Stabilised Moment-Form Junction Tree for Gaussian Bayesian Networks with Semidefinite Models
25 Pages Posted: 29 Jul 2022
Abstract
This paper presents a junction tree algorithm for linear Gaussian probability density functions (pdfs). This new formulation produces exact solutions even if some models are (semi-) deterministic and some posterior marginals have linearly dependent covariance. Given a Bayesian network where all conditional probabilities have positive definite covariance matrices, then a canonical-form or information-form Gaussian junction tree is sufficient, and gives the exact solution by simple sum-product message passing. However, this approach becomes numerically unstable if some conditional pdfs have covariance that is semidefinite or nearly so; the information-form cannot represent deterministic relationships. The moment-form Gaussian representation can manage semidefinite models, but has other numerical limitations that hinder a simple message passing solution. Existing moment-form algorithms employ lazy evaluation, and require considerable bookkeeping and manipulation to propagate messages in decomposed form. We propose a stabilised moment-form algorithm, where each junction tree clique is assigned a prior potential that acts as scaffolding for absorbing conditional pdfs. Judicious removal of this scaffolding during message passing permits a simple non-lazy strategy analogous to that of the information-form.
Keywords: Gaussian Bayesian networks, Junction tree algorithm, Deterministic models, Semidefinite covariance matrix
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