Deep learning has generated diverse perspectives in astronomy, with ongoing discussions between proponents and skeptics motivating this review. We examine how neural networks complement classical statistics, extending our data analytical toolkit for modern surveys. Astronomy offers unique opportunities through encoding physical symmetries, conservation laws, and differential equations directly into architectures, creating models that generalize beyond training data. Yet challenges persist as unlabeled observations number in billions while confirmed examples with known properties remain scarce and expensive. This review demonstrates how deep learning incorporates domain knowledge through architectural design, with built-in assumptions guiding models toward physically meaningful solutions. We evaluate where these methods offer genuine advances versus claims requiring careful scrutiny. - Neural architectures overcome trade-offs between scalability, expressivity, and data efficiency by encoding physical symmetries and conservation laws into network structure, enabling learning from limited labeled data. - Simulation-based inference and anomaly detection extract information from complex, non-Gaussian distributions where analytical likelihoods fail, enabling field-level cosmological analysis and systematic discovery of rare phenomena. - Multi-scale neural modeling bridges resolution gaps in astronomical simulations, learning effective subgrid physics from expensive high-fidelity runs to enhance large-volume calculations where direct computation remains prohibitive. - Emerging paradigms-reinforcement learning for telescope operations, foundation models learning from minimal examples, and large language model agents for research automation-show promise though are still developing in astronomical applications.

Belief systems are rarely globally consistent, yet effective reasoning often persists locally. We propose a novel graph-theoretic framework that cleanly separates credibility--external, a priori trust in sources--from confidence--an internal, emergent valuation induced by network structure. Beliefs are nodes in a directed, signed, weighted graph whose edges encode support and contradiction. Confidence is obtained by a contractive propagation process that mixes a stated prior with structure-aware influence and guarantees a unique, stable solution. Within this dynamics, we define reasoning zones: high-confidence, structurally balanced subgraphs on which classical inference is safe despite global contradictions. We provide a near-linear procedure that seeds zones by confidence, tests balance using a parity-based coloring, and applies a greedy, locality-preserving repair with Jaccard de-duplication to build a compact atlas. To model belief change, we introduce shock updates that locally downscale support and elevate targeted contradictions while preserving contractivity via a simple backtracking rule. Re-propagation yields localized reconfiguration-zones may shrink, split, or collapse--without destabilizing the entire graph. We outline an empirical protocol on synthetic signed graphs with planted zones, reporting zone recovery, stability under shocks, and runtime. The result is a principled foundation for contradiction-tolerant reasoning that activates classical logic precisely where structure supports it.

Diffusion Posterior Sampling (DPS) provides a principled Bayesian approach to inverse problems by sampling from $p(x_0 \mid y)$. However, in practice, the goal of inverse problem solving is not to cover the posterior but to recover the most accurate reconstruction, where optimization-based diffusion solvers often excel despite lacking a clear probabilistic foundation. We introduce Local MAP Sampling (LMAPS), a new inference framework that iteratively solving local MAP subproblems along the diffusion trajectory. This perspective clarifies their connection to global MAP estimation and DPS, offering a unified probabilistic interpretation for optimization-based methods. Building on this foundation, we develop practical algorithms with a probabilistically interpretable covariance approximation, a reformulated objective for stability and interpretability, and a gradient approximation for non-differentiable operators. Across a broad set of image restoration and scientific tasks, LMAPS achieves state-of-the-art performance, including $\geq 2$ dB gains on motion deblurring, JPEG restoration, and quantization, and $>1.5$ dB improvements on inverse scattering benchmarks.

Conditional sampling is a fundamental task in Bayesian statistics and generative modeling. Consider the problem of sampling from the posterior distribution $P_{X|Y=y^*}$ for some observation $y^*$, where the likelihood $P_{Y|X}$ is known, and we are given $n$ i.i.d. samples $D=\{X_i\}_{i=1}^n$ drawn from an unknown prior distribution $π_X$. Suppose that $f(\hatπ_{X^n})$ is the distribution of a posterior sample generated by an algorithm (e.g. a conditional generative model or the Bayes rule) when $\hatπ_{X^n}$ is the empirical distribution of the training data. Although averaging over the randomness of the training data $D$, we have $\mathbb{E}_D\left(\hatπ_{X^n}\right)= π_X$, we do not have $\mathbb{E}_D\left\{f(\hatπ_{X^n})\right\}= f(π_X)$ due to the nonlinearity of $f$, leading to a bias. In this paper we propose a black-box debiasing scheme that improves the accuracy of such a naive plug-in approach. For any integer $k$ and under boundedness of the likelihood and smoothness of $f$, we generate samples $\hat{X}^{(1)},\dots,\hat{X}^{(k)}$ and weights $w_1,\dots,w_k$ such that $\sum_{i=1}^kw_iP_{\hat{X}^{(i)}}$ is a $k$-th order approximation of $f(π_X)$, where the generation process treats $f$ as a black-box. Our generation process achieves higher accuracy when averaged over the randomness of the training data, without degrading the variance, which can be interpreted as improving memorization without compromising generalization in generative models.

