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On Binomial Observations of Continuous-Time Markovian Population Models

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

N. G. Bean ,
R. Elliott ,
A. Eshragh  and
J. V. Ross
N. G. Bean*
Affiliation:
The University of Adelaide
R. Elliott*
Affiliation:
The University of Adelaide
A. Eshragh*
Affiliation:
The University of Newcastle
J. V. Ross*
Affiliation:
The University of Adelaide
*
∗ Postal address: School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia.
∗ Postal address: School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia.
∗∗ Postal address: School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia Email address: ali.eshragh@newcastle.edu.au
∗ Postal address: School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia.
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Abstract

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In this paper we consider a class of stochastic processes based on binomial observations of continuous-time, Markovian population models. We derive the conditional probability mass function of the next binomial observation given a set of binomial observations. For this purpose, we first find the conditional probability mass function of the underlying continuous-time Markovian population model, given a set of binomial observations, by exploiting a conditional Bayes' theorem from filtering, and then use the law of total probability to find the former. This result paves the way for further study of the stochastic process introduced by the binomial observations. We utilize our results to show that binomial observations of the simple birth process are non-Markovian.

Keywords

Continuous-time Markovian population modelbinomial observationsimple birth processfiltering

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 62M09: Non-Markovian processes 62M20: Prediction
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 457 - 472
Copyright
© Applied Probability Trust 

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