**Abstract**

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 different
low-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 on unobserved 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 fairly efficient way to train a product of
models. This training algorithm suggests a biologically plausible way of learning neural
population codes.

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*Proceedings of the Ninth International Conference on
Artificial Neural Networks [ICANN 99 Vol 1 pages 1-6]*

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