[Paper Review] Inference for Belief Networks Using Coupling From the Past

Mike Harvey, Radford M. Neal|arXiv (Cornell University)|Jan 16, 2013

Bayesian Modeling and Causal Inference8 references3 citations

TL;DR

This paper proposes a novel exact inference method for belief networks using Coupling From the Past (CFTP) combined with a compact state summary for layered noisy-or networks. By simulating Gibbs samplers from all possible past states and leveraging a summarized representation of state sets, the method achieves exact sampling from the true posterior distribution—though at the cost of potentially more simulation steps than standard Gibbs sampling—while requiring only about twice the computational time per step.

ABSTRACT

Inference for belief networks using Gibbs sampling produces a distribution for unobserved variables that differs from the correct distribution by a (usually) unknown error, since convergence to the right distribution occurs only asymptotically. The method of "coupling from the past" samples from exactly the correct distribution by (conceptually) running dependent Gibbs sampling simulations from every possible starting state from a time far enough in the past that all runs reach the same state at time t=0. Explicitly considering every possible state is intractable for large networks, however. We propose a method for layered noisy-or networks that uses a compact, but often imprecise, summary of a set of states. This method samples from exactly the correct distribution, and requires only about twice the time per step as ordinary Gibbs sampling, but it may require more simulation steps than would be needed if chains were tracked exactly.

Motivation & Objective

• To address the issue of approximate inference in belief networks due to asymptotic convergence in Gibbs sampling.
• To develop a method that ensures exact sampling from the true posterior distribution in belief networks.
• To make CFTP feasible for large belief networks by replacing explicit enumeration of all states with a compact, summarized representation.
• To maintain computational efficiency by limiting the per-step cost to roughly double that of standard Gibbs sampling.

Proposed method

• The method applies Coupling From the Past (CFTP) to belief networks, conceptually running Gibbs samplers from all possible initial states in the distant past until all chains coalesce at time t=0.
• For large networks where tracking all states is infeasible, it uses a compact summary of sets of states instead of tracking individual states explicitly.
• The summary is designed for layered noisy-or networks, exploiting their structure to represent multiple states efficiently using a minimal set of parameters.
• The algorithm ensures that all chains couple by time t=0, guaranteeing exact samples from the target distribution.
• It maintains detailed balance and reversibility by using standard Gibbs transition kernels within the CFTP framework.
• The method relies on the monotonicity of the noisy-or structure to ensure coalescence of chains across the entire state space.

Experimental results

Research questions

• RQ1 Can Coupling From the Past be adapted to perform exact inference in large belief networks without explicitly tracking all possible states?
• RQ2 How can the computational cost of CFTP be reduced in belief networks while preserving exactness of the sampling?
• RQ3 What kind of state summary is sufficient to maintain correctness in CFTP for layered noisy-or networks?
• RQ4 Does the use of a compact state summary lead to a significant increase in the number of simulation steps compared to exact chain tracking?
• RQ5 Can the method achieve near-constant time per step relative to standard Gibbs sampling while ensuring exact inference?

Key findings

• The proposed method achieves exact inference by ensuring all coupled chains coalesce at time t=0, guaranteeing samples from the true posterior distribution.
• The use of a compact state summary enables the method to scale to larger networks where full enumeration of states is infeasible.
• The computational cost per step is approximately double that of standard Gibbs sampling, making it efficient in practice.
• Despite the efficiency per step, the method may require more simulation steps than exact chain tracking due to the imprecision of the state summary.
• The method is applicable to layered noisy-or networks, leveraging their structural properties to maintain correctness under summarization.