scholarly journals Phylogeography by Diffusion on a Sphere

2015 ◽  
Author(s):  
Remco Bouckaert
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Diffusion Process ◽  
Open Source ◽  
Efficient Algorithm ◽  
Diffusion Processes ◽  
Mcmc Algorithm ◽  
Large Samples ◽  
Mercator Projection

Techniques for reconstructing geographical history along a phylogeny can answer many interesting question about the geographical origins of species. Bayesian models based on the assumption that taxa move through a diffusion process have found many applications. However, these methods rely on diffusion processes on a plane, and do not take the spherical nature of our planet in account. This makes it hard to perform analysis that cover the whole world, or do not take in account the distortions caused by projections like the Mercator projection. In this paper, we introduce a Bayesian phylogeographical method based on diffusion on a sphere. When the area where taxa are sampled from is small, a sphere can be approximated by a plane and the model results in the same inferences as with models using diffusion on a plane. For taxa samples from the whole world, we obtain substantial differences. We present an efficient algorithm for performing inference in an Markov Chain Monte Carlo (MCMC) algorithm, and show applications to small and large samples areas. Availability: The method is implemented in the GEO_SPHERE package in BEAST 2, which is open source licensed under LGPL.

scholarly journals Phylogeography by diffusion on a sphere: whole world phylogeography

PeerJ ◽  
2016 ◽  
Vol 4 ◽  
pp. e2406 ◽  
Author(s):  
Remco Bouckaert
Keyword(s):  
Monte Carlo ◽  
Prior Information ◽  
Diffusion Processes ◽  
Mcmc Algorithm ◽  
Map Projections ◽  
Large Samples ◽  
Joint Tree ◽  
Wide Range ◽  
Mercator Projection ◽  
Spherical Diffusion

BackgroundTechniques for reconstructing geographical history along a phylogeny can answer many questions of interest about the geographical origins of species. Bayesian models based on the assumption that taxa move through a diffusion process have found many applications. However, these methods rely on diffusion processes on a plane, and do not take the spherical nature of our planet in account. Performing an analysis that covers the whole world thus does not take in account the distortions caused by projections like the Mercator projection.ResultsIn this paper, we introduce a Bayesian phylogeographical method based on diffusion on a sphere. When the area where taxa are sampled from is small, a sphere can be approximated by a plane and the model results in the same inferences as with models using diffusion on a plane. For taxa sampled from the whole world, we obtain substantial differences. We present an efficient algorithm for performing inference in a Markov Chain Monte Carlo (MCMC) algorithm, and show applications to small and large samples areas. We compare results between planar and spherical diffusion in a simulation study and apply the method by inferring the origin of Hepatitis B based on sequences sampled from Eurasia and Africa.ConclusionsWe describe a framework for performing phylogeographical inference, which is suitable when the distortion introduced by map projections is large, but works well on a smaller scale as well. The framework allows sampling tips from regions, which is useful when the exact sample location is unknown, and placing prior information on locations of clades in the tree. The method is implemented in the GEO_SPHERE package in BEAST 2, which is open source licensed under LGPL and allows joint tree and geography inference under a wide range of models.

Markov Chain Monte Carlo

2015 ◽  
Author(s):  
N. Thompson Hobbs ◽  
Mevin B. Hooten
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Analysis ◽  
Posterior Distribution ◽  
Mcmc Algorithm ◽  
Bayesian Analyses ◽  
Seminal Paper ◽  
Formal Treatment ◽  
High Level

This chapter explains how to implement Bayesian analyses using the Markov chain Monte Carlo (MCMC) algorithm, a set of methods for Bayesian analysis made popular by the seminal paper of Gelfand and Smith (1990). It begins with an explanation of MCMC with a heuristic, high-level treatment of the algorithm, describing its operation in simple terms with a minimum of formalism. In this first part, the chapter explains the algorithm so that all readers can gain an intuitive understanding of how to find the posterior distribution by sampling from it. Next, the chapter offers a somewhat more formal treatment of how MCMC is implemented mathematically. Finally, this chapter discusses implementation of Bayesian models via two routes—by using software and by writing one's own algorithm.

scholarly journals Marathon: An Open Source Software Library for the Analysis of Markov-Chain Monte Carlo Algorithms

PLoS ONE ◽  
2016 ◽  
Vol 11 (1) ◽  
pp. e0147935 ◽  
Author(s):  
Steffen Rechner ◽  
Annabell Berger
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Open Source ◽  
Open Source Software ◽  
Software Library ◽  
Monte Carlo Algorithms

scholarly journals Convergence of Conditional Metropolis-Hastings Samplers

2014 ◽  
Vol 46 (2) ◽  
pp. 422-445 ◽  
Author(s):  
Galin L. Jones ◽  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Diffusion Process ◽  
Sample Path ◽  
Discrete Observations ◽  
Entire Sample ◽  
Bayesian Inferences ◽  
Monte Carlo Algorithms

We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in a conditional Metropolis-Hastings sampler (CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.

