Bayesian Error-Bars for Belief Net Inference
  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.
