We propose a logical/mathematical framework for statistical parameter learning of parameterized logic programs, i.e. denite clause programs containing probabilistic facts with a parameterized distribution. It extends the traditional least Herbrand model semantics in logic programming to distribution semantics , possible world semantics with a probability distribution which is unconditionally applicable to arbitrary logic programs including ones for HMMs, PCFGs and Bayesian networks.We also propose a new EM algorithm, the graphical EM algorithm, t h a t r u n s f o r a class of parameterized logic programs representing sequential decision processes where each decision is exclusive and independent. It runs on a new data structure called support graphs describing the logical relationship between observations and their explanations, and learns parameters by computing inside and outside probability generalized for logic programs.The complexity analysis shows that when combined with OLDT search for all explanations for observations, the graphical EM algorithm, despite its generality, has the same time complexity as existing EM algorithms, i.e. the Baum-Welch algorithm for HMMs, the Inside-Outside algorithm for PCFGs, and the one for singly connected Bayesian networks that have b e e n d e v eloped independently in each research eld. Learning experiments with PCFGs using two corpora of moderate size indicate that the graphical EM algorithm can signicantly outperform the Inside-Outside algorithm.

keywords: machine learning, EM algorithm, binary decision diagram (BDD), propositonalized probability computation SummaryWe propose an Expectation-Maximization (EM) algorithm which works on binary decision diagrams (BDDs). The proposed algorithm, BDD-EM algorithm, opens a way to apply BDDs to statistical learning. The BDD-EM algorithm makes it possible to learn probabilities in statistical models described by Boolean formulas, and the time complexity is proportional to the size of BDDs representing them. We apply the BDD-EM algorithm to prediction of intermittent errors in logic circuits and demonstrate that it can identify error gates in a 3bit adder circuit.

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Abstract. We review a logic-based modeling language PRISM and report recent developments including belief propagation by the generalized inside-outside algorithm and generative modeling with constraints. The former implies PRISM subsumes belief propagation at the algorithmic level. We also compare the performance of PRISM with state-of-theart systems in statistical natural language processing and probabilistic inference in Bayesian networks respectively, and show that PRISM is reasonably competitive.

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