A two-step inversion approach for seismic-reservoir characterization and a comparison with a single-loop Markov-chain Monte Carlo algorithm

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. R227-R244 ◽  
Author(s):  
Mattia Aleardi ◽  
Fabio Ciabarri ◽  
Timur Gukov
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Reservoir Characterization ◽  
Gaussian Mixture ◽  
Petrophysical Properties ◽  
Mcmc Algorithm ◽  
Prior Model ◽  
Single Loop ◽  
Seismic Reservoir

We have evaluated a two-step Bayesian algorithm for seismic-reservoir characterization, which, thanks to some simplifying assumptions, is computationally very efficient. The applicability and reliability of this method are assessed by comparison with a more sophisticated and computer-intensive Markov-chain Monte Carlo (MCMC) algorithm, which in a single loop directly estimates petrophysical properties and lithofluid facies from prestack data. The two-step method first combines a linear rock-physics model (RPM) with the analytical solution of a linearized amplitude versus angle (AVA) inversion, to directly estimate the petrophysical properties, and related uncertainties, from prestack data under the assumptions of a Gaussian prior model and weak elastic contrasts at the reflecting interface. In particular, we use an empirical, linear RPM, properly calibrated for the investigated area, to reparameterize the linear time-continuous P-wave reflectivity equation in terms of petrophysical contrasts instead of elastic constants. In the second step, a downward 1D Markov-chain prior model is used to infer the lithofluid classes from the outcomes of the first step. The single-loop (SL) MCMC algorithm uses a convolutional forward modeling based on the exact Zoeppritz equations, and it adopts a nonlinear RPM. Moreover, it assumes a more realistic Gaussian mixture distribution for the petrophysical properties. Both approaches are applied on an onshore 3D seismic data set for the characterization of a gas-bearing, clastic reservoir. Notwithstanding the differences in the forward-model parameterization, in the considered RPM, and in the assumed a priori probability density functions, the two methods yield maximum a posteriori solutions that are consistent with well-log data, although the Gaussian mixture assumption adopted by the SL method slightly improves the description of the multimodal behavior of the petrophysical parameters. However, in the considered reservoir, the main difference between the two approaches remains the very different computational times, the SL method being much more computationally intensive than the two-step approach.

scholarly journals Parameters Identification for Inverse Option Problems Using Markov Chain Monte Carlo Methods

2019 ◽  
Author(s):  
Yasushi Ota ◽  
Yu Jiang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Inference ◽  
Financial Markets ◽  
Sampling Strategy ◽  
Measured Data ◽  
Mcmc Algorithm ◽  
Unknown Parameters ◽  
Posterior Probability Density

This paper investigates the inverse option problems (IOP) in the extended Black--Scholes model arising in financial markets. We identify the volatility and the drift coefficient from the measured data in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the measured data.

A two-stage Markov chain Monte Carlo method for seismic inversion and uncertainty quantification

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R1003-R1020 ◽  
Author(s):  
Georgia K. Stuart ◽  
Susan E. Minkoff ◽  
Felipe Pereira
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Uncertainty Quantification ◽  
Bayesian Methods ◽  
Posterior Distributions ◽  
Mcmc Algorithm ◽  
Two Stage ◽  
Fine Grid ◽  
Highest Posterior Density

Bayesian methods for full-waveform inversion allow quantification of uncertainty in the solution, including determination of interval estimates and posterior distributions of the model unknowns. Markov chain Monte Carlo (MCMC) methods produce posterior distributions subject to fewer assumptions, such as normality, than deterministic Bayesian methods. However, MCMC is computationally a very expensive process that requires repeated solution of the wave equation for different velocity samples. Ultimately, a large proportion of these samples (often 40%–90%) is rejected. We have evaluated a two-stage MCMC algorithm that uses a coarse-grid filter to quickly reject unacceptable velocity proposals, thereby reducing the computational expense of solving the velocity inversion problem and quantifying uncertainty. Our filter stage uses operator upscaling, which provides near-perfect speedup in parallel with essentially no communication between processes and produces data that are highly correlated with those obtained from the full fine-grid solution. Four numerical experiments demonstrate the efficiency and accuracy of the method. The two-stage MCMC algorithm produce the same results (i.e., posterior distributions and uncertainty information, such as medians and highest posterior density intervals) as the Metropolis-Hastings MCMC. Thus, no information needed for uncertainty quantification is compromised when replacing the one-stage MCMC with the more computationally efficient two-stage MCMC. In four representative experiments, the two-stage method reduces the time spent on rejected models by one-third to one-half, which is important because most of models tried during the course of the MCMC algorithm are rejected. Furthermore, the two-stage MCMC algorithm substantially reduced the overall time-per-trial by as much as 40%, while increasing the acceptance rate from 9% to 90%.

Experiences With Markov Chain Monte Carlo Convergence Assessment in Two Psychometric Examples

2004 ◽  
Vol 29 (4) ◽  
pp. 461-488 ◽  
Author(s):  
Sandip Sinharay
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Diagnostic Tool ◽  
Statistical Models ◽  
Real Data ◽  
Mcmc Algorithm ◽  
Mcmc Algorithms ◽  
Convergence Diagnostics ◽  
Number Of Iterations

There is an increasing use of Markov chain Monte Carlo (MCMC) algorithms for fitting statistical models in psychometrics, especially in situations where the traditional estimation techniques are very difficult to apply. One of the disadvantages of using an MCMC algorithm is that it is not straightforward to determine the convergence of the algorithm. Using the output of an MCMC algorithm that has not converged may lead to incorrect inferences on the problem at hand. The convergence is not one to a point, but that of the distribution of a sequence of generated values to another distribution, and hence is not easy to assess; there is no guaranteed diagnostic tool to determine convergence of an MCMC algorithm in general. This article examines the convergence of MCMC algorithms using a number of convergence diagnostics for two real data examples from psychometrics. Findings from this research have the potential to be useful to researchers using the algorithms. For both the examples, the number of iterations required (suggested by the diagnostics) to be reasonably confident that the MCMC algorithm has converged may be larger than what many practitioners consider to be safe.

scholarly journals Convergence of Conditional Metropolis-Hastings Samplers

2014 ◽  
Vol 46 (02) ◽  
pp. 422-445 ◽  
Author(s):  
Galin L. Jones ◽  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Diffusion Process ◽  
Sample Path ◽  
Discrete Observations ◽  
Entire Sample ◽  
Bayesian Inferences ◽  
Monte Carlo Algorithms

We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in aconditional Metropolis-Hastings sampler(CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.

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