Many-query analyses, in which a computational model is repeatedly evaluated, play an important role in uncertainty quantification and optimization for engineering design. However, many-query analyses can be prohibitively expensive when considering high-fidelity numerical models, especially those governed by partial differential equations (PDEs). Surrogate modeling replaces costly simulations with cheap approximations, enabling practical uncertainty quantification and design workflows. Machine learning (ML) approaches to surrogate modeling have gained significant traction in the development of surrogate models due to their non-intrusive nature; they do not require access to the underlying simulation code. The Operator Inference (OpInf) approach, in particular, fits reduced operators to state and corresponding time derivative data. However, high-fidelity training data is scarce due to limited computational budgets, and OpInf models trained on such scarce data often exhibit high variance, a critical liability in high-risk engineering design. We propose Multifidelity Operator Inference (MF-OpInf), which reduces this variance by supplementing scarce high-fidelity snapshots with abundant low-fidelity data drawn from coarser discretizations of the same problem. MF-OpInf handles data across varying fidelity levels by utilizing a multifidelity linear regression framework to estimate the reduced operators. Numerical experiments on model PDE problems demonstrate that MF-OpInf predicts PDE solution fields with higher accuracy compared to the high-fidelity-only OpInf with the same computational budget.
GARCH-LSTM hybrid models have become increasingly popular in the volatility forecasting literature. To date, these models have almost exclusively focused on point estimation and accuracy measures such as RMSE and MAPE, while failing to assess the uncertainty of these point estimates. We approach this gap in two ways. First, we extend the existing GARCH-LSTM model by modeling the conditional mean with an ARMA structure. Second, we develop a forward bootstrap algorithm to build multi-step-ahead prediction intervals for future realized volatility. In order to address the notorious undercoverage of prediction intervals, we implement two solutions. One draws bootstrap innovations from the empirical residual distribution to mimic the fat-tail property of financial series. The other utilizes principles from conformal prediction to systematically extend the prediction interval length. We evaluate the methods in simulations with heavy-tailed and time-varying GARCH processes and in rolling-window forecasts of four U.S. stock indices from 2020 to 2025. The ARMA-based hybrid gives the lowest point-forecast errors in most settings, and combining both remedies with the mean-based hybrid yields intervals closest to the nominal level among competing methods.
Pretrained diffusion models serve as frozen teachers for pipelines such as text-to-3D, single-step distillation, and data attribution. The teacher gradients these pipelines consume are expectations over noise levels and Gaussian noise, estimated by Monte Carlo (MC). Each MC draw requires expensive upstream work (rendering, simulation, encoding), so estimator variance dominates compute cost. We introduce CARV, a compute-aware variance-accounting framework that casts these gradients as a classical two-stage sampling problem with asymmetric per-level costs. This motivates a hierarchical MC estimator that amortizes each expensive upstream computation by reusing it across cheap diffusion draws, with within-unit variance further reduced by timestep importance sampling (IS) and stratified inverse-CDF sampling. CARV delivers 2 to 3x effective compute multipliers on text-to-3D and data attribution without changing the objective. In text-to-3D, reuse provides most of the gain and IS with stratification adds about 25%. In single-step distillation, the same techniques cut gradient variance by an order of magnitude but do not improve downstream FID, marking the regime where MC variance is no longer the bottleneck.
In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. This work is a review paper that brings together these different methodological contributions within a unified framework, highlighting the connections among noise contrastive estimation (NCE), reverse logistic regression (RLR), Bennett's acceptance-ratio (BAR) method, multiple importance sampling (MIS), and bridge sampling within the context of EBMs. Some of them are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Our aim is to provide a self-contained and accessible treatment, with clear and straightforward derivations, that brings together these contributions within a common framework and makes the main results and their underlying ideas understandable to a broader readership. The contribution is twofold. First, this work serves as a comprehensive tutorial for researchers entering the field, introducing the fundamental concepts and derivations. Second, it offers a thorough review for experienced practitioners. The code used in the numerical experiments is available to support the reproducibility of the results.
