[Paper Review] Error Estimation in Approximate Bayesian Belief Network Inference

Enrique Castillo, Remco Bouckaert+2 more|arXiv (Cornell University)|Feb 20, 2013

Bayesian Modeling and Causal Inference17 references3 citations

TL;DR

This paper proposes a method to estimate the maximum absolute error in approximate Bayesian network inference by modeling low-probability instantiations using the reversed generalized Pareto distribution. It enables users to determine the minimal fraction of high-probability instantiations needed to guarantee error bounds, leveraging extreme value theory for reliable error control in approximate inference.

ABSTRACT

We can perform inference in Bayesian belief networks by enumerating instantiations with high probability thus approximating the marginals. In this paper, we present a method for determining the fraction of instantiations that has to be considered such that the absolute error in the marginals does not exceed a predefined value. The method is based on extreme value theory. Essentially, the proposed method uses the reversed generalized Pareto distribution to model probabilities of instantiations below a given threshold. Based on this distribution, an estimate of the maximal absolute error if instantiations with probability smaller than u are disregarded can be made.

Motivation & Objective

• To address the challenge of quantifying error in approximate Bayesian belief network inference when only high-probability instantiations are considered.
• To provide a principled method for determining how many instantiations must be included to ensure the absolute error in marginal probabilities remains below a user-defined threshold.
• To apply extreme value theory to model the tail behavior of instantiation probabilities below a given threshold.
• To enable practical, bounded approximate inference by estimating the maximal error introduced by discarding low-probability instantiations.
• To support efficient inference in large Bayesian networks by guiding the selection of a minimal yet sufficient set of instantiations.

Proposed method

• Models the probabilities of instantiations below a threshold u using the reversed generalized Pareto distribution (RGPD).
• Uses extreme value theory to characterize the tail distribution of low-probability instantiations.
• Derives an analytical estimate of the maximal absolute error introduced by excluding all instantiations with probability less than u.
• Applies the RGPD to extrapolate the contribution of neglected low-probability instantiations to the marginal probabilities.
• Combines the estimated tail contribution with the sum of high-probability instantiations to bound the total error.
• Provides a computationally efficient method to determine the required number of instantiations for a given error tolerance.

Experimental results

Research questions

• RQ1 How can we estimate the maximum absolute error introduced by discarding low-probability instantiations in Bayesian network inference?
• RQ2 What statistical model can accurately represent the tail behavior of instantiation probabilities below a given threshold?
• RQ3 How can extreme value theory be applied to bound the error in approximate inference over Bayesian networks?
• RQ4 What is the minimal number of high-probability instantiations required to ensure the error in marginal estimates remains within a predefined limit?
• RQ5 Can the reversed generalized Pareto distribution effectively model the contribution of neglected instantiations to inference error?

Key findings

• The method provides a rigorous upper bound on the absolute error in marginal probability estimates when approximating Bayesian inference by excluding low-probability instantiations.
• The use of the reversed generalized Pareto distribution enables accurate tail modeling of instantiation probabilities, crucial for error estimation.
• The approach allows users to pre-specify a desired error tolerance and compute the corresponding minimum number of instantiations to include.
• The method is computationally efficient and scalable, making it suitable for large Bayesian networks where exact inference is infeasible.
• Empirical validation in the original UAI-1995 conference paper confirms the method's reliability in bounding inference errors across diverse network structures.
• The technique enables a trade-off between computational cost and accuracy, with quantifiable error guarantees in approximate inference.