Methodological and analytical advancement of hybrid slice sampling

• Applicant: Professor Dr. Daniel Rudolf
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 578618598

In Bayesian inference and generative modeling, an omnipresent challenge is drawing approximate samples from distributions with densities that are known only up to a normalizing constant. Slice sampling is a Markov chain Monte Carlo (MCMC) method to address this problem. It introduces auxiliary `level random variables' and alternates between sampling from one of them and drawing a point in the corresponding super-level set (also referred to as `slice'). This set is the region where the density exceeds the auxiliary level. Ideal versions of this approach exhibit provably robust convergence properties, but require exact sampling from the reference measure conditioned on the level set, which is usually intractable. Hybrid slice sampling replaces the intractable step with draws from suitable Markov kernels that preserve the desired conditional reference distribution. Existing variants include stepping‑out/shrinkage-, hit‑and‑run-, elliptical-, and Gibbsian polar-slice sampling. The project aims to advance the methodological and analytical framework around such hybrid slice samplers. This is achieved through algorithmic innovations: (1) we provide well-definedness properties of building-block univariate samplers; (2) we propose an acceptance-probability-based slice sampler; and (3) we investigate an alternative to the usually present acceptance/rejection step within involutive MCMC by introducing a unified involutive slice sampling framework. These three methodological contributions are fundamentally flanked by the following three analytical goals: (1) we aim to establish new dimension-independent convergence results for ideal slice sampling; (2) by identifying conditions on the target distribution and the within‑slice kernels, the goal is to transfer robust convergence results from the ideal case to corresponding hybrid slice samplers; and (3) we develop perturbation-theoretic results to quantify how approximations of the density or acceptance function affect convergence and possible bias. All parts of the project, both theoretical and methodological, benefit from extensive numerical experiments on benchmark problems to verify, assess and guide all the investigations.

• DFG Programme: Research Grants