Ensemble samplers with affine invariance
31 Jan 2010-Vol. 5, Iss: 1, pp 65-80
TL;DR: A family of Markov chain Monte Carlo methods whose performance is unaffected by affine tranformations of space is proposed, and computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.
Abstract: We propose a family of Markov chain Monte Carlo methods whose performance is unaffected by affine tranformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particularly useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.
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TL;DR: The emcee algorithm as mentioned in this paper is a Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010).
Abstract: We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to ~N2 for a traditional algorithm in an N-dimensional parameter space. In this document, we describe the algorithm and the details of our implementation. Exploiting the parallelism of the ensemble method, emcee permits any user to take advantage of multiple CPU cores without extra effort. The code is available online at http://dan.iel.fm/emcee under the GNU General Public License v2.
8,805 citations
TL;DR: This document introduces a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010).
Abstract: We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to $\sim N^2$ for a traditional algorithm in an N-dimensional parameter space. In this document, we describe the algorithm and the details of our implementation and API. Exploiting the parallelism of the ensemble method, emcee permits any user to take advantage of multiple CPU cores without extra effort. The code is available online at this http URL under the MIT License.
5,293 citations
Cites methods from "Ensemble samplers with affine invar..."
...I M ] 1 8 Fe b 20 12 We introduce a stable, well tested Python implementation of the affineinvariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010)....
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TL;DR: Stan as discussed by the authors is a probabilistic programming language for specifying statistical models, where a program imperatively defines a log probability function over parameters conditioned on specified data and constants, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration.
Abstract: Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.
4,947 citations
01 Jan 2017
TL;DR: Stan is a probabilistic programming language for specifying statistical models that provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler and an adaptive form of Hamiltonian Monte Carlo sampling.
Abstract: Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.
2,938 citations
Additional excerpts
...(Goodman and Weare 2010) and a differential evolution sampler (Ter Braak 2006)....
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TL;DR: In this paper, a Monte Carlo sampler (The Joker) is used to perform a search for companions to 96,231 red-giant stars observed in the APOGEE survey (DR14) with $ ≥ 3$ spectroscopic epochs.
Abstract: Multi-epoch radial velocity measurements of stars can be used to identify stellar, sub-stellar, and planetary-mass companions. Even a small number of observation epochs can be informative about companions, though there can be multiple qualitatively different orbital solutions that fit the data. We have custom-built a Monte Carlo sampler (The Joker) that delivers reliable (and often highly multi-modal) posterior samplings for companion orbital parameters given sparse radial-velocity data. Here we use The Joker to perform a search for companions to 96,231 red-giant stars observed in the APOGEE survey (DR14) with $\\geq 3$ spectroscopic epochs. We select stars with probable companions by making a cut on our posterior belief about the amplitude of the stellar radial-velocity variation induced by the orbit. We provide (1) a catalog of 320 companions for which the stellar companion properties can be confidently determined, (2) a catalog of 4,898 stars that likely have companions, but would require more observations to uniquely determine the orbital properties, and (3) posterior samplings for the full orbital parameters for all stars in the parent sample. We show the characteristics of systems with confidently determined companion properties and highlight interesting systems with candidate compact object companions.
2,564 citations
Cites methods from "Ensemble samplers with affine invar..."
...In detail, we use an ensemble MCMC sampling algorithm (Goodman & Weare 2010) implemented in Python (emcee; Foreman-Mackey et al. 2013) to perform the samplings....
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References
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TL;DR: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point.
Abstract: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.
27,271 citations
"Ensemble samplers with affine invar..." refers methods in this paper
...Our ensemble methods are motivated in part by the Nelder Mead [11] 1Here xk is walker k in an ensemble of L walkers....
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...Our ensemble methods are motivated in part by the Nelder Mead [11] (1)Here xk is walker k in an ensemble of L walkers....
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...Applying the Nelder Mead algorithm to the ill conditioned optimization problem for the function (3) in Figure 1 is exactly equivalent to applying it to the easier problem of optimizing the well scaled function (4)....
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Book•
01 Jan 2001
TL;DR: This book provides a self-contained and up-to-date treatment of the Monte Carlo method and develops a common framework under which various Monte Carlo techniques can be "standardized" and compared.
