Bayesian

In this work, we propose a flow-matching Markov chain Monte Carlo (FM-MCMC) algorithm for estimating the orbital parameters of exoplanetary systems, especially for those only one exoplanet is involved. Compared to traditional methods that rely on random sampling within the Bayesian framework, our approach first leverages flow matching posterior estimation (FMPE) to efficiently constrain the prior range of physical parameters, and then employs MCMC to accurately infer the posterior distribution. For example, in the orbital parameter inference of beta Pictoris b, our model achieved a substantial speed-up while maintaining comparable accuracy-running 77.8 times faster than Parallel Tempered MCMC (PTMCMC) and 365.4 times faster than nested sampling. Moreover, our FM-MCMC method also attained the highest average log-likelihood among all approaches, demonstrating its superior sampling efficiency and accuracy. This highlights the scalability and efficiency of our approach, making it well-suited for processing the massive datasets expected from future exoplanet surveys. Beyond astrophysics, our methodology establishes a versatile paradigm for synergizing deep generative models with traditional sampling, which can be adopted to tackle complex inference problems in other fields, such as cosmology, biomedical imaging, and particle physics.

This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.

Today's generative models thrive with large amounts of supervised data and informative reward functions characterizing the quality of the generation. They work under the assumptions that the supervised data provides knowledge to pre-train the model, and the reward function provides dense information about how to further improve the generation quality and correctness. However, in the hardest instances of important problems, two problems arise: (1) the base generative model attains a near-zero reward signal, and (2) calls to the reward oracle are expensive. This setting poses a fundamentally different learning challenge than standard reward-based post-training. To address this, we propose BaNEL (Bayesian Negative Evidence Learning), an algorithm that post-trains the model using failed attempts only, while minimizing the number of reward evaluations (NREs). Our method is based on the idea that the problem of learning regularities underlying failures can be cast as another, in-loop generative modeling problem. We then leverage this model to assess whether new data resembles previously seen failures and steer the generation away from them. We show that BaNEL can improve model performance without observing a single successful sample on several sparse-reward tasks, outperforming existing novelty-bonus approaches by up to several orders of magnitude in success rate, while using fewer reward evaluations.

The objectives that Large Language Models (LLMs) implicitly optimize remain dangerously opaque, making trustworthy alignment and auditing a grand challenge. While Inverse Reinforcement Learning (IRL) can infer reward functions from behaviour, existing approaches either produce a single, overconfident reward estimate or fail to address the fundamental ambiguity of the task (non-identifiability). This paper introduces a principled auditing framework that re-frames reward inference from a simple estimation task to a comprehensive process for verification. Our framework leverages Bayesian IRL to not only recover a distribution over objectives but to enable three critical audit capabilities: (i) Quantifying and systematically reducing non-identifiability by demonstrating posterior contraction over sequential rounds of evidence; (ii) Providing actionable, uncertainty-aware diagnostics that expose spurious shortcuts and identify out-of-distribution prompts where the inferred objective cannot be trusted; and (iii) Validating policy-level utility by showing that the refined, low-uncertainty reward can be used directly in RLHF to achieve training dynamics and toxicity reductions comparable to the ground-truth alignment process. Empirically, our framework successfully audits a detoxified LLM, yielding a well-calibrated and interpretable objective that strengthens alignment guarantees. Overall, this work provides a practical toolkit for auditors, safety teams, and regulators to verify what LLMs are truly trying to achieve, moving us toward more trustworthy and accountable AI.

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a Wasserstein gradient flow in the latent space of an expected negative log posterior objective, regularized by a Kullback-Leibler divergence to the diffusion prior. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.

