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Markov chain Monte Carlo (MCMC) methods provide an important means to simulate from almost any probability density. To approximate non-standard discrete distributions, the equation-solving MCMC estimator was developed as an alternative to the classical frequency estimator. The used simulation scheme is the Metropolis–Hastings (M–H) algorithm. Recently, this estimator has been extended to the specific context of 2-step Metropolis-Hastings with delayed rejection (MHDR) algorithm, which allowed a considerable reduction in asymptotic variance. In this paper, we propose an adaptation of equation-solving estimator to the case of general n-step MHDR sampler. The aim is to further improve the precision. An application to a Bayesian hypothesis test problem shows the high performance, in terms of accuracy, of the equation-solving estimator, based on a MHDR algorithm with more than two stages. [ABSTRACT FROM AUTHOR]

Assume that K independent copies are made from a common prototype DNA sequence whose length is a random variable. In this paper, the problem of aligning those copies and therefore the problem of estimating the prototype sequence that produced the copies is addressed. A hidden Markov chain is used to model the copying procedure, and a reversible jump Markov chain Monte Carlo algorithm is used to sample the parameters of the model from their posterior distribution. Using the sample obtained, the Bayesian model and the prototype sequence may be selected using the maximum a posteriori estimate. A prior distribution for the prototype DNA sequence that incorporates a correlation among neighbouring bases is also considered. In addition, an analysis of the performance of the algorithm is presented when different scenarios are taken into account. [ABSTRACT FROM AUTHOR]