The Experts below are selected from a list of 16275 Experts worldwide ranked by ideXlab platform
Håkon Tjelmeland - One of the best experts on this subject based on the ideXlab platform.
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A multiple-try Metropolis–Hastings Algorithm with tailored proposals
Computational Statistics, 2019Co-Authors: Xin Luo, Håkon TjelmelandAbstract:We present a new multiple-try Metropolis–Hastings Algorithm designed to be especially beneficial when a tailored proposal distribution is available. The Algorithm is based on a given acyclic graph $$\mathcal {G}$$ , where one of the nodes in $$\mathcal {G}$$ , k say, contains the current state of the Markov chain and the remaining nodes contain proposed states generated by applying the tailored proposal distribution. The Metropolis–Hastings Algorithm alternates between two types of updates. The first type of update is using the tailored proposal distribution to generate new states for all nodes in $$\mathcal {G}$$ except node k. The second type of update is generating a new value for k, thereby changing the value of the current state. We evaluate the effectiveness of the proposed scheme in two examples with previously defined target and proposal distributions.
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a multiple try metropolis hastings Algorithm with tailored proposals
arXiv: Computation, 2018Co-Authors: Xin Luo, Håkon TjelmelandAbstract:We present a new multiple-try Metropolis-Hastings Algorithm designed to be especially beneficial when a tailored proposal distribution is available. The Algorithm is based on a given acyclic graph $G$, where one of the nodes in $G$, $k$ say, contains the current state of the Markov chain and the remaining nodes contain proposed states generated by applying the tailored proposal distribution. The Metropolis-Hastings Algorithm alternates between two types of updates. The first update type is using the tailored proposal distribution to generate new states in all nodes in $G$ except in node $k$. The second update type is generating a new value for $k$, thereby changing the value of the current state. We evaluate the effectiveness of the proposed scheme in an example with previously defined target and proposal distributions.
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control variates for the metropolis hastings Algorithm
Scandinavian Journal of Statistics, 2008Co-Authors: Hugo Lewi Hammer, Håkon TjelmelandAbstract:We propose new control variates for variance reduction in estimation of mean values using the Metropolis-Hastings Algorithm. Traditionally, states that are rejected in the Metropolis-Hastings Algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis-Hastings and reversible jump Algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state x and the proposal y , but we also consider variates that in addition are functions of the Metropolis-Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%. Copyright (c) Board of the Foundation of the Scandinavian Journal of Statistics 2008.
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Control Variates for the Metropolis–Hastings Algorithm
Scandinavian Journal of Statistics, 2008Co-Authors: Hugo Lewi Hammer, Håkon TjelmelandAbstract:Abstract. We propose new control variates for variance reduction in estimation of mean values using the Metropolis–Hastings Algorithm. Traditionally, states that are rejected in the Metropolis–Hastings Algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis–Hastings and reversible jump Algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state x and the proposal y, but we also consider variates that in addition are functions of the Metropolis–Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%.
Sebastian J. Vollmer - One of the best experts on this subject based on the ideXlab platform.
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the bouncy particle sampler a nonreversible rejection free markov chain monte carlo method
Journal of the American Statistical Association, 2018Co-Authors: Alexandre Bouchardcote, Sebastian J. Vollmer, Arnaud DoucetAbstract:Many Markov chain Monte Carlo techniques currently available rely on discrete-time reversible Markov processes whose transition kernels are variations of the Metropolis–Hastings Algorithm. We explo...
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the bouncy particle sampler a nonreversible rejection free markov chain monte carlo method
Journal of the American Statistical Association, 2018Co-Authors: Alexandre Bouchardcote, Sebastian J. Vollmer, Arnaud DoucetAbstract:ABSTRACTMany Markov chain Monte Carlo techniques currently available rely on discrete-time reversible Markov processes whose transition kernels are variations of the Metropolis–Hastings Algorithm. We explore and generalize an alternative scheme recently introduced in the physics literature (Peters and de With 2012) where the target distribution is explored using a continuous-time nonreversible piecewise-deterministic Markov process. In the Metropolis–Hastings Algorithm, a trial move to a region of lower target density, equivalently of higher “energy,” than the current state can be rejected with positive probability. In this alternative approach, a particle moves along straight lines around the space and, when facing a high energy barrier, it is not rejected but its path is modified by bouncing against this barrier. By reformulating this Algorithm using inhomogeneous Poisson processes, we exploit standard sampling techniques to simulate exactly this Markov process in a wide range of scenarios of interest. ...
