Exact and computationally efficient Bayesian inference for generalized Markov modulated Poisson processes

Abstract We describe a new and computationally efficient Bayesian methodology for inferring species trees and demographics from unlinked binary markers. Likelihood calculations are carried out using diffusion models of allele frequency dynamics combined with novel numerical algorithms. The diffusion approach allows for analysis of data sets containing hundreds or thousands of individuals. The method, which we call Snapper, has been implemented as part of the BEAST2 package. We conducted simulation experiments to assess numerical error, computational requirements, and accuracy recovering known model parameters. A reanalysis of soybean SNP data demonstrates that the models implemented in Snapp and Snapper can be difficult to distinguish in practice, a characteristic which we tested with further simulations. We demonstrate the scale of analysis possible using a SNP data set sampled from 399 fresh water turtles in 41 populations. [Bayesian inference; diffusion models; multi-species coalescent; SNP data; species trees; spectral methods.]

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A new traffic aggregation technique based on Markov modulated Poisson processes

GLOBECOM '05. IEEE Global Telecommunications Conference, 2005. ◽

10.1109/glocom.2005.1577945 ◽

2005 ◽

Author(s):

Ming Yu ◽

D.G. Daut

Keyword(s):

Poisson Processes ◽

Markov Modulated ◽

Aggregation Technique

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On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

Journal of Applied Probability ◽

10.2307/3215346 ◽

1996 ◽

Vol 33 (3) ◽

pp. 640-653 ◽

Cited By ~ 24

Author(s):

Tobias Rydén

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Continuous Time ◽

Poisson Processes ◽

Hitting Times ◽

Phase Type ◽

Phase Type Distributions ◽

Minimum Number ◽

Markov Modulated ◽

Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

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