Objective: Sparse Bayesian learning provides an effective framework to solve high-dimensional problems in brain signal decoding. However, conventional likelihoods regarding data distributions, such as Gaussian or Bernoulli, are potentially inadequate for handling the noisy recordings of brain activity. Hence, this work aims to formulate a robust sparse Bayesian learning framework to address noisy high-dimensional brain activity decoding. Methods: Motivated by the commendable robustness of the minimum error entropy learning criterion for addressing non-Gaussian signals, this study reformulated the sparse Bayesian learning framework under a generalized Bayesian paradigm, in which the model parameter is regulated with the minimum error entropy loss rather than a conventional likelihood function. Results: Our developed SBL-MEE algorithm was evaluated with two real-world brain decoding tasks of regression and classification scenarios, respectively. Experimental results demonstrated that our approach not only realizes superior brain decoding performance than existing methods, but also presents more physiologically interpretable decoder patterns. Conclusion: Although minimum error entropy is not constructed from an arbitrary probabilistic distribution, it is effective to establish noise-robust inference in sparse Bayesian learning method. Significance: This work provides a powerful tool to improve brain activity decoding capability, particularly regarding the noisy high-dimensional setting, thus promoting biomedical engineering applications such as brain-computer interface.

Existing methods for transfer learning struggle to deal with situations where the source datasets are limited and not guaranteed to be well-aligned with the target dataset. A typical strategy is to use the empirical loss minimizer on the source data as a prior mean for the target parameters. Our key conceptual contribution is to use a risk minimizer conditional on source parameters instead. This allows us to construct a single joint prior distribution for all parameters from the source datasets as well as the target dataset. As a consequence, we benefit from full Bayesian uncertainty quantification and can perform model averaging via Gibbs sampling over indicator variables governing the inclusion of each source dataset. We show how a particular instantiation of our prior leads to a Bayesian Lasso in a transformed coordinate system and discuss computational techniques to scale our approach to moderately sized datasets. We discuss connections between the Maximum a Posteriori estimate associated with our approach and the recently proposed Trans-Lasso method and demonstrate that the MAP estimator MSE-dominates the Trans-Lasso in the normal means setting when there is no regularization on the source datasets. Finally, we perform numerical experiments finding that full Bayesian inference provides superior predictive performance relative to Trans-Lasso on a genetics application, especially when the source data are limited.

Bayesian accounts of in-context learning face a direct objection: exact posterior predictives for exchangeable data are invariant to task-preserving order, yet transformers change next-token probabilities when the same examples are serialized differently. We show this objection targets a structural invariant rather than the quantity scoring online prediction. For any Bayesian reference, excess prequential code length is exactly cumulative predictive KL. For unordered support sets that must be serialized, the expected regret of a single admissible ordering decomposes into that of the order-averaged predictor plus an order-averaging gain. Exchangeability violations are therefore not binary refutations; they are priced by log loss. We instantiate the theory with KT/Dirichlet finite-alphabet prediction and coarsened Bayesian linear-regression (BLR) predictive distributions. On Qwen2.5-7B/14B, floored candidate distributions at support $256$ have one-step excess code lengths of $0.020/0.011$ bits for Bernoulli and $0.039/0.022$ bits for four-way categorical prediction, with candidate mass above $0.999$; coarsened BLR continuations increasingly match the posterior-predictive digit distribution as support grows. A frequentist plug-in baseline sharpens the reading: the predictive distributions sit closer to the Bayesian posterior predictive than to the maximum-likelihood plug-in, by a margin largest at small support, where the plug-in is degenerate, and vanishing as the references converge. Position interventions and a from-scratch ablation localize order sensitivity to the positional encoding, activation patching tests causal use of decoded sufficient statistics, and permutation mixtures quantify the downstream log-loss cost of arbitrary orderings. Transformers need not realize exchangeable posterior predictives for every serialization to be Bayes-competitive prequential predictors.

