Article,
Quantifying the uncertainty of a belief net response: Bayesian error-bars for belief net inference
Artificial Intelligence, 172 (4-5): 483 - 513 (2008)
DOI: DOI: 10.1016/j.artint.2007.09.004
Abstract
A Bayesian belief network models a joint distribution over variables
using a DAG to represent variable dependencies and network parameters
to represent the conditional probability of each variable given an
assignment to its immediate parents. Existing algorithms assume each
network parameter is fixed. From a Bayesian perspective, however,
these network parameters can be random variables that reflect uncertainty
in parameter estimates, arising because the parameters are learned
from data, or because they are elicited from uncertain experts. Belief
networks are commonly used to compute responses to queries--i.e.,
return a number for . Parameter uncertainty induces uncertainty in
query responses, which are thus themselves random variables. This
paper investigates this query response distribution, and shows how
to accurately model this distribution for any query and any network
structure. In particular, we prove that the query response is asymptotically
Gaussian and provide its mean value and asymptotic variance. Moreover,
we present an algorithm for computing these quantities that has the
same worst-case complexity as inference in general, and also describe
straight-line code when the query includes all n variables. We provide
empirical evidence that (1) our approximation of the variance is
very accurate, and (2) a Beta distribution with these moments provides
a very accurate model of the observed query response distribution.
We also show how to use this to produce accurate error bars around
these responses--i.e., to determine that the response to is x�y with
confidence 1-delta.
