On the mixing time of Markov Chain Monte Carlo for integer least-square problems

Summary The availability of datasets with large numbers of variables is rapidly increasing. The effective application of Bayesian variable selection methods for regression with these datasets has proved difficult since available Markov chain Monte Carlo methods do not perform well in typical problem sizes of interest. We propose new adaptive Markov chain Monte Carlo algorithms to address this shortcoming. The adaptive design of these algorithms exploits the observation that in large-$p$, small-$n$ settings, the majority of the $p$ variables will be approximately uncorrelated a posteriori. The algorithms adaptively build suitable nonlocal proposals that result in moves with squared jumping distance significantly larger than standard methods. Their performance is studied empirically in high-dimensional problems and speed-ups of up to four orders of magnitude are observed.

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On the convergence rates of some adaptive Markov chain Monte Carlo algorithms

Journal of Applied Probability ◽

10.1239/jap/1445543848 ◽

2015 ◽

Vol 52 (3) ◽

pp. 811-825

Author(s):

Yves Atchadé ◽

Yizao Wang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Convergence Rates ◽

Mixing Time ◽

Regularity Conditions ◽

Monte Carlo Algorithms ◽

Mcmc Algorithms ◽

Variation Distance ◽

Importance Resampling

In this paper we study the mixing time of certain adaptive Markov chain Monte Carlo (MCMC) algorithms. Under some regularity conditions, we show that the convergence rate of importance resampling MCMC algorithms, measured in terms of the total variation distance, is O(n-1). By means of an example, we establish that, in general, this algorithm does not converge at a faster rate. We also study the interacting tempering algorithm, a simplified version of the equi-energy sampler, and establish that its mixing time is of order O(n-1/2).

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Ensuring Rapid Mixing and Low Bias for Asynchronous Gibbs Sampling

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2017/672 ◽

2017 ◽

Cited By ~ 2

Author(s):

Christopher De Sa ◽

Kunle Olukotun ◽

Christopher Ré

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Gibbs Sampling ◽

Mixing Time ◽

Markov Chain Analysis ◽

Empirical Results ◽

Chain Analysis ◽

Speed Up ◽

Theoretical Results

Gibbs sampling is a Markov chain Monte Carlo technique commonly used for estimating marginal distributions. To speed up Gibbs sampling, there has recently been interest in parallelizing it by executing asynchronously. While empirical results suggest that many models can be efficiently sampled asynchronously, traditional Markov chain analysis does not apply to the asynchronous case, and thus asynchronous Gibbs sampling is poorly understood. In this paper, we derive a better understanding of the two main challenges of asynchronous Gibbs: bias and mixing time. We show experimentally that our theoretical results match practical outcomes.

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Optimized Markov Chain Monte Carlo for Signal Detection in MIMO Systems: An Analysis of the Stationary Distribution and Mixing Time

IEEE Transactions on Signal Processing ◽

10.1109/tsp.2014.2334558 ◽

2014 ◽

Vol 62 (17) ◽

pp. 4436-4450 ◽

Cited By ~ 15

Author(s):

Babak Hassibi ◽

Morten Hansen ◽

Alexandros G. Dimakis ◽

Haider Ali Jasim Alshamary ◽

Weiyu Xu

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Signal Detection ◽

Stationary Distribution ◽

Mimo Systems ◽

Mixing Time

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Biomass Hydrothermal Carbonization: Markov-Chain Monte Carlo Data Analysis and Modeling

Frontiers in Chemical Engineering ◽

10.3389/fceng.2021.643041 ◽

2021 ◽

Vol 3 ◽

Author(s):

Alberto Gallifuoco ◽

Alessandro Antonio Papa ◽

Luca Taglieri

Keyword(s):

Experimental Data ◽

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Biomass Conversion ◽

Hydrothermal Carbonization ◽

Statistical Simulation ◽

Reaction Scheme ◽

Least Square ◽

Monte Carlo Data

This paper introduces Bayesian statistical methods for studying the kinetics of biomass hydrothermal carbonization. Two simple, specially developed computer programs implement Markov-chain Monte Carlo methods to illustrate these techniques' potential, long since established in other areas of chemical reaction engineering. A range of experimental data, both from this study and the literature, test the soundness of a Bayesian approach to modeling biomass hydrothermal carbonization kinetics. The first program carries out parameter estimations and performs better or equal than the traditional deterministic methods (R2 as high as 0.9998). For three out of the 22 datasets, the program detected the global minima of the parameter space, while the deterministic least-square found local values. The second program uses Gillespie's algorithm for the statistical simulation of the reactions occurring in hydrothermal carbonization. Comparing six basic kinetic models with literature data tested the stochastic simulation as a tool for assessing biomass conversion reaction networks rapidly. Among the simple models discussed, reaction scheme 3 fitted better to the experimental data (R2 > 0.999). The proposed approach is worth extending to more complex, time-consuming computer models and could support other techniques for studying hydrothermal conversions.

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On the convergence rates of some adaptive Markov chain Monte Carlo algorithms

Journal of Applied Probability ◽

10.1017/s0021900200113452 ◽

2015 ◽

Vol 52 (03) ◽

pp. 811-825

Author(s):

Yves Atchadé ◽

Yizao Wang

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Convergence Rates ◽

Mixing Time ◽

Regularity Conditions ◽

Monte Carlo Algorithms ◽

Mcmc Algorithms ◽

Variation Distance ◽

Importance Resampling

In this paper we study the mixing time of certain adaptive Markov chain Monte Carlo (MCMC) algorithms. Under some regularity conditions, we show that the convergence rate of importance resampling MCMC algorithms, measured in terms of the total variation distance, isO(n-1). By means of an example, we establish that, in general, this algorithm does not converge at a faster rate. We also study the interacting tempering algorithm, a simplified version of the equi-energy sampler, and establish that its mixing time is of orderO(n-1/2).

Download Full-text