Continual learning is an online paradigm where a learner continually accumulates knowledge from different tasks encountered over sequential time steps. Importantly, the learner is required to extend and update its knowledge without forgetting about the learning experience acquired from the past, and while avoiding the need to retrain from scratch. Given its sequential nature and its resemblance to the way humans think, continual learning offers an opportunity to address several challenges which currently stand in the way of widening the range of applicability of deep models to further real-world problems. The continual need to update the learner with data arriving sequentially strikes inherent congruence between continual learning and Bayesian inference which provides a principal platform to keep updating the prior beliefs of a model given new data, without completely forgetting the knowledge acquired from the old data. This survey inspects different settings of Bayesian continual learning, namely task-incremental learning and class-incremental learning. We begin by discussing definitions of continual learning along with its Bayesian setting, as well as the links with related fields, such as domain adaptation, transfer learning and meta-learning. Afterwards, we introduce a taxonomy offering a comprehensive categorization of algorithms belonging to the Bayesian continual learning paradigm. Meanwhile, we analyze the state-of-the-art while zooming in on some of the most prominent Bayesian continual learning algorithms to date. Furthermore, we shed some light on links between continual learning and developmental psychology, and correspondingly introduce analogies between both fields. We follow that with a discussion of current challenges, and finally conclude with potential areas for future research on Bayesian continual learning.

Invariant prediction [Peters et al., 2016] analyzes feature/outcome data from multiple environments to identify invariant features - those with a stable predictive relationship to the outcome. Such features support generalization to new environments and help reveal causal mechanisms. Previous methods have primarily tackled this problem through hypothesis testing or regularized optimization. Here we develop Bayesian Invariant Prediction (BIP), a probabilistic model for invariant prediction. BIP encodes the indices of invariant features as a latent variable and recover them by posterior inference. Under the assumptions of Peters et al. [2016], the BIP posterior targets the true invariant features. We prove that the posterior is consistent and that greater environment heterogeneity leads to faster posterior contraction. To handle many features, we design an efficient variational approximation called VI-BIP. In simulations and real data, we find that BIP and VI-BIP are more accurate and scalable than existing methods for invariant prediction.

Heavy-tailed posteriors are routine in Euclidean variational autoencoders, where the Student family relaxes the Gaussian without new machinery. The sphere has had no comparable option. Von Mises-Fisher distribution needs modified Bessel functions and a rejection sampler, and Power Spherical buys its closed forms by forcing the density to vanish at the antipode. We develop the spherical Cauchy distribution as a hyperspherical posterior that needs neither compromise. Stereographic projection carries it to a multivariate Student law, and a Möbius transformation turns a uniform spherical draw into an exact posterior sample from inner products, norms, and scalar arithmetic. The same transformation settles the regularizer. Evaluating the density along the sampling map reduces the Kullback-Leibler (KL) divergence to the uniform prior to a scalar expectation whose expansion terminates in every even ambient dimension, leaving one logarithm and a polynomial with finitely many correction terms. Odd dimensions admit certified truncation of value and gradient, the KL is increasing and convex in concentration, and the same function gives the pairwise KL. At matched modal curvature it has broader angular tails and a smaller KL penalty than both alternatives, so equal local precision costs less regularization. In dimension 128 the fused evaluator runs 1.5 times faster per latent-layer step than Power Spherical and 4.2 times faster than robust von Mises-Fisher on CPU, with factors of 1.6 and 5.4 on CUDA. Across five paired seeds it attains the lowest MNIST reconstruction loss at every tested dimension and lowers held-out viewpoint-gap negative log-likelihood on smallNORB by 3.6 percent.

Magnetic resonance spectroscopy (MRS) is a non-invasive technique to measure the metabolic composition of tissues, offering valuable insights into neurological disorders, tumor detection, and other metabolic dysfunctions. However, accurate metabolite quantification is hindered by challenges such as spectral overlap, low signal-to-noise ratio, and various artifacts. Traditional methods like linear-combination modeling are susceptible to ambiguities and commonly only provide a theoretical lower bound on estimation accuracy in the form of the Cramér-Rao bound. This work introduces a Bayesian inference framework using Sylvester normalizing flows (SNFs) to approximate posterior distributions over metabolite concentrations, enhancing quantification reliability. A physics-based decoder incorporates prior knowledge of MRS signal formation, ensuring realistic distribution representations. We validate the method on simulated 7T proton MRS data, demonstrating accurate metabolite quantification, well-calibrated uncertainties, and insights into parameter correlations and multi-modal distributions.