Generalized linear mixed models (GLMMs) are widely used for analyzing correlated data, particularly in large-scale biomedical and social science applications. Scalable Bayesian inference for GLMMs is challenging due to an intractable marginal likelihood and a high computational cost incurred by conventional Markov chain Monte Carlo (MCMC) methods. We develop a stochastic gradient MCMC (SGMCMC) algorithm tailored to GLMMs that enables accurate posterior inference in the large-sample regime. Our approach uses Fisher's identity to construct a (biased) Monte Carlo estimator of the gradient of the marginal log-likelihood, making SGMCMC feasible when direct gradient computation is impossible. We analyze the additional variability, introduced by both data subsampling and gradient approximation, to derive a post-hoc covariance correction that yields properly calibrated posterior uncertainty. We show through simulated studies that the proposed method provides accurate posterior means and variances in settings with a large number of groups, outperforming existing approaches, including control variate methods. We further demonstrate the method's practical utility in an analysis of electronic health records data, where accounting for variance inflation materially changes scientific conclusions.
The idea of testing whether the past of one signal improves the prediction of another can be traced back to Wiener. It was Granger who formalized the concept. Granger causality has become one of the most widely used tools to infer directed interactions between the components of a system. The name however can be misleading: it detects predictive relations among the measured signals, not causal relations among the mechanisms that generate them. This article starts by reviewing recent advances in observation-space Granger analysis. It then follows a path to directed structure in latent dynamical models, and asks under which conditions that structure can be read causally.
Theoretical results in statistics and machine learning rest on long, technically delicate proofs, and fluent arguments generated by large language models are not necessarily correct. Automated formalization makes such arguments mechanically checkable, and automated proving goes further by discovering machine-checked proofs for statements whose proofs are incomplete or missing altogether. Both require reconstructing assumptions from source materials, accessing specialized mathematical infrastructure and managing long proof developments. We present Statopia, an ecosystem that jointly develops Statopia Prover, an independent agent specialized for formalizing and proving results in statistics and machine learning, and Statlib, a Lean~4 library built with the prover and organized for automated use. The prover controls its own execution without relying on external coding agents. It is built for formal proof discovery and checking existing proofs. A natural-language proof, when available, guides its search but is not required: from a statement alone, the prover develops the argument itself and constructs a Lean proof. Its workflow integrates source preparation, statement review, library supported planning and proof construction. Dynamic proof dependency graphs and task groups support parallel, resumable proof development. Statlib's shared definitions and theorem interfaces are organized for retrieval and composition by agents, and its coverage of statistics keeps expanding. Verified results and usage feedback from the prover inform reviewed library updates, so that proof development drives this expansion. A cross-platform application provides an interactive workspace for uploading source files and running tasks locally or in the cloud.
The logarithmic vector multiplicative error model (log-vMEM) provides a methodology for modeling and forecasting multivariate positive-valued time series. However, the number of autoregressive parameters increases rapidly with the system dimension and lag order, posing challenges for estimation and model selection. This paper develops a regularized estimation framework for the log-vMEM with multivariate Gamma errors by incorporating hierarchical lag structures to induce sparsity and accommodate heterogeneous lag dynamics. We consider three hierarchical lag structures: componentwise, own-other, and elementwise, in combination with group lasso, adaptive group lasso, group MCP, and group SCAD penalties. Parameter estimation is carried out through an alternating block-coordinate optimization algorithm with sequential Gauss-Seidel updates, while GPU-accelerated numerical quadrature is employed to address the computational burden of evaluating the multivariate Gamma likelihood. Simulation experiments investigate parameter recovery and lag selection under different sparsity structures. The real data analysis is carried out on the realized volatility measures for Microsoft (NASDAQ:MSFT) which allows a comparison of the competing regularization approaches in an empirical financial setting. Together, these developments provide a computational framework for sparse multivariate volatility modeling with flexible lag structures and dependent Gamma errors.
Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \citep{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theory and methodology at scale and under DAG misspecification. We then model a latent-collider DAG, a protein signalling network, and a multi-fidelity \mbox{heavy-ion} collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability over the simulator hierarchy.
Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a \emph{biased} sampler, as it discretises a continuous diffusion process. In this work, we consider \emph{adjusted} Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fr\'echet Inception Distance (FID) on the latter.