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Spectral gaps for a Metropolis–Hastings Algorithm in infinite dimensions
Annals of Applied Probability, 2014Co-Authors: Martin Hairer, Andrew M. Stuart, Sebastian J. VollmerAbstract:We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings Algorithm that adds an accept–reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein–Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN Algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an L^2-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM Algorithm applied to the reference measures degenerates as the dimension tends to infinity.
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spectral gaps for a metropolis hastings Algorithm in infinite dimensions
Annals of Applied Probability, 2014Co-Authors: Martin Hairer, Andrew M. Stuart, Sebastian J. VollmerAbstract:We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings Algorithm that adds an accept–reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein–Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN Algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an L^2-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM Algorithm applied to the reference measures degenerates as the dimension tends to infinity.
Benjamin Jourdain - One of the best experts on this subject based on the ideXlab platform.
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does waste recycling really improve the multi proposal metropolis hastings Algorithm an analysis based on control variates
Journal of Applied Probability, 2009Co-Authors: Jeanfranccois Delmas, Benjamin JourdainAbstract:The waste-recycling Monte Carlo (WRMC) Algorithm introduced by physicists is a modification of the (multi-proposal) Metropolis-Hastings Algorithm, which makes use of all the proposals in the empirical mean, whereas the standard (multi-proposal) Metropolis-Hastings Algorithm uses only the accepted proposals. In this paper we extend the WRMC Algorithm to a general control variate technique and exhibit the optimal choice of the control variate in terms of the asymptotic variance. We also give an example which shows that, in contradiction to the intuition of physicists, the WRMC Algorithm can have an asymptotic variance larger than that of the Metropolis-Hastings Algorithm. However, in the particular case of the Metropolis-Hastings Algorithm called the Boltzmann Algorithm, we prove that the WRMC Algorithm is asymptotically better than the Metropolis-Hastings Algorithm. This last property is also true for the multi proposal Metropolis-Hastings Algorithm. In this last framework we consider a linear parametric generalization of WRMC, and we propose an estimator of the explicit optimal parameter using the proposals.
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Does waste-recycling really improve Metropolis-Hastings Monte Carlo Algorithm?
2009Co-Authors: Jean-françois Delmas, Benjamin JourdainAbstract:The Metropolis Hastings Algorithm and its multi-proposal extensions are aimed at the computation of the expectation $\langle \pi,f\rangle$ of a function $f$ under a probability measure $\pi$ difficult to simulate. They consist in constructing by an appropriate acceptation/rejection procedure a Markov chain $(X_k,k\geq 0)$ with transition matrix $P$ such that $\pi$ is reversible with respect to $P$ and in estimating $\langle \pi,f\rangle$ by the empirical mean $I_n(f)=\inv{n}\sum_{k=1}^n f(X_k)$. The waste-recycling Monte Carlo (WR) Algorithm introduced by physicists is a modification of the Metropolis-Hastings Algorithm, which makes use of all the proposals in the empirical mean, whereas the standard Metropolis-Hastings Algorithm only uses the accepted proposals. In this paper, we extend the WR Algorithm into a general control variate technique and exhibit the optimal choice of the control variate in terms of asymptotic variance. We also give an example which shows that in contradiction to the intuition of physicists, the WR Algorithm can have an asymptotic variance larger than the one of the Metropolis-Hastings Algorithm. However, in the particular case of the Metropolis-Hastings Algorithm called Boltzmann Algorithm, we prove that the WR Algorithm is asymptotically better than the Metropolis-Hastings Algorithm.
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Does Waste Recycling Really Improve the Multi-Proposal Metropolis–Hastings Algorithm? an Analysis Based on Control Variates
Journal of Applied Probability, 2009Co-Authors: Jeanfranccois Delmas, Benjamin JourdainAbstract:The waste-recycling Monte Carlo (WRMC) Algorithm introduced by physicists is a modification of the (multi-proposal) Metropolis–Hastings Algorithm, which makes use of all the proposals in the empirical mean, whereas the standard (multi-proposal) Metropolis–Hastings Algorithm uses only the accepted proposals. In this paper we extend the WRMC Algorithm to a general control variate technique and exhibit the optimal choice of the control variate in terms of the asymptotic variance. We also give an example which shows that, in contradiction to the intuition of physicists, the WRMC Algorithm can have an asymptotic variance larger than that of the Metropolis–Hastings Algorithm. However, in the particular case of the Metropolis–Hastings Algorithm called the Boltzmann Algorithm, we prove that the WRMC Algorithm is asymptotically better than the Metropolis–Hastings Algorithm. This last property is also true for the multi-proposal Metropolis–Hastings Algorithm. In this last framework we consider a linear parametric generalization of WRMC, and we propose an estimator of the explicit optimal parameter using the proposals.
James Cussens - One of the best experts on this subject based on the ideXlab platform.