We present a bifidelity method for uncertainty quantification of parameter estimates in complex systems, leveraging generative models trained to sample the target conditional distribution. In the Bayesian inference setting, traditional parameter estimation methods rely on repeated simulations of potentially expensive forward models to determine the posterior distribution of the parameter values, which may result in computationally intractable workflows. Furthermore, methods such as Markov Chain Monte Carlo (MCMC) necessitate rerunning the entire algorithm for each new data observation, further increasing the computational burden. Hence, we propose a novel method for efficiently obtaining posterior distributions of parameter estimates for high-fidelity models given data observations of interest. The method first constructs a low-fidelity, conditional generative model capable of amortized Bayesian inference and hence rapid posterior density approximation over a wide-range of data observations. When higher accuracy is needed for a specific data observation, the method employs adaptive refinement of the density approximation. It uses outputs from the low-fidelity generative model to refine the parameter sampling space, ensuring efficient use of the computationally expensive high-fidelity solver. Subsequently, a high-fidelity, unconditional generative model is trained to achieve greater accuracy in the target posterior distribution. Both low- and high- fidelity generative models enable efficient sampling from the target posterior and do not require repeated simulation of the high-fidelity forward model. We demonstrate the effectiveness of the proposed method on several numerical examples, including cases with multi-modal densities, as well as an application in plasma physics for a runaway electron simulation model.

Pareto fronts are useful to find good task-mixing strategies for multitask finetuning, but they are also costly to compute. To reduce costs, recent works have used existing model merging methods to help train cheap surrogate models to estimate the Pareto fronts. However, no work has yet considered designing new model-merging methods to directly, and provably, improve the quality of Pareto fronts. Here, we fill this gap by proposing a new Bayesian approach called Variational Model Merging. In this approach, existing model-merging methods are obtained as special cases of "posterior-merging" when Gaussian posteriors are used and new model-merging strategies can be derived by using non-Gaussian posteriors. Our main theoretical result is to show that more flexible posteriors necessarily yield better estimates of Pareto fronts. For instance, a Pareto front estimate obtained by merging full-Gaussian posteriors is expected to be better than that obtained by using isotropic Gaussian posteriors. We validate the theory through extensive empirical results on vision and language transformers where better Gaussian families consistently yields better or comparable Pareto fronts. Our work is a rare instance where Bayesian ideas are used to improve Pareto analysis.

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

Despite being widely adopted as a canonical framework for learning robust models, adversarial training suffers from robust overfitting. Existing empirical and theoretical explorations fail to provide a satisfactory mechanistic interpretation of the phenomenon. By modeling adversarial training with momentum SGD as a discrete-time dynamical system, we propose a PAC-Bayesian analytical framework that proves time-resolved robust generalization bounds. Specifically, our framework tracks the closed-form evolution of the posterior mean and covariance under both stationary and non-stationary transient regimes, connecting the model's robust generalization performance to learning rate, local loss geometry, and mini-batch stochastic gradients. By estimating the key quantities associated with the bound, we illustrate the underlying mechanism of robust overfitting. Our framework also shows how adversarial weight perturbation reduces robust generalization gaps by suppressing dominant loss-curvature modes, while suggesting that excessive penalization can be sub-optimal for optimization.

Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.

Large language models (LLMs) may encode biased associations from heterogeneous training corpora that are not immediately visible under ordinary prompting, but can surface when the model is steered toward particular demographic personas. Such behavior often manifests not as explicit toxic output, but as systematic performance differences across semantically equivalent tasks, making the resulting bias difficult to detect and mitigate. To address this issue, we formalize the implicit bias problem as persona-induced performance disparity and argue that bias evidence should be treated as a graded signal rather than a binary label. Motivated by this observation, we model biased knowledge as a fuzzy subset equipped with an explicit membership function that reflects the strength of bias evidence for each candidate example. Building on this formulation, we propose Bayesian-Theory-based Bias Removal (BTBR), a hybrid probabilistic-fuzzy framework for identifying and removing latent bias traces from model parameters. BTBR first performs likelihood-ratio screening to measure how strongly candidate samples align with a target biased persona, then converts high-membership samples into structured knowledge triples, and finally applies targeted model editing with a lightweight fuzzy rule scheduler to reduce collateral performance degradation under high entanglement risk. Extensive experiments across multiple bias sources, tasks, model families and editing backends show that BTBR consistently reduces persona-induced performance gaps while preserving general reasoning ability. These results demonstrate that combining probabilistic evidence with fuzzy degree modeling provides an effective and practical approach for mitigating implicit bias in large language models.

