## Abstract

Graphical models are important and useful, but come with a serious practical problem. For many models, we cannot compute either the normalizing constant or the maximum a posteriori state. It will help to have some notation. Write *X* for a set of observed values, *H*_{1}, …, *H*_{N} for the unknown (hidden) values of interest. We will assume that these are discrete. We seek the values of *H*_{1}, …, *H*_{N} that maximizes *P*(*H*_{1}, …, *H*_{N}|*X*). There is an exponential number of such possible values, so we must exploit some kind of structure in the problem to find the maximum. In the case of a model that could be drawn as a forest, this structure was easily found; for models which can’t, mostly that structure isn’t there. This means the model is formally intractable—there is no practical prospect of an efficient algorithm for finding the maximum.