Second Order Approximations for Probability Models

Abstract

In this paper, we derive a second order mean field theory for directed graphical probability models. By using an information theoretic argu(cid:173) ment it is shown how this can be done in the absense of a partition function. This method is a direct generalisation of the well-known TAP approximation for Boltzmann Machines. In a numerical example, it is shown that the method greatly improves the first order mean field ap(cid:173) proximation. For a restricted class of graphical models, so-called single overlap graphs, the second order method has comparable complexity to the first order method. For sigmoid belief networks, the method is shown to be particularly fast and effective.