A Proposed Bayesian Estimation of Transitional Probability for a Markov Chain with Random Times via Swarm Algorithm

Abstract

The transition matrix estimators of the Markov chain are not accurate and the transition matrix is considered given. There are many methods that are used to estimate the transition probabilities matrix for different cases, the most famous of which is the Maximum Likelihood Method, in order to find a good and new estimator for the transition probabilities matrix of the Markov chain, a method was proposed, which is a modification of the Bayes method, to reach the transition probabilities with the least variance. This method assumes that the values of  in the initial probability are estimated by two methods: Maximum Likelihood Method (MLE), and the algorithm of particle swarm (PSO), The Escherichia Coli (E.Coli) gene chain was chosen as an applied aspect of the study due to its importance in medical research and for the purpose of discovering and manufacturing treatments by knowing the final form of its gene chain. After testing the E.Coli gene chain, it was found that is represents a Markov chain, and then both the transition probabilities matrix and the transition probabilities variance were estimated, and it was found that the proposed method for transitional probabilities is better than the method of greatest possibility depending on the variance.