A Generalized Mean Field Algorithm for Variational Inference in Exponential Families (1212.2512v1)

Abstract: The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equations and unclear inference quality in various models, it is not widely used as a generic approximate inference algorithm. In this paper, we discuss a generalized mean field theory on variational approximation to a broad class of intractable distributions using a rich set of tractable distributions via constrained optimization over distribution spaces. We present a class of generalized mean field (GMF) algorithms for approximate inference in complex exponential family models, which entails limiting the optimization over the class of cluster-factorizable distributions. GMF is a generic method requiring no model-specific derivations. It factors a complex model into a set of disjoint variable clusters, and uses a set of canonical fix-point equations to iteratively update the cluster distributions, and converge to locally optimal cluster marginals that preserve the original dependency structure within each cluster, hence, fully decomposed the overall inference problem. We empirically analyzed the effect of different tractable family (clusters of different granularity) on inference quality, and compared GMF with BP on several canonical models. Possible extension to higher-order MF approximation is also discussed.

• The paper introduces a GMF algorithm that approximates posterior distributions by optimizing a lower likelihood bound.
• It leverages nonoverlapping clusters to simplify message-passing while preserving intra-cluster dependencies.
• Empirical results demonstrate improved marginal probability estimates and competitive performance compared to BP and GBP.

A Generalized Mean Field Approach for Variational Inference in Exponential Family Graphical Models

The paper by Xing et al. introduces a novel class of algorithms—Generalized Mean Field (GMF) methods—designed to tackle variational inference within exponential family graphical models using nonoverlapping clusters. This approach offers an alternative to the Generalized Belief Propagation (GBP) methods. While GBP operates with overlapping clusters, GMF is predicated on utilizing disjoint clusters, and it is demonstrated to ensure convergence to a globally consistent set of marginals along with a lower likelihood bound.

Theoretical Foundation and Methodological Advancements

The GMF algorithm capitalizes on the architectural choice of nonoverlapping clusters within a graphical model. This strategic decision simplifies the iterative message-passing mechanics and provides computational efficiency. This advantage is articulated through the GMF theorem, which ensures that, given some conditions, the resultant variational distributions maintain the intrinsic intra-cluster dependencies of the original model, independent of the external structures.

From a technical perspective, the GMF approach pivots on approximating the posterior probability distributions by transforming an inference problem into an optimization challenge. This transformation involves maximizing a lower bound of the likelihood with respect to the chosen cluster marginals, where each marginal can be independently computed once the Markov blanket messages are conveyed, reminiscent of Expectation Propagation (EP) algorithms.

Notably, GMF circumvents the necessity for manually configuring clusters, presenting potential for automatized cluster selection using graph partitioning techniques, which remains an area for exploration.

Empirical Validation and Comparative Analysis

The paper's authors validate GMF's competence through empirical evaluations across canonical models: the Ising model, sigmoid networks, and factorial hidden Markov models (fHMMs). The experiments underscore GMF's superior accuracy in estimating marginal probabilities when compared to the Belief Propagation (BP) algorithm, particularly in strongly coupled Ising models. Results further indicate that the GMF algorithm, contingent on cluster selection strategy, offers a magnified flexibility to balance between computational cost and precision.

Interestingly, GMF's performance rivals GBP for Ising models by leveraging nonoverlapping clusters, while avoiding GBP's computational complexities associated with overlapping clusters. Additionally, the algorithm delivers plausible outcomes even in dense dependency models like sigmoid networks, where BP may underperform due to heavy computational overhead.

Implications for Future Research

The research contributions articulated by Xing et al. present significant theoretical implications by providing a robust variational inference framework adaptable to a variety of graphical models. Practically, GMF holds promise for scalable inference tasks typical in complex models deployed in machine learning and statistical domains.

Future research may delve into automating cluster selection based on spectral graph partitioning or other optimization methods, potentially enhancing GMF's operational efficacy. Furthermore, incorporating higher-order variational approximations could strengthen the delivered bounds on likelihood estimates, thereby improving inference accuracy in even more intricate models.

In conclusion, this work lays an important foundation in variational inference, proposing an uncomplicated yet effective alternative to traditional approximation methods, and heralds a path toward more adaptive and scalable probabilistic modeling solutions.