Bayesian inference for statistical abduction using Markov chain Monte Carlo

Abduction is one of the basic logical inferences (deduction, induction and abduction) and derives the best explanations for our observation. Statistical abduction attempts to define a probability distribution over explanations and to evaluate them by their probabilities. The framework of statistical abduction is general since many well-known probabilistic models, i.e., BNs, HMMs and PCFGs, are formulated as statistical abduction. Logic-based probabilistic models (LBPMs) have been developed as a way to combine probabilities and logic, and it enables us to perform statistical abduction. However, most of existing LBPMs impose restrictions on explanations (logical formulas) to realize efficient probability computation and learning. To relax those restrictions, we propose two MCMC (Markov chain Monte Carlo) methods for Bayesian inference on LBPMs using binary decision diagrams. The main advantage of our methods over existing methods is that it has no restriction on formulas. In the context of statistical abduction with Bayesian inference, whereas our deterministic knowledge can be described by logical formulas as rules and facts, our non-deterministic knowledge like frequency and preference can be reflected in a prior distribution in Bayesian inference. To illustrate our methods, we first formulate LDA (latent Dirichlet allocation) which is a well-known generative probabilistic model for bag-of-words as a form of statistical abduction, and compare the learning result of our methods with that of an MCMC method called collapsed Gibbs sampling specialized for LDA. We also apply our methods to diagnosis for failure in a logic circuit and evaluate explanations using a posterior distribution approximated by our method. The experiment shows Bayesian inference achieves better predicting accuracy than that of Maximum likelihood estimation.