Bayesian retrieval in associative memories with storage errors

It is well known that for finite-sized networks, onestep retrieval in the autoassociative Willshaw net is a suboptimal way to extract the information stored in the synapses. Iterative retrieval strategies are much better, but have hitherto only had heuristic justification. We show how they emerge naturally from considerations of probabilistic inference under conditions of noisy and partial input and a corrupted weight matrix. We start from the conditional probability distribution over possible patterns for retrieval. This contains all possible information that is available to an observer of the network and the initial input. Since this distribution is over exponentially many patterns, we use it to develop two approximate, but tractable, iterative retrieval methods. One performs maximum likelihood inference to find the single most likely pattern, using the (negative log of the) conditional probability as a Lyapunov function for retrieval. In physics terms, if storage errors are present, then the modified iterative update equations contain an additional antiferromagnetic interaction term and site dependent threshold values. The second method makes a mean field assumption to optimize a tractable estimate of the full conditional probability distribution. This leads to iterative mean field equations which can be interpreted in terms of a network of neurons with sigmoidal responses but with the same interactions and thresholds as in the maximum likelihood update equations. In the absence of storage errors, both models become very similiar to the Willshaw model, where standard retrieval is iterated using a particular form of linear threshold strategy.