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.

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.

The property of learning-curve monotonicity, highlighted in a recent series of work by Loog, Mey and Viering, describes algorithms which only improve in average performance given more data, for any underlying data distribution within a given family. We establish the first nontrivial monotonicity guarantees for the maximum likelihood estimator in a variety of well-specified parametric settings. For sequential prediction with log loss, we show monotonicity (in fact complete monotonicity) of the forward KL divergence for Gaussian vectors with unknown covariance and either known or unknown mean, as well as for Gamma variables with unknown scale parameter. The Gaussian setting was explicitly highlighted as open in the aforementioned works, even in dimension 1. Finally we observe that for reverse KL divergence, a folklore trick yields monotonicity for very general exponential families. All results in this paper were derived by variants of GPT-5.2 Pro. Humans did not provide any proof strategies or intermediate arguments, but only prompted the model to continue developing additional results, and verified and transcribed its proofs.

Causal discovery methods have traditionally been developed under two distinct regimes: independent and identically distributed (i.i.d.) and timeseries data, each governed by separate modelling assumptions. In this paper, we argue that the i.i.d. setting can and should be reframed in terms of exchangeability, a strictly more general symmetry principle. We present the implications of this reframing, alongside two core arguments: (1) a conceptual argument, based on extending the dependency of experimental causal inference on exchangeability to causal discovery; and (2) an empirical argument, showing that many existing i.i.d. causal-discovery methods are predicated on exchangeability assumptions, and that the sole extensive widely-used real-world "i.i.d." benchmark (the Tübingen dataset) consists mainly of exchangeable (and not i.i.d.) examples. Building on this insight, we introduce a novel synthetic dataset that enforces only the exchangeability assumption, without imposing the stronger i.i.d. assumption. We show that our exchangeable synthetic dataset mirrors the statistical structure of the real-world "i.i.d." dataset more closely than all other i.i.d. synthetic datasets. Furthermore, we demonstrate the predictive capability of this dataset by proposing a neural-network-based causal-discovery algorithm trained exclusively on our synthetic dataset, and which performs similarly to other state-of-the-art i.i.d. methods on the real-world benchmark.

This article looks at how reasoning works in current Large Language Models (LLMs) that function using the token-completion method. It examines their stochastic nature and their similarity to human abductive reasoning. The argument is that these LLMs create text based on learned patterns rather than performing actual abductive reasoning. When their output seems abductive, this is largely because they are trained on human-generated texts that include reasoning structures. Examples are used to show how LLMs can produce plausible ideas, mimic commonsense reasoning, and give explanatory answers without being grounded in truth, semantics, verification, or understanding, and without performing any real abductive reasoning. This dual nature, where the models have a stochastic base but appear abductive in use, has important consequences for how LLMs are evaluated and applied. They can assist with generating ideas and supporting human thinking, but their outputs must be critically assessed because they cannot identify truth or verify their explanations. The article concludes by addressing five objections to these points, noting some limitations in the analysis, and offering an overall evaluation.

Reproducibility is a cornerstone of scientific validation and of the authority it confers on its results. Reproducibility in machine learning evaluations leads to greater trust, confidence, and value. However, the ground truth responses used in machine learning often necessarily come from humans, among whom disagreement is prevalent, and surprisingly little research has studied the impact of effectively ignoring disagreement in these responses, as is typically the case. One reason for the lack of research is that budgets for collecting human-annotated evaluation data are limited, and obtaining more samples from multiple raters for each example greatly increases the per-item annotation costs. We investigate the trade-off between the number of items ($N$) and the number of responses per item ($K$) needed for reliable machine learning evaluation. We analyze a diverse collection of categorical datasets for which multiple annotations per item exist, and simulated distributions fit to these datasets, to determine the optimal $(N, K)$ configuration, given a fixed budget ($N \times K$), for collecting evaluation data and reliably comparing the performance of machine learning models. Our findings show, first, that accounting for human disagreement may come with $N \times K$ at no more than 1000 (and often much lower) for every dataset tested on at least one metric. Moreover, this minimal $N \times K$ almost always occurred for $K > 10$. Furthermore, the nature of the tradeoff between $K$ and $N$, or if one even existed, depends on the evaluation metric, with metrics that are more sensitive to the full distribution of responses performing better at higher levels of $K$. Our methods can be used to help ML practitioners get more effective test data by finding the optimal metrics and number of items and annotations per item to collect to get the most reliability for their budget.

We propose geodesic-based optimization methods on dually flat spaces, where the geometric structure of the parameter manifold is closely related to the form of the objective function. A primary application is maximum likelihood estimation in statistical models, especially exponential families, whose model manifolds are dually flat. We show that an m-geodesic update, which directly optimizes the log-likelihood, can theoretically reach the maximum likelihood estimator in a single step. In contrast, an e-geodesic update has a practical advantage in cases where the parameter space is geodesically complete, allowing optimization without explicitly handling parameter constraints. We establish the theoretical properties of the proposed methods and validate their effectiveness through numerical experiments.

Applied researchers now routinely work with regression models that feature a large number of covariates. A primary inferential goal in econometrics is to estimate the ceteris paribus effect of a specific variable while controlling for other variables (Belloni et al., 2013a, 2018). The prevailing practice interprets the coefficient on a regressor as a causal effect, conditional on the included controls. As the plausibility of conditional unconfoundedness is often argued using a large set of covariates, practitioners have increasingly embraced high-dimensional regression models. This setting has been extensively studied, predominantly using frequentist methods. Bayesian inference, on the other hand, has long been valued for its coherent framework for handling uncertainty in statistical analysis. As highlighted by Rubin (1984), Bayesian methods provide direct answers to many empirical questions by quantifying uncertainty about unknown parameters conditional on the observed data.