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Markov Chain Monte Carlo using Tree-Based Priors on Model Structure
arXiv: Artificial Intelligence, 2013Co-Authors: Nicos Angelopoulos, James CussensAbstract:We present a general framework for defining priors on model structure and sampling from the posterior using the Metropolis-Hastings Algorithm. The key idea is that structure priors are defined via a probability tree and that the proposal mechanism for the Metropolis-Hastings Algorithm operates by traversing this tree, thereby defining a cheaply computable acceptance probability. We have applied this approach to Bayesian net structure learning using a number of priors and tree traversal strategies. Our results show that these must be chosen appropriately for this approach to be successful.
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UAI - Markov chain monte carlo using tree-based priors on model structure
2001Co-Authors: Nicos Angelopoulos, James CussensAbstract:We present a general framework for defining priors on model structure and sampling from the posterior using the Metropolis-Hastings Algorithm. The key ideas are that structure priors are defined via a probability tree and that the proposal distribution for the Metropolis-Hastings Algorithm is defined using the prior, thereby defining a cheaply computable acceptance probability. We have applied this approach to Bayesian net structure learning using a number of priors and proposal distributions. Our results show that these must be chosen appropriately for this approach to be successful.
L. Gottschalk - One of the best experts on this subject based on the ideXlab platform.
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Assessing uncertainties in a conceptual water balance model using Bayesian methodology
Hydrological Sciences Journal, 2005Co-Authors: K. Engeland, X. Chong Yu, L. GottschalkAbstract:The aim of this study was to estimate the uncertainties in the streamflow simulated by a rainfall-runoff model. Two sources of uncertainties in hydrological modelling were considered: The uncertainties in model parameters and those in model structure. The uncertainties were calculated by Bayesian statistics, and the Metropolis-Hastings Algorithm was used to simulate the posterior parameter distribution. The parameter uncertainty calculated by the Metropolis-Hastings Algorithm was compared to maximum likelihood estimates which assume that both the parameters and model residuals are normally distributed. The study was performed using the model WASMOD on 25 basins in central Sweden. Confidence intervals in the simulated discharge due to the parameter uncertainty and the total uncertainty were calculated. The results indicate that (a) the Metropolis-Hastings Algorithm and the maximum likelihood method give almost identical estimates concerning the parameter uncertainty, and (b) the uncertainties in the simulated streamflow due to the parameter uncertainty are less important than uncertainties originating from other sources for this simple model with fewer parameters.
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Assessing uncertainties in a conceptual water balance model using Bayesian methodology
Hydrological Sciences Journal, 2005Co-Authors: K. Engeland, L. GottschalkAbstract:Abstract Abstract The aim of this study was to estimate the uncertainties in the streamflow simulated by a rainfall–runoff model. Two sources of uncertainties in hydrological modelling were considered: the uncertainties in model parameters and those in model structure. The uncertainties were calculated by Bayesian statistics, and the Metropolis-Hastings Algorithm was used to simulate the posterior parameter distribution. The parameter uncertainty calculated by the Metropolis-Hastings Algorithm was compared to maximum likelihood estimates which assume that both the parameters and model residuals are normally distributed. The study was performed using the model WASMOD on 25 basins in central Sweden. Confidence intervals in the simulated discharge due to the parameter uncertainty and the total uncertainty were calculated. The results indicate that (a) the Metropolis-Hastings Algorithm and the maximum likelihood method give almost identical estimates concerning the parameter uncertainty, and (b) the uncertain...
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68. Assessing Uncertainties in a Conceptual Water Balance Model Using Bayesian Methodology
Tunnelling and Underground Space Technology, 2005Co-Authors: K. Engeland, L. GottschalkAbstract:The aim of this study was to estimate the uncertainties in the streamflow simulated by rainfall-runoff model. Two sources of uncertainties in hydrological modelling were considered: the uncertainties in model parameters and those in model structure. The uncertainties were calculated by Bayesian statistics, and the Metropolis-Hastings Algorithm was used to simulate the posterior parameter distribution. The parameter uncertainty calculated by the Metropolis-Hastings Algorithm was compared to maximum likelihood estimates which assume that both the parameters and model residuals are normally distributed. The study was performed using the model WASMOD on 25 basins in central Sweden. Confidence intervals in the simulated discharge due to the parameter uncertainty and the total uncertainty were calculated. The results indicate that (a) the Metropolis-Hastings Algorithm and the maximum likelihood method give almost identical estimates concerning the parameter uncertainty, and (b) the uncertainties in the simulated streamflow due to the parameter uncertainty are less important than uncertainties originating from other sources for this simple model with fewer parameters.