scholarly journals Limit theorems for sequential MCMC methods

2020 ◽  
Vol 52 (2) ◽  
pp. 377-403 ◽  
Author(s):  
Axel Finke ◽  
Arnaud Doucet ◽  
Adam M. Johansen
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Sequential Monte Carlo ◽  
Probability Distributions ◽  
Asymptotic Variance ◽  
Mcmc Methods ◽  
Time Step ◽  
Strong Law ◽  
Large Numbers

AbstractBoth sequential Monte Carlo (SMC) methods (a.k.a. ‘particle filters’) and sequential Markov chain Monte Carlo (sequential MCMC) methods constitute classes of algorithms which can be used to approximate expectations with respect to (a sequence of) probability distributions and their normalising constants. While SMC methods sample particles conditionally independently at each time step, sequential MCMC methods sample particles according to a Markov chain Monte Carlo (MCMC) kernel. Introduced over twenty years ago in [6], sequential MCMC methods have attracted renewed interest recently as they empirically outperform SMC methods in some applications. We establish an $\mathbb{L}_r$ -inequality (which implies a strong law of large numbers) and a central limit theorem for sequential MCMC methods and provide conditions under which errors can be controlled uniformly in time. In the context of state-space models, we also provide conditions under which sequential MCMC methods can indeed outperform standard SMC methods in terms of asymptotic variance of the corresponding Monte Carlo estimators.

scholarly journals Mapping-Linked Quantitative Trait Loci Using Bayesian Analysis and Markov Chain Monte Carlo Algorithms

Genetics ◽  
1997 ◽  
Vol 146 (2) ◽  
pp. 735-743 ◽  
Author(s):  
Pekka Uimari ◽  
Ina Hoeschele
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Quantitative Trait Loci ◽  
Quantitative Trait ◽  
Allele Frequencies ◽  
Mcmc Methods ◽  
Substitution Effects ◽  
Indicator Variable ◽  
Trait Loci

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL and no additional or an unlinked QTL.

scholarly journals A sequential Bayesian approach for the estimation of the age–depth relationship of Dome Fuji ice core

2015 ◽  
Vol 2 (3) ◽  
pp. 939-968
Author(s):  
S. Nakano ◽  
K. Suzuki ◽  
K. Kawamura ◽  
F. Parrenin ◽  
T. Higuchi
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Sequential Monte Carlo ◽  
Ice Core ◽  
Posterior Distributions ◽  
Metropolis Method ◽  
Depth Relationship ◽  
Relationship Of ◽  
Thinning Process

Abstract. A technique for estimating the age–depth relationship in an ice core and evaluating its uncertainty is presented. The age–depth relationship is mainly determined by the accumulation of snow at the site of the ice core and the thinning process due to the horizontal stretching and vertical compression of ice layers. However, since neither the accumulation process nor the thinning process are fully understood, it is essential to incorporate observational information into a model that describes the accumulation and thinning processes. In the proposed technique, the age as a function of depth is estimated from age markers and δ18O data. The estimation is achieved using the particle Markov chain Monte Carlo (PMCMC) method, in which the sequential Monte Carlo (SMC) method is combined with the Markov chain Monte Carlo method. In this hybrid method, the posterior distributions for the parameters in the models for the accumulation and thinning processes are computed using the Metropolis method, in which the likelihood is obtained with the SMC method. Meanwhile, the posterior distribution for the age as a function of depth is obtained by collecting the samples generated by the SMC method with Metropolis iterations. The use of this PMCMC method enables us to estimate the age–depth relationship without assuming either linearity or Gaussianity. The performance of the proposed technique is demonstrated by applying it to ice core data from Dome Fuji in Antarctica.

scholarly journals In search of lost mixing time: adaptive Markov chain Monte Carlo schemes for Bayesian variable selection with very large p

Biometrika ◽  
2020 ◽  
Author(s):  
J E Griffin ◽  
K G Łatuszyński ◽  
M F J Steel
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Variable Selection ◽  
Adaptive Design ◽  
Mixing Time ◽  
Bayesian Variable Selection ◽  
Monte Carlo Algorithms ◽  
Large Numbers ◽  
Large P Small N

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.

scholarly journals Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation

2013 ◽  
Vol 371 (1984) ◽  
pp. 20110541 ◽  
Author(s):  
Vassilios Stathopoulos ◽  
Mark A. Girolami
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Jump Processes ◽  
Mcmc Methods ◽  
Computationally Efficient ◽  
Markov Jump ◽  
Riemann Manifold ◽  
Exact Inference ◽  
Markov Jump Processes

Bayesian analysis for Markov jump processes (MJPs) is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding, thus its applicability is limited to a small class of problems. In this paper, we describe the application of Riemann manifold Markov chain Monte Carlo (MCMC) methods using an approximation to the likelihood of the MJP that is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient whereas the convergence rate and mixing of the chains allow for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology.

scholarly journals Measuring Stellar Radial Velocity using Markov Chain Monte Carlo(MCMC) Method

2013 ◽  
Vol 9 (S298) ◽  
pp. 441-441
Author(s):  
Yihan Song ◽  
Ali Luo ◽  
Yongheng Zhao
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Distribution ◽  
Radial Velocity ◽  
Mcmc Methods ◽  
Mcmc Method ◽  
Stellar Spectra ◽  
Mcmc Simulation ◽  
Template Fitting

AbstractStellar radial velocity is estimated by using template fitting and Markov Chain Monte Carlo(MCMC) methods. This method works on the LAMOST stellar spectra. The MCMC simulation generates a probability distribution of the RV. The RV error can also computed from distribution.

scholarly journals Planning Tunnel Construction Using Markov Chain Monte Carlo (MCMC)

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Juan P. Vargas ◽  
Jair C. Koppe ◽  
Sebastián Pérez ◽  
Juan P. Hurtado
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Probability Distributions ◽  
Financial Risks ◽  
Underground Excavation ◽  
Construction Time ◽  
Tunnel Excavation ◽  
Unit Operations

Tunnels, drifts, drives, and other types of underground excavation are very common in mining as well as in the construction of roads, railways, dams, and other civil engineering projects. Planning is essential to the success of tunnel excavation, and construction time is one of the most important factors to be taken into account. This paper proposes a simulation algorithm based on a stochastic numerical method, the Markov chain Monte Carlo method, that can provide the best estimate of the opening excavation times for the classic method of drilling and blasting. Taking account of technical considerations that affect the tunnel excavation cycle, the simulation is developed through a computational algorithm. Using the Markov chain Monte Carlo method, the unit operations involved in the underground excavation cycle are identified and assigned probability distributions that, with random number input, make it possible to simulate the total excavation time. The results obtained with this method are compared with a real case of tunneling excavation. By incorporating variability in the planning, it is possible to determine with greater certainty the ranges over which the execution times of the unit operations fluctuate. In addition, the financial risks associated with planning errors can be reduced and the exploitation of resources maximized.

scholarly journals Improved Markov Chain Monte Carlo method for cryptanalysis substitution-transposition cipher

2016 ◽  
Vol 22 (4) ◽  
Author(s):  
Behrouz Fathi-Vajargah ◽  
Mohadeseh Kanafchian
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Mcmc Methods ◽  
Random Numbers ◽  
Simple Substitution

AbstractIn this paper we first investigate the use of Markov Chain Monte Carlo (MCMC) methods to attack classical ciphers. MCMC has previously been used to break simple substitution, transposition and substitution-transposition ciphers. Here, we improve the accuracy of obtained results by these algorithms and we show the performance of the algorithms using quasi random numbers such as Faure, Sobol and Niederreiter sequences.

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