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Markov chain Monte Carlo (MCMC) is a simulation method commonly used for estimating expectations with respect to a given distribution. We consider estimating the covariance matrix of the asymptotic multivariate normal distribution of a vector of sample means. Geyer [9] developed a Monte Carlo error estimation method for estimating a univariate mean. We propose a novel multivariate version of Ge...

Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs ...

He obtained his Ph.D. from Stanford University in 1982 and since then has done research in various areas including geometrical probability, multivariate probability inequalities and survival analysis. This work is part of an ongoing project devoted to the modeling the relationship between species distribution and climate variables. Boulder for providing us with species and climate data. Abstrac...

A fast mean field variational Bayes (MFVB) approach to nonparametric regression when the predictors are subject to classical measurement error is investigated. It is shown that the use of such technology to the measurement error setting achieves reasonable accuracy. In tandem with the methodological development, a customized Markov chain Monte Carlo method is developed to facilitate the evaluat...

Hierarchical models are one of the central tools of Bayesian analysis. They offer many advantages, including the ability to borrow strength to estimate individual parameters and the ability to specify complex models that reflect engineering and physical realities. Markov chain Monte Carlo is a set of algorithms that allow Bayesian inference in a variety of models. We illustrate hierarchical mod...

Maximum likelihood (ML) method has been widely used because it allows phylogenetic analysis based on probabilistic models of molecular evolution. However, despite its effectiveness and simplicity, ML method does not work properly in analyses of many species — it even fails with only 20-30 species. To overcome this problem, we propose an approximate version of ML method based on Markov Chain Mon...

The coupling between dark energy and dark matter provides a possible approach to mitigate the coincidence problem of the cosmological standard model. In this paper, we assumed the interacting term was related to the Hubble parameter, energy density of dark energy, and equation of state of dark energy. The interaction rate between dark energy and dark matter was a constant parameter, which was, ...

In this note we attempt to trace the history and development of Markov chain Monte Carlo (MCMC) from its early inception in the late 1940’s through its use today. We see how the earlier stages of the Monte Carlo research have led to the algorithms currently in use. More importantly, we see how the development of this methodology has not only changed our solutions to problems, but has changed th...