Bayesian Error-Bars for Belief Net Inference
This work addresses uncertainty estimation for practitioners using Bayesian networks, but it is incremental as it builds on existing inference methods.
The paper tackles the problem of quantifying uncertainty in Bayesian Belief Network inference by analyzing the variance of query responses due to uncertain parameters, showing it is asymptotically normal and providing an efficient algorithm for computing this variance with the same complexity as computing the mean response.
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.