A maximum entropy principle approach to a joint probability model for sequences with known neighbor and next neighbor pair probabilities

Abstract

A Markov chain can be used to model a joint probability density of a sequence, pR1R2...RNx1,x2,..xN, from neighbor pair probability data: pRnRn+1xn|xn+1n=1,2,...N-1, the Rn′s specifying the type of random variable, xn. A Maximum Entropy Principle (MEP) is used to show that, with knowledge of only these neighbor pair probabilities, the Markov chain maximizes entropy. To introduce more information, it would be useful to consider triplet probabilities, but data limitations may preclude this approach. Instead, the joint probability density function is modeled with neighbor and next neighbor pair probabilities. Optimized triplet probabilities are obtained from these probabilities, again using a MEP method, and an augmented Markov chain is constructed from them. This joint probability density function is the MEP joint probability density function with known neighbor and next neighbor pair probabilities. Based on this information, we construct various length Markov and augmented Markov chains to produce diverging patterns of chain probabilities.