It is possible to combine multiple probabilistic models of the same data by multiplying the probabilities together and then renormalizing. This is a very efficient way to model high-dimensional data which Simultaneously satisfies many differentlow-dimensional constraints. Each individual expert model can focus on giving high probability to data vectors that satisfy just one of the constraints. Data vectors that satisfy this one constraint but violate other constraints will be ruled out by their low probability under the other expert models. Training a product of models appears difficult because, in addition to maximizing the probabilities that the individual models assign to the observed data, it is necessary to make the models disagree onunobserved regions of the data space: It is fine for one model to assign a high probability to an unobserved region as long as some other model assigns it a very low probability. Fortunately, if the individual models are tractable there is a fairlyefficient way to train a product of models. This training algorithm suggests a biologically plausible way of learning neural population codes.
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