A Metropolis-class sampler for targets with non-convex support

We aim to improve upon the exploration of the general-purpose random walk Metropolis algorithm when the target has non-convex support A⊂Rd, by reusing proposals in Ac which would otherwise be rejected. The algorithm is Metropolis-class and under standard conditions the chain satisfies a strong law of large numbers and central limit theorem. Theoretical and numerical evidence of improved performance relative to random walk Metropolis are provided. Issues of implementation are discussed and numerical examples, including applications to global optimisation and rare event sampling, are presented

Geodesic slice sampling on the sphere

Probability measures on the sphere form an important class of statistical models and are used, for example, in modeling directional data or shapes. Due to their widespread use, but also as an algorithmic building block, efficient sampling of distributions on the sphere is highly desirable. We propose a shrinkage based and an idealized geodesic slice sampling Markov chain, designed to generate approximate samples from distributions on the sphere. In particular, the shrinkage based algorithm works in any dimension, is straight-forward to implement and has no tuning parameters. We verify reversibility and show that under weak regularity conditions geodesic slice sampling is uniformly ergodic. Numerical experiments show that the proposed slice samplers achieve excellent mixing on challenging targets including the Bingham distribution and mixtures of von Mises-Fisher distributions. In these settings our approach outperforms standard samplers such as random-walk Metropolis Hastings and Hamiltonian Monte Carlo.Comment: 53 pages, 10 figures in the main text, 3 figures and 1 table in the appendi

Geometric convergence of slice sampling

In Bayesian statistics sampling w.r.t. a posterior distribution, which is given through a prior and a likelihood function, is a challenging task. The generation of exact samples is in general quite difficult, since the posterior distribution is often known only up to a normalizing constant. A standard way to approach this problem is a Markov chain Monte Carlo (MCMC) algorithm for approximate sampling w.r.t. the target distribution. In this cumulative dissertation geometric convergence guarantees are given for two different MCMC methods: simple slice sampling and elliptical slice sampling.2021-10-2