Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis-Hastings for sampling from mixtures of Gaussian distributions. Our complexity bound scales polynomially with the separation between modes, logarithmically with 1/ε, where εdenotes the target accuracy in total variation distance, and exponentially with the dimension d.

Expected information gain (EIG) is a crucial quantity in Bayesian optimal experimental design (BOED), quantifying how useful an experiment is by the amount we expect the posterior to differ from the prior. However, evaluating the EIG can be computationally expensive since it generally requires estimating the posterior normalizing constant. In this work, we leverage two idiosyncrasies of BOED to improve efficiency of EIG estimation via sequential Monte Carlo (SMC). First, in BOED we simulate the data and thus know the true underlying parameters. Second, we ultimately care about the EIG, not the individual normalizing constants. Often we observe that the Monte Carlo variance of standard SMC estimators for the normalizing constant of a single dataset are significantly lower than the variance of the normalizing constants across datasets; the latter thus contributes the majority of the variance for EIG estimates. This suggests the potential to slightly increase variance while drastically decreasing computation time by reducing the SMC population size, which leads us to an EIG-specific SMC estimator that starts with only a single sample from the posterior and tempers backwards towards the prior. Using this single-sample estimator, which we call reverse-annealed SMC (RA-SMC), we show that it is possible to estimate EIG with orders of magnitude fewer likelihood evaluations in three models: a four-dimensional spring-mass, a six-dimensional Johnson-Cook model and a four-dimensional source-finding problem.

Optimal decision-making under partial observability requires agents to balance reducing uncertainty (exploration) against pursuing immediate objectives (exploitation). In this paper, we introduce a novel policy optimization framework for continuous partially observable Markov decision processes (POMDPs) that explicitly addresses this challenge. Our method casts policy learning as probabilistic inference in a non-Markovian Feynman-Kac model that inherently captures the value of information gathering by anticipating future observations, without requiring suboptimal approximations or handcrafted heuristics. To optimize policies under this model, we develop a nested sequential Monte Carlo (SMC) algorithm that efficiently estimates a history-dependent policy gradient under samples from the optimal trajectory distribution induced by the POMDP. We demonstrate the effectiveness of our algorithm across standard continuous POMDP benchmarks, where existing methods struggle to act under uncertainty.

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Sampling from high dimensional distributions is a computational bottleneck in many scientific applications. Hamiltonian Monte Carlo (HMC), and in particular the No-U-Turn Sampler (NUTS), are widely used, yet they struggle on problems with a very large number of parameters or a complicated geometry. Microcanonical Langevin Monte Carlo (MCLMC) has been recently proposed as an alternative which shows striking gains in efficiency over NUTS, especially for high-dimensional problems. However, it produces biased samples, with a bias that is hard to control in general. We introduce the Metropolis-Adjusted Microcanonical sampler (MAMS), which relies on the same dynamics as MCLMC, but introduces a Metropolis-Hastings step and thus produces asymptotically unbiased samples. We develop an automated tuning scheme for the hyperparameters of the algorithm, making it applicable out of the box. We demonstrate that MAMS outperforms NUTS across the board on benchmark problems of varying complexity and dimensionality, achieving up to a factor of seven speedup.

Optimal decision-making under partial observability requires agents to balance reducing uncertainty (exploration) against pursuing immediate objectives (exploitation). In this paper, we introduce a novel policy optimization framework for continuous partially observable Markov decision processes (POMDPs) that explicitly addresses this challenge. Our method casts policy learning as probabilistic inference in a non-Markovian Feynman--Kac model that inherently captures the value of information gathering by anticipating future observations, without requiring suboptimal approximations or handcrafted heuristics. To optimize policies under this model, we develop a nested sequential Monte Carlo (SMC) algorithm that efficiently estimates a history-dependent policy gradient under samples from the optimal trajectory distribution induced by the POMDP. We demonstrate the effectiveness of our algorithm across standard continuous POMDP benchmarks, where existing methods struggle to act under uncertainty.

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.

Capped-usage SaaS products -- LLM subscriptions such as Claude Code and ChatGPT, cloud platforms such as Vercel and Cloudflare Workers, corporate benefit platforms, identity-verification services with liability transfer -- share a structural signature with insurance products: a fixed premium decoupled from realized consumption, stochastic per-user demand with heavy-tailed severity, a non-fungible cap that resets on a fixed schedule, and a portfolio-level exposure that requires reserve adequacy under tail risk. We argue that this is not an analogy. It is the same operational problem actuarial science has been tooled for decades to address, restated with new dependent variables (tokens, bandwidth bytes, function-invocations, gym check-ins) in place of medical claims. This paper proposes a modeling framework for capped-usage SaaS pricing built from frequency-severity decomposition, premium calculation principles, and Monte Carlo reserve adequacy. We map the framework to publicly observable subscription tiers in two domains (LLM services and cloud platforms), ground it in canonical health-insurance economics (Arrow 1963; Pauly 1968; Manning et al. 1987; Brot-Goldberg et al. 2017), and demonstrate divergence from traditional unit economics through a worked example. The contribution is operational rather than theoretical: not a new theorem, but vocabulary and tools currently absent from cs.LG/stat.ML practice.

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.