypic analyses (Hripcsak and Albers (2013)). Bayesian approaches to patient phenotyping in clinical observational studies have been limited by the computational challenges associated with applying the Markov Chain Monte Carlo (MCMC) approach to real-world data. Hubbard, Huang, Harton, Oganisian, Choi, Utidjian, Eneli, Bailey, and Chen (2019) proposed a Bayes latent class model that could be used in a general context for observational studies that use EHR data. They consider the common clinical context where gold-standard phenotype information, such as genetic and laboratory data, is not fully available. A general model of this form has high potential applicability for use in clinical decision support across disease areas for both primary and secondary clinical databases. Latent Class Analysis (LCA) is widely used when we want to identify patient phenotypes or subgroups given multivariate data (Lanza and Rhoades (2013)). A challenge in clinical LCA is the prevalence of mixed data, where we may have combinations of continuous, nominal, ordinal and count data.

Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.

Covariate adjustment is an approach to improve the precision of trial analyses by adjusting for baseline variables that are prognostic of the primary endpoint. Motivated by the SEARCH Universal HIV Test-and-Treat Trial (2013-2017), we tell our story of developing, evaluating, and implementing a machine learning-based approach for covariate adjustment. We provide the rationale for as well as the practical concerns with such an approach for estimating marginal effects. Using schematics, we illustrate our procedure: targeted machine learning estimation (TMLE) with Adaptive Pre-specification. Briefly, sample-splitting is used to data-adaptively select the combination of estimators of the outcome regression (i.e., the conditional expectation of the outcome given the trial arm and covariates) and known propensity score (i.e., the conditional probability of being randomized to the intervention given the covariates) that minimizes the cross-validated variance estimate and, thereby, maximizes empirical efficiency. We discuss our approach for evaluating finite sample performance with parametric and plasmode simulations, pre-specifying the Statistical Analysis Plan, and unblinding in real-time on video conference with our colleagues from around the world. We present the results from applying our approach in the primary, pre-specified analysis of 8 recently published trials (2022-2024). We conclude with practical recommendations and an invitation to implement our approach in the primary analysis of your next trial.

Mixture models are widely used in modeling heterogeneous data populations. A standard approach of mixture modeling is to assume that the mixture component takes a parametric kernel form, while the flexibility of the model can be obtained by using a large or possibly unbounded number of such parametric kernels. In many applications, making parametric assumptions on the latent subpopulation distributions may be unrealistic, which motivates the need for nonparametric modeling of the mixture components themselves. In this paper we study finite mixtures with nonparametric mixture components, using a Bayesian nonparametric modeling approach. In particular, it is assumed that the data population is generated according to a finite mixture of latent component distributions, where each component is endowed with a Bayesian nonparametric prior such as the Dirichlet process mixture. We present conditions under which the individual mixture component's distributions can be identified, and establish posterior contraction behavior for the data population's density, as well as densities of the latent mixture components. We develop an efficient MCMC algorithm for posterior inference and demonstrate via simulation studies and real-world data illustrations that it is possible to efficiently learn complex distributions for the latent subpopulations. In theory, the posterior contraction rate of the component densities is nearly polynomial, which is a significant improvement over the logarithm convergence rate of estimating mixing measures via deconvolution.

We propose a robust and scalable variational Bayes (VB) framework designed to effectively handle contamination and outliers in dataset. Our approach partitions the data into $m$ disjoint subsets and formulates a joint optimization problem based on robust aggregation principles. A key insight is that the full posterior distribution is equivalent to the minimizer of the mean Kullback-Leibler (KL) divergence from the $m$-powered local posterior distributions. To enhance robustness, we replace the mean KL divergence with a min-max median formulation. The min-max formulation not only ensures consistency between the KL minimizer and the Evidence Lower Bound (ELBO) maximizer but also facilitates the establishment of improved statistical rates for the mean of variational posterior. We observe a notable discrepancy in the $m$-powered marginal log likelihood function contingent on the presence of local latent variables. To address this, we treat these two scenarios separately to guarantee the consistency of the aggregated variational posterior. Specifically, when local latent variables are present, we introduce an aggregate-and-rescale strategy. Theoretically, we provide a non-asymptotic analysis of our proposed posterior, incorporating a refined analysis of Bernstein-von Mises (BvM) theorem to accommodate a diverging number of subsets $m$. Our findings indicate that the two-stage approach yields a smaller approximation error compared to directly aggregating the $m$-powered local posteriors. Furthermore, we establish a nearly optimal statistical rate for the mean of the proposed posterior, advancing existing theories related to min-max median estimators. The efficacy of our method is demonstrated through extensive simulation studies.

