Backward probabilistic logic reasoning algorithm for decision problem with conditional event algebra on Bayesian networks

For processing backward reasoning and resolving the discrepancy between logic and probability in sequential decision problem with the implied probabilistic casual relations among interval-valued variables, we propose and implement a backward Bayesian probabilistic logic reasoning approach, which combings Conditional Event Algebra, weak conditional probability, and Markov Monte Carlo simulating algorithm. By partly changing casual relations in a decision problem, we make the problem of backward reasoning possible, and then we bring logic consistent with probability in denoting casual relation by extending normal measurable space with conditional event. We transform a conditional event to normal events and corresponding logical combination events via Conditional Event Algebra, and use Gibbs simulation to sample the events with interval probabilities parameters to be a stationary state. By computing the quantitative values of the stationary events, we can evaluate the quantitative of the conditional event and finish backward reasoning process finally. An example is included to illustrate our study in the paper.