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A Tighter Bound for Graphical Models

Department of Biophysics, University of Nijmegen, NL 6525 EZ Nijmegen, The Netherlands

H. J. Kappen

Department of Biophysics, University of Nijmegen, NL 6525 EZ Nijmegen, The Netherlands

We present a method to bound the partition function of a Boltzmann machine neural network with any odd-order polynomial. This is a direct extension of the mean-field bound, which is first order. We show that the third-order bound is strictly better than mean field. Additionally, we derive a third-order bound for the likelihood of sigmoid belief networks. Numerical experiments indicate that an error reduction of a factor of two is easily reached in the region where expansion-based approximations are useful.