Monte Carlo EM加速算法

Acceleration of Monte Carlo EM Algorithm

EM算法是近年来常用的求后验众数的估计的一种数据增广算法,但由于求出其E步中积分的显示表达式有时很困难,甚至不可能,限制了其应用的广泛性.而Monte Carlo EM算法很好地解决了这个问题,将EM算法中E步的积分用Monte Carlo模拟来有效实现,使其适用性大大增强.但无论是EM算法,还是Monte Carlo EM算法,其收敛速度都是线性的,被缺损信息的倒数所控制,当缺损数据的比例很高时,收敛速度就非常缓慢.而Newton-Raphson算法在后验众数的附近具有二次收敛速率.本文提出Monte Carlo EM加速算法,将Monte Carlo EM算法与Newton-Raphson算法结合,既使得EM算法中的E步用Monte Carlo模拟得以实现,又证明了该算法在后验众数附近具有二次收敛速度.从而使其保留了Monte Carlo EM算法的优点,并改进了Monte Carlo EM算法的收敛速度.本文通过数值例子,将Monte Carlo EM加速算法的结果与EM算法、Monte Carlo EM算法的结果进行比较,进一步说明了Monte Carlo EM加速算法的优良性.

EM algorithm is one of the data augmentation algorithms, which usually are used to obtain estimate of the posterior mode of observed data recent years. However, because of its difficulty in calculating the explicit expression of the integral in E step, the application of EM algorithm is limited. While Monte Carlo EM algorithm solves the problem well. Owing to effectively facilitating the integral in E step of EM algorithm by Monte Carlo simulating, Monte Carlo EM algorithm has been successfully used to a wide range of applications. There is, however, the same shortage for EM algorithm and Monte Carlo EM algorithm, that the convergence rate of the two algorithms is linear. So this paper proposes the acceleration of Monte Carlo EM Algorithm, which is based on Monte Carlo EM Algorithm and Newton-Raphson algorithm, to improve the convergence rate. Thus the acceleration of Monte Carlo EM Algorithm has the advantages of both Monte Carlo EM Algorithm and Newton-Raphson algorithm, that is to say it facilitates E step by Monte Carlo simulation and also has quadratic convergence rate in a neighborhood of the posterior mode. Later its excellence in convergence rate is illustrated by a classical example.