Studies in Lower Bounding Probabilities of Evidence using the Markov Inequality

TL;DR

This paper introduces a randomized importance sampling method using the Markov inequality to efficiently compute high-confidence lower bounds on evidence probabilities, addressing an NP-hard problem with practical heuristics.

Contribution

It presents a novel approximation scheme that leverages the Markov inequality and heuristics to improve lower bound estimation of evidence probabilities.

Findings

Empirical results show competitive performance with state-of-the-art methods.

The proposed heuristics significantly improve lower bound quality.

The method offers a practical approach for a computationally hard problem.

Abstract

Computing the probability of evidence even with known error bounds is NP-hard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on probability of evidence but does not have any guarantees in terms of relative or absolute error. Our proposed approximation is a randomized importance sampling scheme that uses the Markov inequality. However, a straight-forward application of the Markov inequality may lead to poor lower bounds. We therefore propose several heuristic measures to improve its performance in practice. Empirical evaluation of our scheme with state-of- the-art lower bounding schemes reveals the promise of our approach.