The Super Learner (SL) is a widely used ensemble method that combines point predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformal prediction sets via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. We present an application of this methodology to predict creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.
We address the brittleness of Bayesian experimental design under model misspecification by formulating the problem as a max--min game between the experimenter and an adversarial nature subject to information-theoretic constraints. We demonstrate that this approach yields a robust objective governed by Sibson's $\alpha$-mutual information~(MI), which identifies the $\alpha$-tilted posterior as the robust belief update and establishes the R\'enyi divergence as the appropriate measure of conditional information gain. To mitigate the bias and variance of nested Monte Carlo estimators needed to estimate Sibson's $\alpha$-MI, we adopt a PAC-Bayes framework to search over stochastic design policies, yielding rigorous high-probability lower bounds on the robust expected information gain that explicitly control finite-sample error.
We develop diffusion-based samplers for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a simple base distribution and the target, popularised by diffusion models. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, providing principled score and density estimates for time-varying distributions. To control the variance of score estimates, we further propose practical control variate schedules that incur minimal overhead. We adapt this general framework to paths induced by the Ornstein-Uhlenbeck (OU) time-reversal process, stochastic interpolants, and diffusion annealed Langevin dynamics, outlining their trade-offs. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets. Our code is available at: https://github.com/akyildiz-group/dpsmc.
Generalized Bayes product partition models provide a probabilistic counterpart to loss-based clustering. They update a prior over partitions through an exponentiated clustering loss rather than a likelihood, so allocation uncertainty can be quantified without a full generative model for the data. However, their performance depends on the space in which the loss is evaluated. For high-dimensional observations, a squared Euclidean loss on the raw features can be dominated by irrelevant coordinates or fail to reflect cluster structure lying near a nonlinear low-dimensional manifold. We propose an extension in which an autoencoder and the partition are learned together, so that the space in which the clustering loss is evaluated is shaped by the clustering itself. Our generalized posterior combines a reconstruction loss, which keeps the low-dimensional representation faithful to the observed data, with a within-cluster sum of squares in the latent representation, which encourages cluster separation. Posterior computation alternates between Gibbs sampling over partitions and gradient-based updates of the autoencoder network, with the learning rate estimated jointly. We illustrate the approach through simulation studies and applications to four benchmark datasets.
This work develops a locally adaptive jackknife+ conformal prediction framework for polynomial chaos expansions. Therein, the non-conformity scores are normalized with an auxiliary polynomial chaos expansion trained on the log-transformed leave-one-out residuals of the base model. We show that the coverage guarantee of jackknife+ conformal prediction carries over to the adaptive intervals if the leave-one-out auxiliary models are trained on nested leave-one-out residuals. For fixed-basis polynomial chaos expansions fitted by ordinary least squares, all required quantities are derived in closed form and the guarantee-preserving adaptive intervals are obtained from a single model fit, without any model retraining. For sparse polynomial chaos expansions, where multiple model fits are necessary, we propose a heuristic simplification that limits the computational overhead of adaptivity at the cost of the formal coverage guarantee. Numerical experiments on an academic benchmark with known heteroscedastic noise and on an engineering simulation dataset show that the suggested method attains the target coverage while providing intervals that adapt to local prediction uncertainty, therefore being considerably more statistically efficient. These advantages are particularly prominent in high-uncertainty regions of the input space, where the performance of non-adaptive methods degrades substantially.
Thermal simulation is becoming increasingly critical in modern IC design and manufacturing. Feynman--Kac random walk methods can estimate localized temperatures without solving the global field, but in practical settings without Dirichlet boundaries they require very long paths to control the truncation residual. This paper presents FasterRW, a continuation-value framework for accelerating Feynman--Kac thermal random walks. We prove a stopping-time residual identity showing that the expected tail after truncation equals the current attenuation weight multiplied by the temperature at the stopping location. This identity enables a prior-corrected single-point estimator: a cheap, noisy prior replaces the unknown continuation value, making the truncation bias scale with prior error rather than absolute temperature and yielding an error-controlled threshold-selection rule. For multi-point queries, FasterRW further converts cross-query pass-through tails into pseudo-samples and refines the resulting observations in prior-error coordinates through kriging. Compared with prior state-of-the-art methods, FasterRW achieves up to $28\times$ acceleration while maintaining accuracy. Our code is open-sourced at https://github.com/ShiningSord/FastRW.
