Rich Neapolitan <[EMAIL PROTECTED]> wrote: > I wrote some papers in the early 90's on computing the variances in > inferred relative frequencies in Bayesian networks. At the time, I was not > thinking in terms of an equivalent sample size as Heckerman et al did not > publish their paper concerning this subject until the mid 90's. Now using > an equivalent sample size, I am getting results similar to what Heckerman > et al proved concerning the probability of data. Namely, the variances are > the same for equivalent networks as long as we use the same equivalent > sample size (and of course the same probability distribution). Of course, > this is what we want, but I don't see an apparent proof of it. I was > wondering if anyone pursued this matter. We have recently pursued a related line of work: computing the variance of a belief net's response to a specific query, given that the CPtables of a fixed structure were obtained from a sample of tuples -- see the abstract given below. The tech report (expanded from a recent submission) can be downloaded from http://www.cs.ualberta.ca/~greiner/ErrorBar/ Error-Bars for Belief Net Inference Tim van Allen and Russ Greiner Many belief nets are built by first obtaining a structure (typically from an expert) then using a random training sample (drawn iid from underlying distribution) to fill in the parameters. The answers to queries produced by the resulting belief net therefore depend on this random sample. We investigate how the expected variability of this sample translates to a variance around the answers produced by the associated belief net. We present both a Frequentist approach using confidence intervals, where integration is done under the sampling distribution, and a Bayesian approach using credible regions, where integration is done under the posterior distribution. Within each approach, we present two methods for computing the error-bars: an analytic method based on a first-order Taylor series approximation, and a simulation method based on Monte Carlo simulation. We compare and contrast all four approaches (the two kinds of error-bars and the two methods for computing them), by explaining empirical results obtained over a range of queries and belief net structures. | R Greiner Phone: (780) 492-5461 | | Dep't of Computing Science FAX: (780) 492-1071 | | University of Alberta Email: [EMAIL PROTECTED] | | Edmonton, AB T6G 2H1 Canada http://www.cs.ualberta.ca/~greiner/ |
