Precision-Aware Hopfield Retrieval: Unifying Population Codes and Memory Retrieval with Information Optimization

Bayesian estimation has been used to explore optimal encoder-decoders for continuous quantities, but no such framework exists for discrete associative memory. We introduce precision-aware Hopfield retrieval, a framework that incorporates the encoder's Fisher information directly into a decoder of memory retrieval. Four results follow. First, optimal encoder-decoder information is shaped only by the prior, across both log and $L^p$ norm loss functions. Second, the optimal information follows a general coupling law across log and $L^p$ loss functions. Third, optimal precision-aware Hopfield retrieval uses a stimulus-dependent inverse temperature. It outperforms the encoder-blind Bayesian decoder on both retrieval loss and required coding resource. Last, the framework yields a closed-form expression for memory recall bias. It reproduces central-tendency and cognitive-load effects observed in cognitive psychology. These effects emerge only when the encoder and decoder are modeled jointly.