Novelty search (NS) aims to uncover diverse system behaviors through simulation or experiment without requiring a pre-specified scalar objective. This capability is especially relevant to modern discovery problems in chemistry, materials science, and molecular design, where researchers often seek broad coverage of attainable property space rather than a single optimum and where each evaluation may require a costly computation or experiment. For such expensive black-box settings, we propose BEACON, a sample-efficient NS strategy inspired by Bayesian optimization principles. BEACON models the input-to-outcome mapping using multi-output Gaussian processes and selects new inputs by scoring how far plausible posterior outcomes lie from a denoised archive of previously observed outcomes. This gives a distance-based novelty acquisition that accounts for predictive uncertainty and observational noise while operating directly in continuous outcome space, rather than requiring direct optimization over a discretized partition of behaviors. By leveraging efficient posterior sampling together with scalable high-dimensional Gaussian process models, the proposed framework can be extended to settings with large data sets and high-dimensional design variables. We demonstrate BEACON on established benchmark problems together with real-world case studies in materials and molecular discovery. Across these settings, BEACON consistently discovers broader sets of distinct behaviors than several competing baselines under limited evaluation budgets.

In applied statistics and machine learning, the gold standards used for training are often biased and almost always noisy. Dawid and Skene's justifiably popular crowdsourcing model adjusts for rater sensitivity and specificity, but fails to capture distributional properties of rating data gathered for training, which in turn biases training. In this study, we introduce a general purpose measurement-error model with which we can infer consensus categories by adding item-level effects for difficulty, discriminativeness, and guessability. We further show how to constrain the bimodal posterior of these models to avoid adversarial raters. We validate our model's goodness of fit with posterior predictive checks, the Bayesian analogue of $χ^2$ tests, and assess its predictive accuracy using leave-one-out cross-validation. We illustrate our new model with two well-studied data sets, binary rating data for caries in dental X-rays and implication in natural language.

We pose uncertainty quantification and exploration in online decision-making as a problem of training and generation from an autoregressive sequence model, an area experiencing rapid innovation. Our approach rests on viewing uncertainty as arising from missing future outcomes that could be revealed through action choices, rather than from unobservable latent parameters of the environment. This reformulation aligns naturally with modern machine learning capabilities: we can i) train generative models through next-outcome prediction rather than fit explicit priors, ii) assess uncertainty through autoregressive generation rather than sampling latent parameters from posteriors, and iii) adapt to new information by extending the sequence model's context rather than explicit posterior updating. Our main theoretical result establishes a reduction from online decision-making to offline next-outcome prediction: Bayesian regret is controlled directly by the sequence model's offline prediction loss, without requiring an explicit latent-variable posterior. Experiments, including a semi-synthetic news recommendation task, show that autoregressive generation produces calibrated epistemic uncertainty and enables effective exploration by using article text as prior information to focus exploration on resolving remaining uncertainties.

Autoregressive large language models (LLMs) compress knowledge from their training data through next-token conditional distributions. This limits tractable querying of this knowledge to start-to-end autoregressive sampling. However, many tasks of interest -- including sequence continuation, infilling, and other forms of constrained generation -- involve sampling from intractable posterior distributions. We address this limitation by using amortized Bayesian inference to sample from these intractable posteriors. Such amortization is algorithmically achieved by fine-tuning LLMs via diversity-seeking reinforcement learning algorithms: generative flow networks (GFlowNets). We empirically demonstrate that this distribution-matching paradigm of LLM fine-tuning can serve as an effective alternative to maximum-likelihood training and reward-maximizing policy optimization. As an important application, we interpret chain-of-thought reasoning as a latent variable modeling problem and demonstrate that our approach enables data-efficient adaptation of LLMs to tasks that require multi-step rationalization and tool use.

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.

