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We establish upper bounds for the minimal number of hidden units for which abinary stochastic feedforward network with sigmoid activation probabilities anda single hidden layer is a universal approximator of Markov kernels. We showthat each possible probabilistic assignment of the states of $n$ output units,given the states of $k\geq1$ input units, can be approximated arbitrarily wellby a network with $2^{k-1}(2^{n-1}-1)$ hidden units.

Universal Approximation of Markov Kernels by Shallow Stochastic Feedforward Networks