A Dynamic Programming Approach to Evaluating Multivariate Gaussian Probabilities

TL;DR

This paper introduces a dynamic programming method to approximate multivariate Gaussian probabilities by linking them to Markov decision processes, providing explicit error bounds for the approximation.

Contribution

It presents a novel dynamic programming approach for Gaussian probability evaluation, connecting it to MDPs with non-Lipschitz costs and deriving error bounds.

Findings

Approximation scheme for Gaussian probabilities with explicit error bounds

Equivalence between Gaussian integration over polytopes and MDP optimization

Method applicable to non-Lipschitz cost functions in MDPs

Abstract

We propose a method of approximating multivariate Gaussian probabilities using dynamic programming. We show that solving the optimization problem associated with a class of discrete-time finite horizon Markov decision processes with non-Lipschitz cost functions is equivalent to integrating a Gaussian functions over polytopes. An approximation scheme for this class of MDPs is proposed and explicit error bounds under the supremum norm for the optimal cost to go functions are derived.