Sum-product algorithm

Belief propagation, also known as sum-product message passing, is a message passing algorithm for performing inference on graphical models, such as Bayesian networks and Markov random fields. It calculates the marginal distribution for each unobserved node, conditional on any observed nodes. Belief propagation is commonly used in artificial intelligence and information theory and has demonstrated empirical success in numerous applications including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability.

The algorithm was first proposed by Judea Pearl in 1982, who formulated this algorithm on trees, and was later extended to polytrees. It has since been shown to be a useful approximate algorithm on general graphs.

If X={X_i} is a set of discrete random variables with a joint mass function p, the marginal distribution of a single X_i is simply the summation of p over all other variables:

However, this quickly becomes computationally prohibitive: if there are 100 binary variables, then one needs to sum over 2⁹⁹ ≈ 6.338 × 10²⁹ possible values. By exploiting the polytree structure, belief propagation allows the marginals to be computed much more efficiently.

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