Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard CD categories. To go further, we then study enrichments of CD categories over commutative monoids. This gives an expressive setting for reasoning abstractly about a range of important probabilistic concepts, including substochastic kernels, finite and σ-finite measures, absolute continuity, singular measures, and Lebesgue decompositions. Using these tools, we give synthetic necessary and sufficient conditions for a general MH-type sampler to be reversible with respect to a given target distribution.

We propose SymDiff, a method for constructing equivariant diffusion models using the framework of stochastic symmetrisation. SymDiff resembles a learned data augmentation that is deployed at sampling time, and is lightweight, computationally efficient, and easy to implement on top of arbitrary off-the-shelf models. In contrast to previous work, SymDiff typically does not require any neural network components that are intrinsically equivariant, avoiding the need for complex parameterisations or the use of higher-order geometric features. Instead, our method can leverage highly scalable modern architectures as drop-in replacements for these more constrained alternatives. We show that this additional flexibility yields significant empirical benefit for E(3)-equivariant molecular generation. To the best of our knowledge, this is the first application of symmetrisation to generative modelling, suggesting its potential in this domain more generally.

Off-Policy Evaluation (OPE) in contextual bandits is crucial for assessing new policies using existing data without costly experimentation. However, current OPE methods, such as Inverse Probability Weighting (IPW) and Doubly Robust (DR) estimators, suffer from high variance, particularly in cases of low overlap between target and behaviour policies or large action and context spaces. In this paper, we introduce a new OPE estimator for contextual bandits, the Marginal Ratio (MR) estimator, which focuses on the shift in the marginal distribution of outcomes $Y$ instead of the policies themselves. Through rigorous theoretical analysis, we demonstrate the benefits of the MR estimator compared to conventional methods like IPW and DR in terms of variance reduction. Additionally, we establish a connection between the MR estimator and the state-of-the-art Marginalized Inverse Propensity Score (MIPS) estimator, proving that MR achieves lower variance among a generalized family of MIPS estimators. We further illustrate the utility of the MR estimator in causal inference settings, where it exhibits enhanced performance in estimating Average Treatment Effects (ATE). Our experiments on synthetic and real-world datasets corroborate our theoretical findings and highlight the practical advantages of the MR estimator in OPE for contextual bandits.

Most off-policy evaluation methods for contextual bandits have focused on the expected outcome of a policy, which is estimated via methods that at best provide only asymptotic guarantees. However, in many applications, the expectation may not be the best measure of performance as it does not capture the variability of the outcome. In addition, particularly in safety-critical settings, stronger guarantees than asymptotic correctness may be required. To address these limitations, we consider a novel application of conformal prediction to contextual bandits. Given data collected under a behavioral policy, we propose \emph{conformal off-policy prediction} (COPP), which can output reliable predictive intervals for the outcome under a new target policy. We provide theoretical finite-sample guarantees without making any additional assumptions beyond the standard contextual bandit setup, and empirically demonstrate the utility of COPP compared with existing methods on synthetic and real-world data.

We propose a variational autoencoder architecture to model both ignorable and nonignorable missing data using pattern-set mixtures as proposed by Little (1993). Our model explicitly learns to cluster the missing data into missingness pattern sets based on the observed data and missingness masks. Underpinning our approach is the assumption that the data distribution under missingness is probabilistically semi-supervised by samples from the observed data distribution. Our setup trades off the characteristics of ignorable and nonignorable missingness and can thus be applied to data of both types. We evaluate our method on a wide range of data sets with different types of missingness and achieve state-of-the-art imputation performance. Our model outperforms many common imputation algorithms, especially when the amount of missing data is high and the missingness mechanism is nonignorable.

Standard variational schemes for training deep latent variable models rely on biased gradient estimates of the target objective. Techniques based on the Evidence Lower Bound (ELBO), and tighter variants obtained via importance sampling, produce biased gradient estimates of the true log-likelihood. The family of Reweighted Wake-Sleep (RWS) methods further relies on a biased estimator of the inference objective, which biases training of the encoder also. In this work, we show how Multilevel Monte Carlo (MLMC) can provide a natural framework for debiasing these methods with two different estimators. We prove rigorously that this approach yields unbiased gradient estimators with finite variance under reasonable conditions. Furthermore, we investigate methods that can reduce variance and ensure finite variance in practice. Finally, we show empirically that the proposed unbiased estimators outperform IWAE and other debiasing method on a variety of applications at the same expected cost.