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Simulation Approaches to General Probabilistic Inference on Belief Networks
Ross D. Shachter, Mark Alan Peot
TL;DR
The paper addresses the need for probabilistic inference algorithms that work across multiply connected networks and extreme probabilities. It investigates forward Monte Carlo methods related to Logic Sampling, adding variance-reduction and importance-sampling enhancements. The Basic and Markov Blanket Algorithms appear competitive and simple and robust across a wide variety of inference problems, although the paper lacks theoretical convergence bounds for most variants.
Problem
The paper seeks simulation algorithms suitable for probabilistic inference in networks with multiple connectedness, extreme probabilities, and deterministic logical functions.
Method
The paper develops forward Monte Carlo algorithms related to Logic Sampling, including a Basic Algorithm, Markov Blanket Scoring, and importance-distribution enhancements.
Results
The Basic and Markov Blanket Algorithms were winners for each case considered and overall, while Self-Importance and Heuristic Importance performed poorly compared with the other new algorithms.
Takeaways & Limitations
The proposed algorithms appear simple and robust for a wide variety of probabilistic inference problems, while enhancements can be applied selectively or adaptively and support parallel, interruptible computation.
Takeaways & Limitations
The paper lacks a theoretical upper bound on convergence rates for the algorithms beyond the Basic Algorithm without observed experimental evidence.
Abstract
from arXiv · showhide
A number of algorithms have been developed to solve probabilistic inference problems on belief networks. These algorithms can be divided into two main groups: exact techniques which exploit the conditional independence revealed when the graph structure is relatively sparse, and probabilistic sampling techniques which exploit the "conductance" of an embedded Markov chain when the conditional probabilities have non-extreme values. In this paper, we investigate a family of "forward" Monte Carlo sampling techniques similar to Logic Sampling [Henrion, 1988] which appear to perform well even in some multiply connected networks with extreme conditional probabilities, and thus would be generally applicable. We consider several enhancements which reduce the posterior variance using this approach and propose a framework and criteria for choosing when to use those enhancements.
1. Introduction
The paper targets probabilistic inference methods that remain practical on multiply connected belief networks with extreme conditional probabilities. It develops forward Monte Carlo algorithms and enhancements intended to reduce posterior variance and support problem-sensitive selection.
- Motivation: Exact inference methods can become intractable on medium-sized general networks because their complexity is sensitive to graph connectedness and the problem is NP-hard.Monte Carlo methods offer an alternative, but Markov chain approaches can degrade with near-zero conditional probabilities, while Logic Sampling degrades exponentially with the number of evidence variables.
- Motivation: The research goal is simulation algorithms suitable for multiple connectedness, extreme probabilities, and deterministic logical functions.The authors seek generally applicable methods rather than algorithms superior on every problem.
- Basic Algorithm: Each sample receives a score based on the original probability divided by its sampling probability, with scores accumulated and normalized to estimate posterior marginals.This weighting corrects for the difference between the probability used to select a sample and its probability in the original distribution.
- Basic Algorithm: The Basic Algorithm modifies Logic Sampling by conditioning samples on observed evidence and simulating successors of evidence nodes accordingly.Unlike Logic Sampling, it rarely discards cases when the conditional probabilities for evidence variables are nonzero, even when the evidence was unlikely beforehand.
- Enhancements: Markov Blanket Scoring reduces score variance by scoring all possible states of a variable using probabilities proportional to its Markov Blanket distribution.It can be applied selectively to individual variables when the additional accuracy justifies the additional computation.
- Enhancements: Importance sampling replaces the Basic Algorithm’s sampling distribution with an alternative distribution that approximates the posterior, including Self-Importance and Heuristic-Importance variants.The importance distribution must be independent of the samples it scores and must assign nonzero sampling probability to every possible value.
2. The Chavez algorithm is really designed
The paper evaluates inference algorithms on a standard multiply connected cancer network and a network containing deterministic logical nodes, using repeated trials and error-based comparisons. The tests include evidence conditions and acknowledge that some algorithms require more iterations than the experiments allowed.
- Evaluation limitations: Some algorithms were not tested long enough for a representative showing because they required substantially more iterations than the experiments permitted.The implementation used ten restarts on every trial.
- Experimental procedure: Each experiment uses 250- and 1000-iteration trials, repeated 25 times because results vary substantially across trials.The reported observations are means over the 25 repetitions.
- Evaluation measures: Performance is assessed using error against exact marginal probabilities, mean runtime, error standard deviation, and squared mean error multiplied by mean time.The last product is reported because it appears fairly invariant for some algorithms as iteration counts increase.
- Experimental problems: The evaluation uses two networks: Cooper’s standard multiply connected cancer problem and a constructed network containing deterministic OR and AND nodes.The cancer problem is tested with and without evidence; the logical-node problem examines performance when deterministic structure is present.
- Test conditions: The tested conditional probabilities include extreme values, and the study examines both no-evidence and E-true conditions for the logical-network problem.The stated model assigns probabilities such as 0.9, 0.8, 0.2, and 0.1 to its conditional cases.
4. Conclusions
The Basic and Markov Blanket Algorithms performed competitively overall, while the proposed methods remain simple and robust across varied probabilistic inference problems. Their enhancements can be applied selectively, parallelized, and interrupted, but important performance and convergence limitations remain.
- The Basic Algorithm and Markov Blanket Algorithm were winners for each case considered and overall.
- Among Markov chain algorithms, Pearl with Markov Blanket Algorithm was best, but deterministic or near-deterministic functions caused expected problems.
- Markov Blanket modifications can be applied selectively to individual nodes when additional accuracy warrants additional computation.
- These enhancements can be applied adaptively to improve simulation accuracy, and the algorithm is suitable for parallel processing.
- Basic Algorithm performance can be poor when likelihood products vary greatly between iterations, because most iterations are then ignored.
- A theoretical upper bound on convergence rate is unavailable for the other algorithms and for the Basic Algorithm with observed evidence.
- The Basic and Markov Blanket Algorithms appear competitive, while the proposed algorithms are simple and robust for a wide variety of probabilistic inference problems.