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.

Markov Chain Monte Carlo (MCMC), Laplace approximation (LA) and variational inference (VI) methods are popular approaches to Bayesian inference, each with trade-offs between computational cost and accuracy. However, a theoretical understanding of these differences is missing, particularly when both the sample size $n$ and the dimension $d$ are large. LA and Gaussian VI are justified by Bernstein-von Mises (BvM) theorems, and recent work has derived the characteristic condition $n\gg d^2$ for their validity, improving over the condition $n\gg d^3$. In this paper, we show for linear, logistic and Poisson regression that for $n\gtrsim d$, MCMC attains the same complexity scaling in $n$, $d$ as first-order optimization algorithms, up to sub-polynomial factors. Thus MCMC is competitive with LA and Gaussian VI in complexity, under a scaling between $n$ and $d$ more general than BvM regimes. Our complexities apply to appropriately scaled priors that are not necessarily Gaussian-tailed, including Student-$t$ and flat priors, with log-posteriors that are not necessarily globally concave or gradient-Lipschitz.

We present a framework using variational inference with normalizing flows (VI-NFs) to generate proposals of reversible jump Markov chain Monte Carlo (RJMCMC) for efficient trans-dimensional Bayesian inference. Unlike transport reversible jump methods relying on forward KL minimization with pilot MCMC samples, our approach minimizes the reverse KL divergence which requires only samples from a base distribution, eliminating costly target sampling. The method employs RealNVP-based flows to learn model-specific transport maps, enabling construction of both between-model and within-model proposals. Our framework provides accurate marginal likelihood estimates from the variational approximation. This facilitates efficient model comparison and proposal adaptation in RJMCMC. Experiments on illustrative example, factor analysis and variable selection tasks in linear regression show that TRJ designed by VI-NFs achieves faster mixing and more efficient model space exploration compared to existing baselines. The proposed algorithm can be extended to conditional flows for amortized vairiational inference across models. Code is available at https://github.com/YinPingping111/TRJ_VINFs.

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.

Temporal-difference (TD) methods learn state and action values efficiently by bootstrapping from their own future value predictions, but such a self-bootstrapping mechanism is prone to bootstrapping bias, where the errors in the value targets accumulate across steps and result in biased value estimates. Recent work has proposed to use chunked critics, which estimate the value of short action sequences ("chunks") rather than individual actions, speeding up value backup. However, extracting policies from chunked critics is challenging: policies must output the entire action chunk open-loop, which can be sub-optimal for environments that require policy reactivity and also challenging to model especially when the chunk length grows. Our key insight is to decouple the chunk length of the critic from that of the policy, allowing the policy to operate over shorter action chunks. We propose a novel algorithm that achieves this by optimizing the policy against a distilled critic for partial action chunks, constructed by optimistically backing up from the original chunked critic to approximate the maximum value achievable when a partial action chunk is extended to a complete one. This design retains the benefits of multi-step value propagation while sidestepping both the open-loop sub-optimality and the difficulty of learning action chunking policies for long action chunks. We evaluate our method on challenging, long-horizon offline goal-conditioned tasks and show that it reliably outperforms prior methods. Code: github.com/ColinQiyangLi/dqc.

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.