Hypergraphs model higher-order interactions, but realistic hypergraph generation remains difficult because incidence, hyperedge-size heterogeneity, and overlap structure are not faithfully captured by pairwise reductions. We propose \HEDGE, a generative model defined directly on relaxed incidence matrices via a structured stochastic diffusion. The forward process combines a hypergraph-specific two-sided heat operator with an Ornstein--Uhlenbeck component, preserving structure-aware noising near the data while yielding an explicit Gaussian terminal law. Conditional on an observed hypergraph, this forward process is linear-Gaussian, so conditional means, covariances, scores, and reverse-drift targets are available in closed form. We therefore learn a permutation-equivariant state-only reverse-drift field in incidence space by regressing onto exact conditional targets, and generate samples by simulating a learned reverse-time SDE from the Gaussian base law. We establish exactness in the ideal state-only setting together with finite-horizon stability guarantees, and empirically show improved hypergraph generation quality relative to strong baselines.
Variational approaches to learning transport maps for multimodal sampling can suffer from mode collapse. One way to mitigate this problem is to learn transport along a prescribed path of intermediate distributions defined by geometric annealing. We show that annealing can take a substantial detour in Wasserstein space: when mass moves between modes, the path can be much longer than the distance between its endpoints. We further link path geometry to transport learning, showing that long annealing paths can slow population optimization, while sufficiently regular paths permit rapid convergence. Motivated by these findings, we propose componentwise annealed transport (CAT) to reduce the need for mass transfer between modes. Starting from a mode-informed mixture reference, CAT constructs an exact mixture decomposition of the target and learns separate transport maps along componentwise annealing paths. We estimate the component weights separately and show that these estimates remain consistent even when the learned flows are inexact. We also develop adaptive annealing schedules guided by transport effort. Experiments on multimodal synthetic distributions and Bayesian inference problems demonstrate improvements in transport learning and sampling efficiency.
Aggregating independently computed sub-sample inferences is a well-known challenge in large-scale Bayesian computation, federated learning, and opinion pooling. The core problem is to approximate a distribution whose density is proportional to a product of individual densities. We propose Product Monte Carlo (PMC), an algorithm for computing arbitrary expectations under a product of densities using only Monte Carlo draws from individual densities. We turn the intractable product expectation into a regression problem by expressing it through a family of computable conditional expectations. Besides the point estimate, our fully Bayesian regression formulation provides coherent uncertainty quantification and principled diagnostics. We establish theoretical guarantees on consistency and characterize convergence rates for the PMC estimator. Across simulated and real-data experiments varying the number, dimensionality, and heterogeneity of the component distributions, PMC achieves higher estimation accuracy, better-calibrated uncertainty coverage, and improved scalability at a lower computational cost than existing methods. In addition, PMC requires no density evaluation, making it applicable to simulation-based inference, as we demonstrate in an application with protein images.
Black-box variational inference (BBVI) is a methodology for posterior approximation that relies on stochastic optimization. In practice, the stochastic optimizers underpinning BBVI generally require extensive problem-specific tuning, which undermines its promise as a truly "black box" inference algorithm. However, over the past decade, many new adaptive stochastic optimization algorithms have been developed that reduce or remove entirely the need for tuning. In this work, we investigate this new collection of adaptive methods in the context of BBVI, with the goal of establishing the current state of the art in tuning-free optimization-based inference. In particular, we present a large-scale empirical evaluation of 56 stochastic gradient-based optimization algorithms applied to 1092 Bayesian inference optimization problems, involving over 550,000 individual optimization runs and 15 core-years of compute. The optimization algorithms we evaluate are chosen to represent a wide spectrum of recent approaches and the benchmark problems are chosen to span a range of difficulty, with posterior target dimension 1-10^4, condition number 1-10^8, and a range of variational families. Our results show that no single method dominates, but running a selection of 5 algorithms suffices to reliably get close to the best-possible observed performance. We thus provide a strong baseline for applications where expert tuning is not possible and for comparison when developing new stochastic optimization algorithms.