  We present two sampling algorithms for probabilistic confidence inference in
Bayesian networks. These two algorithms (we call them AIS-BN-mu and
AIS-BN-sigma algorithms) guarantee that estimates of posterior probabilities
are with a given probability within a desired precision bound. Our algorithms
are based on recent advances in sampling algorithms for (1) estimating the mean
of bounded random variables and (2) adaptive importance sampling in Bayesian
networks. In addition to a simple stopping rule for sampling that they provide,
the AIS-BN-mu and AIS-BN-sigma algorithms are capable of guiding the learning
process in the AIS-BN algorithm. An empirical evaluation of the proposed
algorithms shows excellent performance, even for very unlikely evidence.
