Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions

TL;DR

This paper presents a formal method to approximate the probability distribution of Markov processes using finite abstractions, enabling faster computation with guaranteed error bounds, applicable even for unbounded state spaces.

Contribution

It introduces a novel abstraction technique with formal guarantees, including procedures for truncating unbounded spaces and extending to higher-order approximations.

Findings

Effective approximation of Markov process distributions.

Error bounds depend on partition diameters and process properties.

Compared favorably with existing methods in probabilistic invariance computation.

Abstract

The goal of this work is to formally abstract a Markov process evolving in discrete time over a general state space as a finite-state Markov chain, with the objective of precisely approximating its state probability distribution in time, which allows for its approximate, faster computation by that of the Markov chain. The approach is based on formal abstractions and employs an arbitrary finite partition of the state space of the Markov process, and the computation of average transition probabilities between partition sets. The abstraction technique is formal, in that it comes with guarantees on the introduced approximation that depend on the diameters of the partitions: as such, they can be tuned at will. Further in the case of Markov processes with unbounded state spaces, a procedure for precisely truncating the state space within a compact set is provided, together with an error bound that depends on the asymptotic properties of the transition kernel of the original process. The overall abstraction algorithm, which practically hinges on piecewise constant approximations of the density functions of the Markov process, is extended to higher-order function approximations: these can lead to improved error bounds and associated lower computational requirements. The approach is practically tested to compute probabilistic invariance of the Markov process under study, and is compared to a known alternative approach from the literature.