Mathematical Genetics in Post Genome Era

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Stochastic networks with and without are common frameworks in many genetical problems. In particular, They are basis of linkage analysis of family genetic data. The LBP (Loopy Belief Propagation) algorithm is an efficient algorithm for estimating marginal probabilities of nodes of stochastic networks with loops. In order to apply this algorithm to linkage analysis, we studied the following basic theoretical problem :(1)Taking Cayley trees as examples, convergence and marginal probability recovery problems were studied. Using theoretical and numerical results, we show that the convergence is closely related with the existence of phase transitions. Ising models on Cayley trees have two kind of phase transitions. LBP converges on one phase transition region, but does not converge on another phase transition region. If converged, beliefs may not coincide with true marginal probabilities. Nevertheless, it is observed that states that both give highest values are coincide.(2)As an application of stochastic networks, we consider an application of credit-rating of companies. It is shown that a naive Bayesian networks can give better predictions than common subjective networks employed by analysts.(3)With an applications to bioinformatics in mind, Kanamori studied some properties of learning theory. In particular, boosting methods.