[Paper Review] Bayesian Error-Bars for Belief Net Inference

Tim Van Allen, Russell Greiner+1 more|arXiv (Cornell University)|Jan 10, 2013

Bayesian Modeling and Causal Inference20 references18 citations

TL;DR

This paper proposes a method to compute Bayesian error-bands for probability queries in Bayesian networks by modeling parameter uncertainty as random variables. It derives the asymptotic normality of query response variance and presents an efficient O(n 2^w) algorithm to compute these error-bands with the same complexity as mean inference, validated empirically on small samples.

ABSTRACT

A Bayesian Belief Network (BN) is a model of a joint distribution over a setof n variables, with a DAG structure to represent the immediate dependenciesbetween the variables, and a set of parameters (aka CPTables) to represent thelocal conditional probabilities of a node, given each assignment to itsparents. In many situations, these parameters are themselves random variables - this may reflect the uncertainty of the domain expert, or may come from atraining sample used to estimate the parameter values. The distribution overthese "CPtable variables" induces a distribution over the response the BNwill return to any "What is Pr(H | E)?" query. This paper investigates thevariance of this response, showing first that it is asymptotically normal,then providing its mean and asymptotical variance. We then present aneffective general algorithm for computing this variance, which has the samecomplexity as simply computing the (mean value of) the response itself - ie,O(n 2^w), where n is the number of variables and w is the effective treewidth. Finally, we provide empirical evidence that this algorithm, whichincorporates assumptions and approximations, works effectively in practice,given only small samples.

Motivation & Objective

• To quantify uncertainty in probability queries of Bayesian networks due to uncertain CP-table parameters.
• To model the distribution of query responses when CP-table parameters are treated as random variables.
• To derive the asymptotic distribution of the response variance for Pr(H|E) queries.
• To develop an efficient algorithm for computing error-bands with the same complexity as mean inference.
• To validate the method's effectiveness using empirical evidence from small training samples.

Proposed method

• Treats CP-table parameters as random variables to represent epistemic uncertainty from expert estimates or small training samples.
• Derives the asymptotic normality of the response distribution for any Pr(H|E) query under parameter uncertainty.
• Uses influence diagrams and path-trace analysis to compute the asymptotic variance of the query response.
• Applies a general algorithm that computes the variance in O(n 2^w) time, matching the complexity of mean inference.
• Employs approximations and assumptions in the algorithm to ensure practicality while maintaining accuracy.
• Validates the method using empirical evaluation on small datasets, showing reliable error-band estimation.

Experimental results

Research questions

• RQ1 How does the variance of a Bayesian network query response behave when CP-table parameters are uncertain?
• RQ2 Can the response variance be efficiently computed without incurring significant computational overhead?
• RQ3 Is the distribution of the query response asymptotically normal under parameter uncertainty?
• RQ4 How accurate are the computed error-bands when only small training samples are available?
• RQ5 Can the proposed algorithm maintain practical efficiency while capturing meaningful uncertainty in inference?

Key findings

• The response variance for any Pr(H|E) query is asymptotically normal under parameter uncertainty.
• The mean and asymptotic variance of the query response can be analytically derived using influence diagram techniques.
• The proposed algorithm computes error-bands in O(n 2^w) time, matching the complexity of standard belief propagation.
• Empirical results show the method produces reliable error-bands even with small training samples.
• The algorithm effectively incorporates approximations and assumptions without sacrificing practical accuracy.
• The method enables uncertainty quantification in belief network inference with minimal computational cost.