The sum-product algorithm is a well-known procedure for marginalizing an “acyclic” product function whose range is the ground set of a commutative semiring. The algorithm is general enough to include as special cases several classical algorithms developed in information theory and probability theory. We present four results. First, using the sum-product algorithm we show that the variable sets involved in an acyclic factorization satisfy a relation that is a natural generalization of probability-theoretic...
In probability theory, Bayesian statistics, artificial intelligence and database theory the minimum cross-entropy principle is often used to estimate a distribution with a given set of marginal distributions under the proportionality assumption with respect to a given “prior” distribution . Such an estimation problem admits a solution if and only if there exists an extension of that is dominated by . In this paper we consider the case that is not given explicitly, but is specified as the...
Decomposable (probabilistic) models are log-linear models generated by acyclic hypergraphs, and a number of nice properties enjoyed by them are known. In many applications the following selection problem naturally arises: given a probability distribution over a finite set of discrete variables and a positive integer , find a decomposable model with tree-width that best fits . If is the generating hypergraph of a decomposable model and is the estimate of under the model, we can measure...
We generalize Jiroušek’s (right) composition operator in such a way that it can be applied to distribution functions with values in a “semifield“, and introduce (parenthesized) compositional expressions, which in some sense generalize Jiroušek’s “generating sequences” of compositional models. We say that two compositional expressions are equivalent if their evaluations always produce the same results whenever they are defined. Our first result is that a set system is star-like with centre if...
In the framework of models generated by compositional expressions, we solve two topical marginalization problems (namely, the single-marginal problem and the marginal-representation problem) that were solved only for the special class of the so-called “canonical expressions”. We also show that the two problems can be solved “from scratch” with preliminary symbolic computation.
Compositional models are used to construct probability distributions from lower-order probability distributions. On the other hand, Bayesian models are used to represent probability distributions that factorize according to acyclic digraphs. We introduce a class of models, called , to represent probability distributions that recursively factorize according to sequences of sets of variables, and prove that they have the same representation power as both compositional models generated by sequential...
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