Analyzing the behavior of complex interdependent networks requires complete information about the network topology and the interdependent links across networks. For many applications such as critical infrastructure systems, understanding network interdependencies is crucial to anticipate cascading failures and plan for disruptions. However, data on the topology of individual networks are often publicly unavailable due to privacy and security concerns. Additionally, interdependent links are often only revealed in the aftermath of a disruption as a result of cascading failures. We propose a scalable nonparametric Bayesian approach to reconstruct the topology of interdependent infrastructure networks from observations of cascading failures. Metropolis-Hastings algorithm coupled with the infrastructure-dependent proposal are employed to increase the efficiency of sampling possible graphs. Results of reconstructing a synthetic system of interdependent infrastructure networks demonstrate that the proposed approach outperforms existing methods in both accuracy and computational time. We further apply this approach to reconstruct the topology of one synthetic and two real-world systems of interdependent infrastructure networks, including gas-power-water networks in Shelby County, TN, USA, and an interdependent system of power-water networks in Italy, to demonstrate the general applicability of the approach.

Reranking methods, such as Minimum Bayes Risk (MBR) decoding and Quality Estimation (QE) reranking, have been widely used in modern neural machine translation (NMT) to select an output from a set of candidate hypotheses. However, the performance gains come at the cost of high inference latency. Existing acceleration methods target MBR decoding and reduce only the reranking computation, leaving QE reranking unaddressed and candidate generation---which can be the larger computational bottleneck---largely untouched. In this work, we propose Quit (Quantifying Uncertainty for Incremental Termination), a novel early-stopping strategy for the entire generation--reranking pipeline. Quit treats candidate generation as a sequential decision-making process under uncertainty. It incrementally generates and reranks candidates, stopping when the best reranking score stabilizes. Comprehensive experiments with three NMT models across 19 language pairs show that Quit achieves end-to-end speedups of $1.47$--$2.66\times$ for MBR decoding and $3.43$--$4.12\times$ for QE reranking while preserving translation quality for nearly all external quality metrics.

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to $64$, where the recovered prior is evaluated in a single forward pass.

Modern artificial intelligence is supported by machine learning models (e.g., foundation models) that are pretrained on a massive data corpus and then adapted to solve a variety of downstream tasks. To summarize performance across multiple tasks, evaluation metrics are often aggregated into a summary metric, e.g., average accuracy across 10 question-answering tasks. When aggregating evaluation metrics, it is useful to incorporate uncertainty in the aggregate metric in order to gain a more realistic understanding of model performance. Our objective in this work is to demonstrate how statistical methodology can be used for quantifying uncertainty in metrics that have been aggregated across multiple tasks. The methods we emphasize are bootstrapping, Bayesian hierarchical (i.e., multilevel) modeling, and the visualization of task weightings that consider standard errors. These techniques reveal insights such as the dominance of a specific model for certain types of tasks despite an overall poor performance. We use a popular ML benchmark, the Visual Task Adaptation Benchmark (VTAB), to demonstrate the usefulness of our approaches.

Variational inference, as an alternative to Markov chain Monte Carlo sampling, has played a transformative role in enabling scalable computation for complex Bayesian models. Nevertheless, existing approaches often depend on either rigid model-specific formulations or stochastic black-box optimization routines. Tangent approximation is a principled class of structured variational methods that exploits the geometry of the underlying probability model. However, its utility has largely been confined to logistic regression and related modeling regimes. In this article, we propose a novel variational framework based on tangent transformation for a broad class of probability models characterized by strongly super-Gaussian likelihoods. Our method leverages convex duality to construct tangent minorants of the log-likelihood, thereby inducing conjugacy with Gaussian priors over model parameters in an otherwise intractable setup. Under mild assumptions on the data-generating mechanism, we establish algorithmic convergence guarantees, a contribution that stands in contrast to the limited theoretical assurances typically available for black-box variational methods. Additionally, we derive near-parametric variational risk bounds. Superior performance of our proposed methodology is illustrated on simulated and real-data scenarios that challenge state-of-the-art variational algorithms in terms of scalability and their ability to consistently capture complex underlying data structure.