This paper develops a finite-sample statistical theory for in-context learning (ICL), analyzed within a meta-learning framework that accommodates mixtures of diverse task types. We introduce a principled risk decomposition that separates the total ICL risk into two orthogonal components: Bayes Gap and Posterior Variance. The Bayes Gap quantifies how well the trained model approximates the Bayes-optimal in-context predictor. For a uniform-attention Transformer, we derive a non-asymptotic upper bound on this gap, which explicitly clarifies the dependence on the number of pretraining prompts and their context length. The Posterior Variance is a model-independent risk representing the intrinsic task uncertainty. Our key finding is that this term is determined solely by the difficulty of the true underlying task, while the uncertainty arising from the task mixture vanishes exponentially fast with only a few in-context examples. Together, these results provide a unified view of ICL: the Transformer selects the optimal meta-algorithm during pretraining and rapidly converges to the optimal algorithm for the true task at test time.

We provide a unified algorithmic framework for ensemble sampling in nonlinear contextual bandits and develop corresponding regret bounds for two most common nonlinear contextual bandit settings: Generalized Linear Ensemble Sampling (\texttt{GLM-ES}) for generalized linear bandits and Neural Ensemble Sampling (\texttt{Neural-ES}) for neural contextual bandits. Both methods maintain multiple estimators for the reward model parameters via maximum likelihood estimation on randomly perturbed data. We prove high-probability frequentist regret bounds of $\mathcal{O}(d^{3/2} \sqrt{T} + d^{9/2})$ for \texttt{GLM-ES} and $\mathcal{O}(\widetilde{d} \sqrt{T})$ for \texttt{Neural-ES}, where $d$ is the dimension of feature vectors, $\widetilde{d}$ is the effective dimension of a neural tangent kernel matrix, and $T$ is the number of rounds. These regret bounds match the state-of-the-art results of randomized exploration algorithms in nonlinear contextual bandit settings. In the theoretical analysis, we introduce techniques that address challenges specific to nonlinear models. Practically, we remove fixed-time horizon assumptions by developing anytime versions of our algorithms, suitable when $T$ is unknown. Finally, we empirically evaluate \texttt{GLM-ES}, \texttt{Neural-ES}, and their anytime variants, demonstrating strong performance. Overall, our results establish ensemble sampling as a provable and practical randomized exploration approach for nonlinear contextual bandits.

We derive a novel PAC-Bayesian generalization bound for reinforcement learning that explicitly accounts for Markov dependencies in the data, through the chain's mixing time. This contributes to overcoming challenges in obtaining generalization guarantees for reinforcement learning, where the sequential nature of data breaks the independence assumptions underlying classical bounds. Our bound provides non-vacuous certificates for modern off-policy algorithms like Soft Actor-Critic. We demonstrate the bound's practical utility through PB-SAC, a novel algorithm that optimizes the bound during training to guide exploration. Experiments across continuous control tasks show that our approach provides meaningful confidence certificates while maintaining competitive performance.

The empirical success of multi-agent reinforcement learning (MARL) has motivated the search for more efficient and scalable algorithms for large scale multi-agent systems. However, existing state-of-the-art algorithms do not fully exploit inter-agent coupling information to develop MARL algorithms. In this paper, we propose a systematic approach to leverage structures in the inter-agent couplings for efficient model-free reinforcement learning. We model the cooperative MARL problem via a Bayesian network and characterize the subset of agents, termed as the value dependency set, whose information is required by each agent to estimate its local action value function exactly. Moreover, we propose a partially decentralized training decentralized execution (P-DTDE) paradigm based on the value dependency set. We theoretically establish that the total variance of our P-DTDE policy gradient estimator is less than the centralized training decentralized execution (CTDE) policy gradient estimator. We derive a multi-agent policy gradient theorem based on the P-DTDE scheme and develop a scalable actor-critic algorithm. We demonstrate the efficiency and scalability of the proposed algorithm on multi-warehouse resource allocation and multi-zone temperature control examples. For dense value dependency sets, we propose an approximation scheme based on truncation of the Bayesian network and empirically show that it achieves a faster convergence than the exact value dependence set for applications with a large number of agents.

Over the past couple of decades, many active learning acquisition functions have been proposed, leaving practitioners with an unclear choice of which to use. Bayesian Decision Theory (BDT) offers a universal principle to guide decision-making. In this work, we derive BDT for (Bayesian) active learning in the myopic framework, where we imagine we only have one more point to label. This derivation leads to effective algorithms such as Expected Error Reduction (EER), Expected Predictive Information Gain (EPIG), and other algorithms that appear in the literature. Furthermore, we show that BAIT (active learning based on V-optimal experimental design) can be derived from BDT and asymptotic approximations. A key challenge of such methods is the difficult scaling to large batch sizes, leading to either computational challenges (BatchBALD) or dramatic performance drops (top-$B$ selection). Here, using a particular formulation of the decision process, we derive Partial Batch Label Sampling (ParBaLS) for the EPIG algorithm. We show experimentally for several datasets that ParBaLS EPIG gives superior performance for a fixed budget and Bayesian Logistic Regression on Neural Embeddings. Our code is available at https://github.com/ADDAPT-ML/ParBaLS.

Real-world clinical problems are often characterized by multimodal data, usually associated with incomplete views and limited sample sizes in their cohorts, posing significant limitations for machine learning algorithms. In this work, we propose a Bayesian approach designed to efficiently handle these challenges while providing interpretable solutions. Our approach integrates (1) a generative formulation to capture cross-view relationships with a semi-supervised strategy, and (2) a discriminative task-oriented formulation to identify relevant information for specific downstream objectives. This dual generative-discriminative formulation offers both general understanding and task-specific insights; thus, it provides an automatic imputation of the missing views while enabling robust inference across different data sources. The potential of this approach becomes evident when applied to the multimodal clinical data, where our algorithm is able to capture and disentangle the complex interactions among biological, psychological, and sociodemographic modalities.