AN IMPROVEMENT OF MARKOVIAN INTEGRATION BY PARTS FORMULA AND APPLICATION TO SENSITIVITY COMPUTATION

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

Shedding light on the underlying characteristics of genomes using Kronecker model families of codon evolution

The mechanistic models of codon evolution rely on some simplistic assumptions in order to reduce the computational complexity of estimating the high number of parameters of the models. This paper is an attempt to investigate how much these simplistic assumptions are misleading when they violate the nature of the biological dataset in hand. We particularly focus on three simplistic assumptions made by most of the current mechanistic codon models including: 1) only single substitutions between nucleotides within codons in the codon transition rate matrix are allowed. 2) mutation is homogenous across nucleotides within a codon. 3) assuming HKY nucleotide model is good enough at the nucleotide level. For this purpose, we developed a framework of mechanistic codon models, each model in the framework hold or relax some of the mentioned simplifying assumptions. Holding or relaxing the three simplistic assumptions results in total to eight different mechanistic models in the framework. Through several experiments on biological datasets and simulations we show that the three simplistic assumptions are unrealistic for most of the biological datasets and relaxing these assumptions lead to accurate estimation of evolutionary parameters such as selection pressure.

BAYESIAN ANALYSIS OF DOUBLY STOCHASTIC MARKOV PROCESSES IN RELIABILITY

Markov processes play an important role in reliability analysis and particularly in modeling the stochastic evolution of survival/failure behavior of systems. The probability law of Markov processes is described by its generator or the transition rate matrix. In this paper, we suppose that the process is doubly stochastic in the sense that the generator is also stochastic. In our model, we suppose that the entries in the generator change with respect to the changing states of yet another Markov process. This process represents the random environment that the stochastic model operates in. In fact, we have a Markov modulated Markov process which can be modeled as a bivariate Markov process that can be analyzed probabilistically using Markovian analysis. In this setting, however, we are interested in Bayesian inference on model parameters. We present a computationally tractable approach using Gibbs sampling and demonstrate it by numerical illustrations. We also discuss cases that involve complete and partial data sets on both processes.

The Invariant Nature of a Morphological Character and Character State: Insights from Gene Regulatory Networks

What constitutes a discrete morphological character versus character state has been long discussed in the systematics literature but the consensus on this issue is still missing. Different methods of classifying organismal features into characters and character states (CCSs) can dramatically affect the results of phylogenetic analyses. Here, I show that, in the framework of Markov models, the modular structure of the gene regulatory network (GRN) underlying trait development, and the hierarchical nature of GRN evolution, essentially remove the distinction between morphological CCS, thus endowing the CCS with an invariant property with respect to each other. This property allows the states of one character to be represented as several individual characters and vice versa. In practice, this means that a phenotype can be encoded using a set of characters or just one complex character with numerous states. The representation of a phenotype using one complex character can be implemented in Markov models of trait evolution by properly structuring transition rate matrix.

Statistical Identification of Markov Chain on Trees

The theoretical study of continuous-time homogeneous Markov chains is usually based on a natural assumption of a known transition rate matrix (TRM). However, the TRM of a Markov chain in realistic systems might be unknown and might even need to be identified by partially observable data. Thus, an issue on how to identify the TRM of the underlying Markov chain by partially observable information is derived from the great significance in applications. That is what we call the statistical identification of Markov chain. The Markov chain inversion approach has been derived for basic Markov chains by partial observation at few states. In the current letter, a more extensive class of Markov chain on trees is investigated. Firstly, a type of a more operable derivative constraint is developed. Then, it is shown that all Markov chains on trees can be identified only by such derivative constraints of univariate distributions of sojourn time and/or hitting time at a few states. A numerical example is included to demonstrate the correctness of the proposed algorithms.

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models

For health-economic analyses that use multistate Markov models, it is often necessary to convert from transition rates to transition probabilities, and for probabilistic sensitivity analysis and other purposes it is useful to have explicit algebraic formulas for these conversions, to avoid having to resort to numerical methods. However, if there are four or more states then the formulas can be extremely complicated. These calculations can be made using packages such as R, but many analysts and other stakeholders still prefer to use spreadsheets for these decision models. We describe a procedure for deriving formulas that use intermediate variables so that each individual formula is reasonably simple. Once the formulas have been derived, the calculations can be performed in Excel or similar software. The procedure is illustrated by several examples and we discuss how to use a computer algebra system to assist with it. The procedure works in a wide variety of scenarios but cannot be employed when there are several backward transitions and the characteristic equation has no algebraic solution, or when the eigenvalues of the transition rate matrix are very close to each other.