Abstract: This paperback edition is a reprint of the 2001 Springer edition. This book provides a self-contained and up-to-date treatment of the Monte Carlo method and develops a common framework under which various Monte Carlo techniques can be "standardized" and compared. Given the interdisciplinary nature of the topics and a moderate prerequisite for the reader, this book should be of interest to a broad audience of quantitative researchers such as computational biologists, computer scientists, econometricians, engineers, probabilists, and statisticians. It can also be used as the textbook for a graduate-level course on Monte Carlo methods. Many problems discussed in the alter chapters can be potential thesis topics for masters or Ph.D. students in statistics or computer science departments. Jun Liu is Professor of Statistics at Harvard University, with a courtesy Professor appointment at Harvard Biostatistics Department. Professor Liu was the recipient of the 2002 COPSS Presidents' Award, the most prestigious one for statisticians and given annually by five leading statistical associations to one individual under age 40. He was selected as a Terman Fellow by Stanford University in 1995, as a Medallion Lecturer by the Institute of Mathematical Statistics (IMS) in 2002, and as a Bernoulli Lecturer by the International Bernoulli Society in 2004. He was elected to the IMS Fellow in 2004 and Fellow of the American Statistical Association in 2005. He and co-workers have published more than 130 research articles and book chapters on Bayesian modeling and computation, bioinformatics, genetics, signal processing, stochastic dynamic systems, Monte Carlo methods, and theoretical statistics. "An excellent survey of current Monte Carlo methods. The applications amply demonstrate the relevance of this approach to modern computing. The book is highly recommended." (Mathematical Reviews) "This book provides comprehensive coverage of Monte Carlo methods, and in the process uncovers and discusses commonalities among seemingly disparate techniques that arose in various areas of application. The book is well organized; the flow of topics follows a logical development. The coverage is up-to-date and comprehensive, and so the book is a good resource for people conducting research on Monte Carlo methods. The book would be an excellent supplementary text for a course in scientific computing ." (SIAM Review) "The strength of this book is in bringing together advanced Monte Carlo (MC) methods developed in many disciplines. Throughout the book are examples of techniques invented, or reinvented, in different fields that may be applied elsewhere. Those interested in using MC to solve difficult problems will find many ideas, collected from a variety of disciplines, and references for further study." (Technometrics)
2,973 citations
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01 Jan 2004
TL;DR: In this paper, the Monte Carlo method is not compelling for one dimensional integration, but it is more compelling for a d-dimensional integral evaluated withM points, so that the error in I goes down as 1/ √ M and is smaller if the variance σ 2 f of f is smaller.
Abstract: so that the error in I goes down as 1/ √ M and is smaller if the variance σ 2 f of f is smaller For a one dimensional integration the Monte Carlo method is not compelling However consider a d dimensional integral evaluated withM points For a uniform mesh each dimension of the integral getsM1/d points, so that the separation is h = M−1/d The error in the integration over one h cube is of order hd+2, since we are approximating the surface by a linear interpolation (a plane) with an O(h2) error The total error in the integral is Mhd+2 = M−2/d The error in the Monte Carlo method remains M−1/2, so that this method wins for d > 4 We can reduce the error in I by reducing the effective σf This is done by concentrating the sampling where f (x) is large, using a weight function w(x) (ie w(x) > 0, ∫ 1 0 w(x) = 1) I = ∫ 1
1,642 citations
16 May 1996
TL;DR: In this article, the authors studied the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces, and special attention was paid to the invariant measures and ergodicity.
Abstract: This book is devoted to the asymptotic properties of solutions of stochastic evolution equations in infinite dimensional spaces. It is divided into three parts: Markovian dynamical systems; invariant measures for stochastic evolution equations; invariant measures for specific models. The focus is on models of dynamical processes affected by white noise, which are described by partial differential equations such as the reaction-diffusion equations or Navier–Stokes equations. Besides existence and uniqueness questions, special attention is paid to the asymptotic behaviour of the solutions, to invariant measures and ergodicity. Some of the results found here are presented for the first time. For all whose research interests involve stochastic modelling, dynamical systems, or ergodic theory, this book will be an essential purchase.
1,147 citations
"Ensemble samplers with affine invar..." refers background in this paper
...with free boundary condition at x = 0 and x = 1 [3; 12]....
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