# Probabilistic inference and learning with Stein’s method

Lester Mackey
Depts. of Statistics and Computer Science, Stanford University; Microsoft Research New England
Meeting: Probabilistic inference and learning with Stein’s method

Stein’s method is a powerful tool from probability theory for bounding the distance between probability distributions. In this talk, I’ll describe how this tool designed to prove central limit theorems can be adapted to assess and improve the quality of practical inference procedures. I’ll highlight applications to Markov chain Monte Carlo sampler selection, goodness-of-fit testing, variational inference, and nonconvex optimization and close with several opportunities for future work.

Marina Riabiz
King's College London
Primer: Optimal thinning of mcmc output with application to cardiac electrophysiology

Calcium is the end-point intracellular signal driving cardiac myocyte contraction, and its dynamics is described through a set of coupled ordinary differential equations (ODEs) [4]. Markov Chain Monte Carlo (MCMC) can be used to characterize the posterior distribution of the parameters of the cardiac ODEs, that can then serve as experimental design for multi-physics and multi-scale models of the whole hearth. However, MCMC suffers from poor mixing caused by the high-dimensional nature of the parameter vector and the correlation of its components, so that post-processing of the MCMC output is required.

The use of existing heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Typically a number of the initial states are attributed to “burn in” and removed [2], whilst the remainder of the chain is “thinned” if compression is also required. In this talk we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy [5, 3, 1], that is suitable when the gradient of the log-target can be evaluated and an approximation using a small number of states is required. Theoretical results guarantee consistency of the method and we demonstrate its effectiveness in the cardiac electrophysiology problem at hand. Software is available at http://stein-thinning.org/.