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The analysis of Belief Propagation and other algorithms for the {\emreconstruction problem} plays a key role in the analysis of community detectionin inference on graphs, phylogenetic reconstruction in bioinformatics, and thecavity method in statistical physics.  We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) whichstates that any bounded memory message passing algorithm is statistically muchweaker than Belief Propagation for the reconstruction problem. More formally,any recursive algorithm with bounded memory for the reconstruction problem onthe trees with the binary symmetric channel has a phase transition strictlybelow the Belief Propagation threshold, also known as the Kesten-Stigum bound.The proof combines in novel fashion tools from recursive reconstruction,information theory, and optimal transport, and also establishes an asymptoticnormality result for BP and other message-passing algorithms near the criticalthreshold.

Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation