Expectation Propagation for approximate Bayesian inference (1301.2294v1)

Abstract: This paper presents a new deterministic approximation technique in Bayesian networks. This method, "Expectation Propagation", unifies two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. All three algorithms try to recover an approximate distribution which is close in KL divergence to the true distribution. Loopy belief propagation, because it propagates exact belief states, is useful for a limited class of belief networks, such as those which are purely discrete. Expectation Propagation approximates the belief states by only retaining certain expectations, such as mean and variance, and iterates until these expectations are consistent throughout the network. This makes it applicable to hybrid networks with discrete and continuous nodes. Expectation Propagation also extends belief propagation in the opposite direction - it can propagate richer belief states that incorporate correlations between nodes. Experiments with Gaussian mixture models show Expectation Propagation to be convincingly better than methods with similar computational cost: Laplace's method, variational Bayes, and Monte Carlo. Expectation Propagation also provides an efficient algorithm for training Bayes point machine classifiers.

• The paper presents the expectation propagation method that iteratively refines approximate Bayesian posteriors for improved inference.
• It bridges assumed-density filtering and loopy belief propagation to outperform traditional variational and Monte Carlo techniques.
• EP shows robust performance on Gaussian mixture models and Bayes Point Machine tasks, highlighting its practical relevance in complex networks.

Expectation Propagation for Approximate Bayesian Inference

Thomas P. Minka's paper, presented at UAI 2001, explores a novel deterministic approximation technique for Bayesian networks called Expectation Propagation (EP). The method effectively extends assumed-density filtering (ADF) and loopy belief propagation, thus bridging the gap between these two previously established techniques. This essay discusses the key aspects of the paper, providing an insightful overview of the proposed methodology, its comparative advantages, results obtained, and potential implications for future research.

Background and Motivation

Bayesian inference often faces significant computational challenges due to the complexity of exact solutions, particularly in belief networks with loops and hybrid structures. Traditional methods, such as variational Bayes and Monte Carlo techniques, either suffer from excessive computational costs or lack accuracy under certain conditions. Consequently, there is a persistent need for efficient and accurate approximation techniques. The paper introduces Expectation Propagation, which combines the strengths of ADF and loopy belief propagation to address these issues.

Expectation Propagation: Methodology

EP extends ADF by incorporating iterative refinement of approximate posteriors instead of a single-pass, sequential update. This iterative process allows the refinement of initial approximations based on later observations, thereby improving the retention of significant information that might be discarded in a pure ADF approach.

Key Steps in Expectation Propagation

1. Initialization: Approximation terms $t_i(x)$ are initialized.
2. Posterior Computation: The posterior for $x$ is computed from the product of approximation terms.
3. Refinement Loop:
   • Choose a term $t_i$ to refine.
   • Remove $t_i$ from the posterior to get an 'old' posterior $q^{-i}(x)$.
   • Combine $q^{-i}(x)$ and $t_i(x)$, and minimize the KL-divergence to obtain a new posterior $q(x)$.
   • Update $t_i$.
4. Normalization: Use the normalizing constant of $q(x)$ as an approximation to $p(D)$.

Comparative Analysis with Other Methods

EP was tested against methods like Laplace's method, variational Bayes, and Monte Carlo techniques on Gaussian mixture models. The numerical results demonstrated that EP achieved significantly better accuracy with comparable computational costs. As indicated in the experiments, EP closely matched the exact posterior, especially when the posterior was more Gaussian in nature, thereby outperforming sampling methods when the latter struggled with complex posterior shapes.

The Clutter Problem: A Case Study

In the clutter problem, EP was shown to yield superior performance in estimating the evidence $p(D)$ and the posterior mean $E[x|D]$. The paper's results highlighted EP's robustness and enhanced accuracy compared to ADF, Laplace's method, and variational Bayes, confirming its practical efficacy.

Loopy Belief Propagation and EP

The paper also connects EP with loopy belief propagation by showing that EP is a generalized form of it. Using EP in a loopy network leads to approximate marginal distributions through iterative updates, analogous to belief propagation in networks with loops. This generalization broadens the applicability of EP to a wider class of belief networks, including hybrid models with both discrete and continuous nodes.

Bayes Point Machine: Application of EP

EP was applied to the Bayes Point Machine (BPM), a Bayesian approach to linear classification. The algorithm approximates the posterior over the parameter vector with a Gaussian distribution. EP's iterative refinement process resulted in efficient computation of the Bayes point, validated by numerical experiments showing superior performance compared to the Support Vector Machine (SVM) and other existing algorithms.

Implications and Future Research

The introduction of Expectation Propagation marks a significant step in approximate Bayesian inference, offering a blend of high accuracy and computational efficiency. This technique is particularly useful in large-scale, complex Bayesian networks where traditional methods fall short. The potential extension of EP to handle arbitrary inner product functions, akin to kernel methods in SVMs, opens avenues for further research in scalable and flexible inference algorithms. Future developments in AI could leverage EP for more sophisticated models and applications, pushing the boundaries of what's achievable in probabilistic inference and machine learning.

Conclusion

Expectation Propagation enriches the toolkit of approximation methods for Bayesian inference by effectively marrying the iterative refinement of ADF with the versatility of loopy belief propagation. Theoretical guarantees of convergence, combined with practical performance benefits, position EP as a valuable method for researchers and practitioners dealing with complex belief networks and probabilistic models. This work sets the stage for future explorations into more adaptive and powerful inference strategies in the field of AI and machine learning.