Measuring the Hardness of Stochastic Sampling on Bayesian Networks with Deterministic Causalities: the k-Test

TL;DR

This paper introduces the k-test, a new method to measure the difficulty of stochastic sampling in Bayesian networks with deterministic causalities, predicting rejection rates and intractability.

Contribution

The k-test extends existing measures to Bayesian networks with zero probabilities, providing a practical tool for assessing sampling efficiency.

Findings

k-test accurately predicts rejection rates in stochastic sampling

It identifies when sampling becomes computationally infeasible

The method applies to networks with deterministic causalities

Abstract

Approximate Bayesian inference is NP-hard. Dagum and Luby defined the Local Variance Bound (LVB) to measure the approximation hardness of Bayesian inference on Bayesian networks, assuming the networks model strictly positive joint probability distributions, i.e. zero probabilities are not permitted. This paper introduces the k-test to measure the approximation hardness of inference on Bayesian networks with deterministic causalities in the probability distribution, i.e. when zero conditional probabilities are permitted. Approximation by stochastic sampling is a widely-used inference method that is known to suffer from inefficiencies due to sample rejection. The k-test predicts when rejection rates of stochastic sampling a Bayesian network will be low, modest, high, or when sampling is intractable.