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Variational Probabilistic Inference and the QMR-DT Network
T. S. Jaakkola, M. I. Jordan
TL;DR
The paper addresses obtaining marginal probabilities from joint distributions. It uses variational bounds on selected local conditionals, yielding bounds on the resulting joint and marginal probabilities.
Problem
The paper concerns obtaining marginal probabilities from a joint probability distribution.
Method
The variational method replaces selected local conditional probabilities with upper- or lower-bounding transformations.
Results
The transformed joint distribution bounds the true joint distribution, and marginalization provides bounds on marginal probabilities.
Takeaways & Limitations
The approach provides a way to obtain bounds on marginal probabilities from variationally transformed joint distributions.
Takeaways & Limitations
Computing the expectations is exponentially costly in the number of exactly treated positive findings.
Abstract
from arXiv · showhide
We describe a variational approximation method for efficient inference in large-scale probabilistic models. Variational methods are deterministic procedures that provide approximations to marginal and conditional probabilities of interest. They provide alternatives to approximate inference methods based on stochastic sampling or search. We describe a variational approach to the problem of diagnostic inference in the `Quick Medical Reference' (QMR) network. The QMR network is a large-scale probabilistic graphical model built on statistical and expert knowledge. Exact probabilistic inference is infeasible in this model for all but a small set of cases. We evaluate our variational inference algorithm on a large set of diagnostic test cases, comparing the algorithm to a state-of-the-art stochastic sampling method.
Ac kno wledgemen ts
The authors acknowledge the University of Pittsburgh and Randy Miller for access to the QMR-DT database, and thank David Heckerman for proposing the variational approach and providing guidance.
- The authors thank the University of Pittsburgh and Randy Miller for use of the QMR-DT database, and David Heckerman for suggesting variational methods and offering helpful counsel.
App endix A. Dualit y
The appendix explains how convex duality yields conjugate representations and supporting half-spaces for convex functions. It then shows how variational transformations produce bounds on joint, marginal, and likelihood probabilities, with parameter optimization tightening those bounds.
- Appendix A. Duality: Convex duality represents a convex function through half-spaces whose intersection forms its epigraph.The conjugate function defines the critical half-spaces, equivalently the tangent planes, needed for this representation.
- Appendix A. Duality: The dual of the dual recovers the original function and provides a computational procedure for calculating dual functions.For concave functions, the analogous results replace maximization with minimization and lower bounds with upper bounds.
- Appendix B. Optimization of the Variational Parameters: Replacing selected local conditional probabilities with upper- or lower-bounding variational transformations makes the transformed joint distribution a corresponding bound on the true joint distribution.This follows because products of bounds remain bounds.
- Appendix B. Optimization of the Variational Parameters: Marginalizing the transformed joint distribution yields bounds on marginal probabilities and likelihood, while optimizing variational parameters tightens the bound and selects a distribution for approximate inference.Upper bounds are minimized and lower bounds are maximized with respect to the variational parameters.
Upp er Bound T ransformations
The method constructs an upper bound on observed-finding likelihood using transformed conditional probabilities and optimizes its variational parameters. Convexity eliminates local minima, while computational shortcuts address exponential costs when many findings are treated exactly.
- Upper-bound construction: The upper bound is formed by introducing bounds for individual node conditional probabilities and minimizing the resulting quantity over variational parameters.Some findings are transformed while remaining positive findings are treated exactly.
- Optimization: Convexity of log P(f|ξ) means the optimization has no local minima, permitting standard Newton-Raphson updates.The derivation establishes convexity from the component terms in the transformed objective.
- Optimization: Fixed-point iterations provide an alternative optimization procedure, with derivatives computable in time linear in the number of diseases associated with each finding.The favorable scaling exploits noisy-OR dependence and marginal independence.
- Computational trade-offs: Expectations become exponentially costly in the number of exactly treated positive findings, motivating optimization after transforming most or all positive findings.This shortcut yields suboptimal variational parameters, but the reported accuracy loss is typically quite small.
- Simulation procedure: In the simulations, variational parameters were optimized after approximately half of the exactly treated findings had been introduced.The procedure was applied during the reported simulations rather than only as a theoretical simplification.
Lo w er Bound T ransformations
Lower-bounding transformations replace selected conditional probabilities to produce a likelihood lower bound, which is optimized through an EM-based variational procedure. The resulting updates are iterative, monotonic, normalized, and extend to mixtures of transformed and unchanged conditionals.
- Lower Bound Transformations: Selected conditional probabilities are replaced by lower-bounding transformations, yielding a lower-bounding expression for the likelihood.The bound is formed by multiplying with P(d) and marginalizing over d.
- Lower Bound Transformations: The optimization is mapped to latent-variable likelihood estimation, with d latent, f observed, and q treated as a parameter vector.The objective or its logarithm can be optimized using the EM algorithm.
- Lower Bound Transformations: EM updates variational parameters by maximizing the expected complete log-likelihood, producing a sequence that monotonically increases log P(f|q).Conditional-probability parameters are independent, allowing each corresponding term to be maximized separately.
mann.
This section is a bibliography of work on probabilistic inference, Bayesian networks, variational methods, simulation, diagnostic reasoning, and the INTERNIST-1/QMR knowledge base.
- Bayesian-network inference: References address Bayesian-network inference through exact, approximate, localized, bucket-elimination, and conditioning-based approaches.The bibliography includes work on computational complexity, mini-buckets, bucket elimination, localized partial evaluation, selective conditioning, and bounded conditioning.
- Stochastic inference: References include sampling and Monte Carlo methods for probabilistic inference and belief networks.Listed topics include sampling-based marginal-density calculation, simulation algorithms, likelihood weighting, blocking-Gibbs sampling, and Monte Carlo methods.
- Variational inference: References cover variational methods for inference and learning in graphical models.Cited works include Jaakkola’s thesis, Jaakkola and Jordan’s recursive approximation algorithms, and an introduction by Jordan and colleagues.
- Medical diagnosis: References include probabilistic diagnosis and diagnostic assistance using reformulations of the INTERNIST-1/QMR knowledge base.The cited work covers the probabilistic model and inference algorithms as well as evaluation of diagnostic performance.