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

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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

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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

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

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


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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...





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