改进的MCMC算法—DSY算法及其在估计IRT模型参数中的应用

Improvement of MCMC Algorithm-DSY Algorithm and its Application in Parameter Estimation under the IRT Model

本文首先简要的阐述了MCMC算法的思想及在IRT参数估计中的操作过程;其次,针对该算法存在的一些问题,提出相应的改进建议;然后,分别运用传统的和改进的MCMC算法进行模拟数据分析和比较,结果显示新的方法表现更好;最后总结新方法的优点所在,并指出下一步的研究方向。

The core issue in IRT is how to estimate the item and person parameters. The common methods used often were the N - R algorithm and the E - M algorithm. Because of their special characteristics in themselves, there were always certain shortcomings to hinder their development. With more and more complex models appearing, it is also difficult to estimate the parameters using those methods. Then, the algorithm called Markov Chain Monte Carlo （MCMC） appeared. The emergence of MCMC algorithm provides the new solution. The MCMC algorithm has been used in statistical physics for more than 50 years. In recent 20 years, it was also widely used in Bayesian estimation, test of significance and maximum likelihood estimation. Albert （1992） is the first statistician to apply the algo-rithm in IRT parameter estimations. Many experts including Albert （ 1992）, Patz and Junker （ 1999a, 1999b）, Kim, Seon and Daniel, Bolt （2007） provide the information about the MCMC algorithm and how to use it in detail. The character of the MCMC algorithm is that it gives full play to the advantages of computer simulation technology, collects a suffi- ciently large sample of state by simulating, uses the elementary method to estimate the model parameters, and thus bypasses the complex calculation of the EM algorithm to improve the success rate of estimation. No algorithm is perfect. Although the traditional MCMC algorithm has been widely used, its shortcomings, such as the serious de- pendence on the prior distribution of the parameters and the extremely long time spent in performing the procedure, still exist. It is the main purpose of this paper to solve the problems. In the paper, the idea of the MCMC a/gorithm is briefly introduced. Then two suggestions are made to improve the algorithm and solve the existing problems. The first suggestion is about the stationary distribution： the traditional MCMC algorithm is largely dependent on the prior distribution of the model parameters. However, in practice, researchers often do not know the prior distribution. So this influences the accuracy of the estimation results. In this paper, we provide another method to avoid the above situation. The second suggestion is about the acceptance probability. In this paper, we believe that in order to reduce the run time of the procedure, only when the stationary distribution value of the new iteration value is greater than that of the original iteration value, does the new iteration value replace the old one to be the value of the iterative chain. Then, the traditional and improved versions of MCMC algorithm are used to simulate and analyze the data. Through the comparison of the results from the two methods, it shows that the new method performs better. Finally, the advantage of the new algorithm is pointed out and the future research direction is suggested.