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Bayesian Analysis of Proportions via a Hidden Markov Model

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Abstract

Time series of proportions arise in many contexts. In this paper, we consider a hidden Markov model (HMM) to describe temporal dependence in such series. In so doing, we introduce a Beta-HMM and develop its Bayesian analysis using Markov Chain Monte Carlo Methods (MCMC). Our proposed model is based on a conjugate prior for beta likelihood which enables us develop Bayesian posterior and predictive computations in an efficient manner. We also address the problem of assessing dimension of the HMM using the marginal likelihood of the model which can be evaluated using posterior samples. Finally, we implement our model and the Bayesian methodology using weekly data on market shares.

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Acknowledgements

We would like to thank the Scientific and Technological Research Council of Turkey (TÜBİTAK) for its financial support while the first author was visiting the Department of Decision Sciences at the George Washington University as a research fellow. This paper was completed as a result of this research under TÜBİTAK-2214/A International Doctoral Research Fellowship Programme.

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Authors and Affiliations

  1. Department of Statistics, Hacettepe University, Ankara, 06800, Turkey

    Ceren Eda Can & Gul Ergun

  2. Department of Decision Sciences, The George Washington University, Washington DC, 20052, US

    Refik Soyer

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  1. Ceren Eda Can
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  2. Gul Ergun
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  3. Refik Soyer
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Correspondence to Ceren Eda Can.

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Appendix

Appendix

The posterior probabilities of the hidden states and the MAP Estimation of \({\pmb {X}^{(T)}}\) are given by the following table. Note that the posterior probabilities, which are too small (or large), have been rounded off (see Table 5).

Table 5 Posterior Probabilities of the Hidden State Sequence
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Can, C.E., Ergun, G. & Soyer, R. Bayesian Analysis of Proportions via a Hidden Markov Model. Methodol Comput Appl Probab 24, 3121–3139 (2022). https://doi.org/10.1007/s11009-022-09971-0

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  • DOI: https://doi.org/10.1007/s11009-022-09971-0

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