An EM algorithm for the model fitting of Markovian binary trees (abstract only)

Markovian binary trees are a special class of branching processes in which the lifetime of an individual is controlled by a transient Markovian arrival process. A Markovian binary tree is characterized by the 4-tuple (?,D0,B, d), where ? is the vector of initial phase distribution of the first individual, D0 is the matrix of phase transition rates between birth and death events, B is the matrix of birth rates and d is the vector of death rates.

In order to use the Markovian binary tree to model the evolution of a real population, we need to determine the parameters (?,D0,B, d) from observations of that population. In the absence of migration, the only observable changes in a population are those associated with a birth or a death event; no phase transition in the underlying process can actually been seen. We are thus dealing with a problem of parameter estimation from incomplete data, and one way to solve this statistical problem is to make use of the EM algorithm. Our purpose here is thus to specify this algorithm to the Markovian binary tree setting. In the first part of this paper, we introduce a discrete time terminating marked Markov arrival process (MMAP), based on which a class of discrete multivariate phase-type (MPH) distributions is defined. The discrete MPH-distributions hold many of the properties possessed by continuous MPH-distributions (Assaf, et al. (1983), Kulkarni (1988), and O'Cinneide (1990)). It is known that the joint distribution functions of continuous MPH are fairly complicated and difficult to calculate. In contrast, for the discrete MPH introduced here, we provide recursive formulas the joint probabilities and explicit expressions for means, variances, and co-variances.