Proper quantification of predictive uncertainty is essential for the use of machine learning in safety-critical applications. V arious uncertainty measures have been proposed for this purpose, typically claiming superiority over other measures. In this paper, we argue that there is no single best measure. Instead, uncertainty quantification should be tailored to the specific application. To this end, we use a flexible family of uncertainty measures that distinguishes between total, aleatoric, and epistemic uncertainty of second-order distributions. These measures can be instantiated with specific loss functions, so-called proper scoring rules, to control their characteristics, and we show that different characteristics are useful for different tasks. In particular, we show that, for the task of selective prediction, the scoring rule should ideally match the task loss. On the other hand, for out-of-distribution detection, our results confirm that mutual information, a widely used measure of epistemic uncertainty, performs best. Furthermore, in an active learning setting, epistemic uncertainty based on zero-one loss is shown to consistently outperform other uncertainty measures.

Model selection is a pivotal process in the quantitative sciences, where researchers must navigate between numerous candidate models of varying complexity. Traditional information criteria, such as the corrected Akaike Information Criterion (AICc), Bayesian Information Criterion (BIC), and Mallows's $\mathit{C_p}$, are valuable tools for identifying optimal models. However, the exponential increase in candidate models with each additional model parameter renders the evaluation of these criteria for all models -- a strategy known as exhaustive, or brute-force, searches -- computationally prohibitive. Consequently, heuristic approaches like stepwise regression are commonly employed, albeit without guarantees of finding the globally-optimal model. In this study, we challenge the prevailing notion that non-monotonicity in information criteria precludes bounds on the search space. We introduce a simple but novel bound that enables the development of branch-and-bound algorithms tailored for these non-monotonic functions. We demonstrate that our approach guarantees identification of the optimal model(s) across diverse model classes, sizes, and applications, often with orders of magnitude computational speedups. For instance, in one previously-published model selection task involving $2^{32}$ (approximately 4 billion) candidate models, our method achieves a computational speedup exceeding 6,000. These findings have broad implications for the scalability and effectiveness of model selection in complex scientific domains.

Adaptive behavior in volatile environments requires agents to deploy different value-control regimes across latent contexts, but representing separate preferences, policy biases, and action confidence for every situation is intractable. We introduce value profiles: a small set of reusable bundles of value-related parameters--outcome preferences, policy priors, and policy precision--that are assigned to hidden states in the generative model. As posterior beliefs over states evolve trial-by-trial, effective control parameters emerge through belief-weighted mixing, enabling state-conditional strategy recruitment without maintaining independent parameters for each situation. We evaluate this framework in probabilistic reversal learning, comparing static precision, entropy-coupled dynamic precision, and profile-based models using cross-validated log-likelihood and information criteria. Model comparison using AIC favors the profile-based model over simpler alternatives ( 100-point differences), with consistent parameter recovery demonstrating structural identifiability even when context must be inferred from noisy observations. Model-based inference suggests that, in this task, adaptive control operates primarily through policy prior modulation rather than policy precision modulation, with gradual belief-driven profile recruitment confirming state-conditional rather than merely uncertainty-driven control. Overall, reusable value profiles provide a tractable computational account of belief-conditioned value control in volatile environments, providing a reusable, mode-like representational scheme for behavioral flexibility that yields testable signatures of belief-conditioned control.

For four decades statistical physics has been providing a framework to analyse neural networks. A long-standing question remained on its capacity to tackle deep learning models capturing rich feature learning effects, thus going beyond the narrow networks or kernel methods analysed until now. We positively answer through the study of the supervised learning of a multi-layer perceptron. Importantly, (i) its width scales as the input dimension, making it more prone to feature learning than ultra wide networks, and more expressive than narrow ones or ones with fixed embedding layers; and (ii) we focus on the challenging interpolation regime where the number of trainable parameters and data are comparable, which forces the model to adapt to the task. We consider the matched teacher-student setting. Therefore, we provide the fundamental limits of learning random deep neural network targets and identify the sufficient statistics describing what is learnt by an optimally trained network as the data budget increases. A rich phenomenology emerges with various learning transitions. With enough data, optimal performance is attained through the model's "specialisation" towards the target, but it can be hard to reach for training algorithms which get attracted by sub-optimal solutions predicted by the theory. Specialisation occurs inhomogeneously across layers, propagating from shallow towards deep ones, but also across neurons in each layer. Furthermore, deeper targets are harder to learn. Despite its simplicity, the Bayes-optimal setting provides insights on how the depth, non-linearity and finite (proportional) width influence neural networks in the feature learning regime that are potentially relevant in much more general settings.