Quantifying the Uncertainty of a Belief Net Response
Reports
Information about Bayesian belief nets in general
"Improved Mean and Variance Approximations for Belief Net Responses via Network Doubling" Peter Hooper, Yasin Abbasi-Yadkori, Russ Greiner, Bret Hoehn UAI-2009
UAI'01: Error Bars for Belief Net Inference (see below)
2007 Artificial Intelligence: doi:10.1016/j.artint.2007.09.004
Tech Report (draft version)
If you want to skim:
-Fig 1 (p4) overviews the system -- MeanVar is the main challenge
-Thm 2 (p11-12) gives the actual formal result; the discussion afterwards illustrates some relevant details
-Fig 7 (p23) summarizes the experiment runs to validate this idea
-Fig 8 (p26) shows how close our variance measures are to the "truth"
-Fig 10 and Fig 11 (p28) show why we use the Beta form, rather than Normal
-Appendix A discusses how to implement this FOR COMPLETE QUERIES -- showing straightline code for the computation (see Thm6, p40-41)(Sec 4.2 [p19-21] describes one way to compute the relevant quantities for general queries.)
Applications of "variance"
Combining different BeliefNet classifiers using variance (Using Query-Specific Variance Estimates to Combine Bayesian Classifiers)
Using Bias2+Variance to select the "best" BeliefNet classifier (Discriminative Model Selection for Belief Net Structures)
Auxiliary information related to the experiments run
Actual Alarm network, Queries, Parameters
Information about Bucket Elimination
Additional Details
Bayesian Error-Bars for Belief Net Inference (ps) (PDF)
TIM VAN ALLEN, RUSSELL GREINER, PETER HOOPERProceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI-01), Seattle, August 2001
A Bayesian Belief Network (BN) is a model of a joint distribution over a finite set of variables, with a DAG structure to represent the immediate dependencies between the variables, and a set of parameters (aka CPTables) to represent the local conditional probabilities of a node, given each assignment to its parents. In many situations, the parameters are themselves treated as random variables --- reflecting the uncertainty remaining after drawing on knowledge of domain experts and/or observing data generated by the network. A distribution over the CPtable parameters induces a distribution for the response the BN will return to any ``What is P(H | E)?'' query. This paper investigates the distribution of this response, shows that it is asymptotically normal, and derives closed-form expressions for its mean and asymptotic variance. We show that this computation has the same complexity as simply computing the (mean value of the) response --- ie, O(n * exp(w)), where n is the number of variables and w is the effective tree width. We also provide empirical evidence showing that the error-bars computed from our estimates are fairly accurate in practice, over a wide range of belief net structures and queries.
Other Possible Ways To Use Variance
To make reliable decisions
Related to the 2006 KDD Cup
In the context of active learning:
Related to the Active Learning With Statistical Methods