On directional Metropolis–Hastings algorithms

Eidsvik, Jo; Tjelmeland, Håkon

doi:10.1007/s11222-006-5536-2

Cited by 11 publications

(8 citation statements)

References 36 publications

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“…Gilks et al (1998), Liu et al (2000) and Craiu and Lemieux (2007), Green and Mira (2001), Eidsvik and Tjelmeland (2006). In this paper, we refer to a more global version of adaptation which is based on learning the geography of π "on the fly" from all the samples available up to the current time t. Such an approach violates the Markovian property as the subsequent realizations of the chain depend not only on the current state but also on all past realizations.…”

Section: Introductionmentioning

confidence: 99%

Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC

Craiu

Rosenthal

Yang

2009

Journal of the American Statistical Association

119

147

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Starting with the seminal paper of Haario, Saksman and Tamminen (Haario et al. (2001)), a substantial amount of work has been done to validate adaptive Markov chain Monte Carlo algorithms. In this paper we focus on two practical aspects of adaptive Metropolis samplers. First, we draw attention to the deficient performance of standard adaptation when the target distribution is multi-modal. We propose a parallel chain adaptation strategy that incorporates multiple Markov chains which are run in parallel. Second, we note that 1 the current adaptive MCMC paradigm implicitly assumes that the adaptation is uniformly efficient on all regions of the state space. However, in many practical instances, different "optimal" kernels are needed in different regions of the state space. We propose here a regional adaptation algorithm in which we account for possible errors made in defining the adaptation regions. This corresponds to the more realistic case in which one does not know exactly the optimal regions for adaptation. The methods focus on the random walk Metropolis sampling algorithm but their scope is much wider. We provide theoretical justification for the two adaptive approaches using the existent theory build for adaptive Markov chain Monte Carlo. We illustrate the performance of the methods using simulations and analyze a mixture model for real data using an algorithm that combines the two approaches.

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Section: Introductionmentioning

confidence: 99%

Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC

Craiu

Rosenthal

Yang

2009

Journal of the American Statistical Association

119

147

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“…Adaptive methods include those that use the chain to modify the proposal distribution [17,18] and adaptive direction samplers [7,11] that maintain multiple points in parameter space. While adaptive methods speed convergence by more efficiently sampling the parameter space, other methods accelerate the posterior evaluations required to compute the Hastings ratio at each step.…”

Section: Mcmcmentioning

confidence: 99%

“…Instead, we compute approximations to the moments by generating samples from the posterior distribution and calculating the discrete analogues. Samples may be generated from an implicitly defined posterior by Markov chain Monte Carlo (MCMC) methods [2,7,10,11,17,18,19,28,30] whereby a Markov chain is established whose stationary distribution is the posterior.…”

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confidence: 99%

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Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems

Lieberman¹,

Willcox²,

Ghattas³

2010

SIAM J. Sci. Comput.

216

175

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A greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters. Large-scale models are too costly to evaluate repeatedly, as is required in the statistical setting. Furthermore, these models often have high-dimensional parametric input spaces, which compounds the difficulty of effectively exploring the uncertainty space. We simultaneously address both challenges by constructing a projectionbased reduced model that accepts low-dimensional parameter inputs and whose model evaluations are inexpensive. The associated parameter and state bases are obtained through a greedy procedure that targets the governing equations, model outputs, and prior information. The methodology and results are presented for groundwater inverse problems in one and two dimensions.1. Introduction. Statistical inverse problems governed by partial differential equations (PDEs) with spatially distributed parameters pose a significant computational challenge for existing methods. While the cost of a repeated PDE solutions can be addressed by traditional model reduction techniques, the difficulty in sampling in high-dimensional parameter spaces remains. We present a model reduction algorithm that seeks low-dimensional representations of parameters and states while maintaining fidelity in outputs of interest. The resulting reduced model accelerates model evaluations and facilitates efficient sampling in the reduced parameter space. The result is a tractable procedure for the solution of statistical inverse problems involving PDEs with high-dimensional parametric input spaces.Given a parameterized mathematical model of a certain phenomenon, the forward problem is to compute output quantities of interest for specified parameter inputs. In many cases, the parameters are uncertain, but they can be inferred from observations by solving an inverse problem. Inference is often performed by solving an optimization problem to minimize the disparity between model-predicted outputs and observations. Many inverse problems of this form are ill-posed in the sense that there may be many values of the parameters whose model-predicted outputs reproduce the observations. The set of parameters consistent with the observations may be larger still if we also admit noise in the sensor instruments. In the deterministic setting, a regularization term is often included in the objective function to make the problem well-posed. The *

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“…The implementation of MCMC does not directly find the maximum likelihoods and can avoid monotone convergence. MetropolisHastings algorithms are commonly adopted in MCMC by construct an ergodic Markov chain using rejection mechanism (Eidsvik and Tjelmeland, 2006;Strid, 2010). …”

Section: A Framework For Comprehensive Uncertainty Analysismentioning

confidence: 99%