# Variational inference by message-passing in graphical models

Variational inference where the model factorizes over some graphical independence structure, which means we get cheap and distributed inference. I am currently particularly interested in this for latent GP models. Many things can be expressed as message passing algorithms. The grandparent idea in this unification seems to be “Belief propagation”, a.k.a. “sum-product message-passing”, credited to (Pearl, 1982) for DAGs and then generalised to MRFs, PGMs, factor graphs etc. Apparently this definition subsumes such diverse models as the Viterbi and Baum-Welch algorithms, among others, and refers to more or less any method that allows local computation of a big statistical model using the graphical conditional independence structure. There are many overviews. Dustin Tran does a good one discussing .

The recipe to make a message-passing algorithm has four steps:

• Pick an approximating family for q to be chosen from. For example, the set of fully-factorized distributions, the set of Gaussians, the set of k-component mixtures, etc.
• Pick a divergence measure to minimize. For example, mean-field methods minimize the Kullback-Leibler divergence $$KL(q \| p)$$, expectation propagation minimizes $$KL(p \| q)$$, and power EP minimizes α-divergence, $$D\alpha(p \| q)$$.
• Construct an optimization algorithm for the chosen divergence measure and approximating family. Usually this is a fixed-point iteration obtained by setting the gradients to zero.
• Distribute the optimization across the network, by dividing the network p into factors, and minimizing local divergence at each factor.

There is an overview lecture by Thomas Orton, which connects this with statistical mechanics of statistics.

Last week, we saw how certain computational problems like 3SAT exhibit a thresholding behavior, similar to a phase transition in a physical system. In this post, we’ll continue to look at this phenomenon by exploring a heuristic method, belief propagation (and the cavity method), which has been used to make hardness conjectures, and also has thresholding properties. In particular, we’ll start by looking at belief propagation for approximate inference on sparse graphs as a purely computational problem. After doing this, we’ll switch perspectives and see belief propagation motivated in terms of Gibbs free energy minimization for physical systems.

Interesting projects in this vein:

• GAMP.

• ForneyLab looks especially useful for me:

The message passing paradigm offers a convenient method for leveraging model-specific structures, while remaining generally applicable. Message passing can be conveniently formulated on a Forney-style factor graph (FFG) representation of the model [2]. Inference tasks on the model can then be decomposed in local computations, represented by messages that flow across the graph. This locality allows for storing pre-computed message updates in a look-up table that can be re-used across models. Automated algorithm construction then amounts to scheduling these messages in the order required by the inference task (see also this conference paper at JuliaCon).

ForneyLab (GitHub) is introduced in this paper as a novel Julia package that allows the user to specify a probabilistic model as an FFG and pose inference problems on this FFG. In return, ForneyLab automatically constructs a Julia program that executes a message passing-based (approximate) inference procedure. ForneyLab is designed with a focus on flexibility, extensibility and applicability to biologically plausible models for perception and decision making, such as the hierarchical Gaussian filter (HGF). With ForneyLab, the search for better models for perception and action can be accelerated

This idea has been baked into various probabilistic programming frameworks by now.

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