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Last edited on
Oct 17, 2024 by JJ.
Jan JOHANNES Jul 29,2017 Theses
Thesis:
Bachelor in Mathematics

Author:
Stefanie Döbler

Title:
Gibbs Sampling of DNA Sequences using Bayesian Hidden Markov Models

Supervisors:
Xavier Loizeau and Jan JOHANNES

Abstract:
In this thesis, Gibbs Sampling of Bayesian Hidden Markov Models is presented and the underlying concepts are introduced and clarified. Following previous work by Robert (2007) the convergence properties of the general Gibbs Sampler are investigated and the proof of the uniform ergodic properties of the sampled subsequences of parameter estimates is given. At first, an in-depth literature review is presented. Then, the convergence properties of the general Gibbs sampler are assessed, extending and clarifying the structural proof given in Robert (2007). Finally, an application of the method in bioinformatics is shown to deepen the understanding of the introduced concepts. Assuming a simple two-state hidden Markov Model for the generation of the DNA, the general Gibbs Sampler was used to infer the models parameters employing conjugate priors. Based on the application Gibbs Sampling is shown to be an efficient, simple to use method for inferring the parameters of a hidden Markov model in the Bayesian framework. After only 104 iterations, the Gibbs sampled posterior mean in our example showed a convergence to the original parameter values. The Gibbs sampler showed a dependence on the initial parameters of the Hidden Markov Model generating the observed DNA sequence as well as the length sequence. The introduction of prior knowledge to the sampling of the parameter values improves the Gibbs samplers convergence properties.

References:
J. Diebolt and C.P. Robert. Estimation of Finite Mixture Distributions through Bayesian Sampling. Journal of the Royal Statistical Society. Series B (Methodological), 56(2):363–375, 1994.
C.P. Robert, G. Celeux, and J. Diebolt. Bayesian estimation of hidden Markov Chains: A stochastic implementation. Statistics & Probability Letters, 16:77–83, 1993.
C.P. Robert. The Bayesian Choice. Springer Texts in Statistics, Springer New York, 2007.