# Quantifying the Uncertainty of a Belief Net Response: Bayesian Error-Bars for Belief Net Inference

Quantifying the Uncertainty of a Belief Net Response: Bayesian Error-Bars for Belief Net Inference - Download this document for free, or read online. Document in PDF available to download.

Variance, Credible interval, Error bars, Bucket elimination, Bayesian belief network

Additional contributors:

Subject-Keyword: Variance Credible interval Error bars Bucket elimination Bayesian belief network

Type of item: Computing Science Technical Report

Computing science technical report ID: TR07-11

Language: English

Place:

Time:

Description: Technical report TR07-11. 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 re ect uncertainty in parameter estimates, arising because the parameters are learned from data, or as they are elicited from uncertain experts. Belief networks are commonly used to compute responses to queries - i.e., return a number for PH=h | E=e. 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 it 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 estimate 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 PH=h | E=e is x \pm y with confidence 1 - \delta.

Date created: 2007

DOI: doi:10.7939-R3H70838F

License information: Creative Commons Attribution 3.0 Unported

Rights:

Author: ** Van Allen, Tim Singh, Ajit Greiner, Russell Hooper, Peter **

Source: https://era.library.ualberta.ca/

## Teaser

Quantifying the Uncertainty of a Belief Net
Response: Bayesian Error-Bars for Belief Net
Inference
Tim Van Allen a Ajit Singh b Russell Greiner c,∗ Peter Hooper d
a Apollo
Data Technologies, 12729 N.E.

20th Suite 7, Bellevue, WA 98005 USA
b Center
for Automated Learning and Discovery, Carnegie Mellon University,
Pittsburgh, PA 15213, USA
c Department
d Department
of Computing Science, University of Alberta, Edmonton, Alberta
T6G 2E8, Canada
of Mathematical and Statistical Scienc
1
Introduction
Bayesian belief nets (BNs), which provide a succinct model of a joint probability distribution, are used in an ever increasing range of applications [Dav95].
They are typically built by first finding an appropriate structure (either by
interviewing an expert, or by selecting a good model from training data), and
then using a training sample to estimate the parameters [Hec98].

The resulting
belief net is then used to answer queries — e.g., compute the conditional probability P(Cancer=true | Smoke=true, Gender=male).

These responses clearly
depend on the training sample used to instantiate the parameters, in that
different training samples will produce different parameters, which will lead
to different responses.
This paper investigates how variability within a sample induces variance in a
query response, and presents a technique for estimating the posterior distribution of the query responses produced by a belief net.

Stated informally, our
goal is an algorithm that takes
• A belief net structure that we assume is correct (i.e., an I-map of the true
distribution D [Pea88])
• A prior distribution over the network parameters Θ
• A data sample S generated from D
• A query of the form “What is q(Θ) = P(H = h | E = e, Θ) ?”
and returns both the expected value and the approximate variance of the query
response q(Θ), based on the posterior distribution of parameters given the
sample.

By using these moments and an appropriate distributional form, we
approxi...