Investigation of Variances in Belief Networks

TL;DR

This paper develops a method to quantify the uncertainty in inferred probabilities within belief networks, accounting for the inherent uncertainty in the stored probabilities, and establishes an upper bound for this variance.

Contribution

It introduces a novel approach to estimate variances in belief network inferences assuming prior distributions, and derives an upper bound for these variances using beta distributions.

Findings

Variance in inferred probabilities can be approximated assuming prior distributions.

An upper bound for prior variances is established based on beta distribution properties.

The method is plausible when there is reasonable confidence in stored probabilities.

Abstract

The belief network is a well-known graphical structure for representing independences in a joint probability distribution. The methods, which perform probabilistic inference in belief networks, often treat the conditional probabilities which are stored in the network as certain values. However, if one takes either a subjectivistic or a limiting frequency approach to probability, one can never be certain of probability values. An algorithm should not only be capable of reporting the probabilities of the alternatives of remaining nodes when other nodes are instantiated; it should also be capable of reporting the uncertainty in these probabilities relative to the uncertainty in the probabilities which are stored in the network. In this paper a method for determining the variances in inferred probabilities is obtained under the assumption that a posterior distribution on the uncertainty variables can be approximated by the prior distribution. It is shown that this assumption is plausible if their is a reasonable amount of confidence in the probabilities which are stored in the network. Furthermore in this paper, a surprising upper bound for the prior variances in the probabilities of the alternatives of all nodes is obtained in the case where the probability distributions of the probabilities of the alternatives are beta distributions. It is shown that the prior variance in the probability at an alternative of a node is bounded above by the largest variance in an element of the conditional probability distribution for that node.