Mathematical Genetics in Post Genome Era

随机网络和随机网络是许多遗传学问题中常见的框架。特别是它们是家系遗传数据连锁分析的基础。LBP（Loopy Belief Propagation）算法是一种估计带环随机网络节点边际概率的有效算法。为了将该算法应用于连锁分析，我们研究了以下基本理论问题：（1）以Cayley树为例，研究了该算法的收敛性和边际概率恢复问题。利用理论和数值结果，我们表明，收敛性是密切相关的相变的存在。Cayley树上的Ising模型有两种相变。LBP收敛于一个相变区域，但不收敛于另一个相变区域。如果收敛，信念可能不符合真正的边际概率。然而，观察到两者给出最高值的状态是一致的。(2)As作为随机网络的一个应用，我们考虑了公司信用评级的一个应用。结果表明，朴素贝叶斯网络可以得到更好的预测比常见的主观网络分析师。（3）为了应用于生物信息学，Kanamori研究了学习理论的一些性质。特别是提升方法。

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.