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36
Slice sampling
 Annals of Statistics
, 2000
"... Abstract. Markov chain sampling methods that automatically adapt to characteristics of the distribution being sampled can be constructed by exploiting the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain th ..."
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Cited by 147 (5 self)
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Abstract. Markov chain sampling methods that automatically adapt to characteristics of the distribution being sampled can be constructed by exploiting the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal ‘slice ’ defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Variations on such ‘slice sampling ’ methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling, and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially allow the sampling to adapt to dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done using ‘overrelaxed ’ versions of univariate slice sampling procedures, or by using ‘reflective ’ multivariate slice sampling methods, which bounce off the edges of the slice.
On the Convergence of Monte Carlo Maximum Likelihood Calculations
 Journal of the Royal Statistical Society B
, 1992
"... Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the ..."
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Cited by 59 (3 self)
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Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'eKuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...
Transdimensional Markov chain Monte Carlo
 in Highly Structured Stochastic Systems
, 2003
"... In the context of samplebased computation of Bayesian posterior distributions in complex stochastic systems, this chapter discusses some of the uses for a Markov chain with a prescribed invariant distribution whose support is a union of euclidean spaces of differing dimensions. This leads into a re ..."
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Cited by 56 (0 self)
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In the context of samplebased computation of Bayesian posterior distributions in complex stochastic systems, this chapter discusses some of the uses for a Markov chain with a prescribed invariant distribution whose support is a union of euclidean spaces of differing dimensions. This leads into a reformulation of the reversible jump MCMC framework for constructing such ‘transdimensional ’ Markov chains. This framework is compared to alternative approaches for the same task, including methods that involve separate sampling within different fixeddimension models. We consider some of the difficulties researchers have encountered with obtaining adequate performance with some of these methods, attributing some of these to misunderstandings, and offer tentative recommendations about algorithm choice for various classes of problem. The chapter concludes with a look towards desirable future developments.
ML parameter estimation for Markov random fields, with applications to Bayesian tomography
 IEEE Trans. on Image Processing
, 1998
"... Abstract 1 Markov random fields (MRF) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (so ..."
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Cited by 49 (18 self)
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Abstract 1 Markov random fields (MRF) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: 1) Direct parameter estimation for MRF’s is known to be mathematically and numerically challenging. 2) Parameters can not be directly estimated because the true image crosssection is unavailable. In this paper, we propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as a generalized Gaussian MRF (GGMRF). The first section of the paper derives methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, σ, has a simple closed form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an offline numerical computation of the dependence of the partition function on p.
Markov Chain Monte Carlo Methods Based on `Slicing' the Density Function
, 1997
"... . One way to sample from a distribution is to sample uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal `sl ..."
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Cited by 46 (0 self)
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. One way to sample from a distribution is to sample uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal `slice' defined by the current vertical position. Variations on such `slice sampling' methods can easily be implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling, and may be more efficient than easilyconstructed versions of the Metropolis algorithm. Slice sampling is therefore attractive in routine Markov chain Monte Carlo applications, and for use by software that automatically generates a Markov chain sampler from a model specification. One can also easily devise overrelaxed versions of slice sampling, which sometimes greatly improve sampling effici...
Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered Overrelaxation
 Learning in Graphical Models
, 1995
"... this paper, and by dynamical methods, such as "hybrid Monte Carlo", which I briefly describe next. The hybrid Monte Carlo method, due to Duane, Kennedy, Pendleton, and Roweth (1987), can be seen as an elaborate form of the Metropolis algorithm (in an extended state space) in which candidate states ..."
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Cited by 43 (5 self)
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this paper, and by dynamical methods, such as "hybrid Monte Carlo", which I briefly describe next. The hybrid Monte Carlo method, due to Duane, Kennedy, Pendleton, and Roweth (1987), can be seen as an elaborate form of the Metropolis algorithm (in an extended state space) in which candidate states are found by simulating a trajectory defined by Hamiltonian dynamics. These trajectories will proceed in a consistent direction, until such time as they reach a region of low probability. By using states proposed by this deterministic process, random walk effects can be largely eliminated. In Bayesian inference problems for complex models based on neural networks, I have found (Neal 1995) that the hybrid Monte Carlo method can be hundreds or thousands of times faster than simple versions of the Metropolis algorithm. Hybrid Monte Carlo can be applied to a wide variety of problems where the state variables are continuous, and derivatives of the probability density can be efficiently computed. The method does, however, require that careful choices be made both for the length of the trajectories and for the stepsize used in the discretization of the dynamics. Using too large a stepsize will cause the dynamics to become unstable, resulting in an extremely high rejection rate. This need to carefully select the stepsize in the hybrid Monte Carlo method is similar to the need to carefully select the width of the proposal distribution in simple forms of the Metropolis algorithm. (For example, if a candidate state is drawn from a Gaussian distribution centred at the current state, one must somehow decide what the standard deviation of this distribution should be). Gibbs sampling does not require that the user set such parameters. A Markov chain Monte Carlo method that shared this adva...
Bayesian Estimation for Homogeneous and Inhomogeneous Gaussian Random Fields
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1996
"... This paper investigates Bayesian estimation for Gaussian Markov random fields. In particular, a new class of inhomogeneous model is proposed. This inhomogeneous model uses a Markov random field to describe spatial variation of the smoothing parameter in a second random field which describes the spat ..."
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Cited by 16 (2 self)
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This paper investigates Bayesian estimation for Gaussian Markov random fields. In particular, a new class of inhomogeneous model is proposed. This inhomogeneous model uses a Markov random field to describe spatial variation of the smoothing parameter in a second random field which describes the spatial variation in the observed intensity image. The coupled Markov random fields will be used as prior distributions, and combined with Gaussian noise models to produce posterior distributions on which estimation will be based. All model parameters are estimated, in a fully Bayesian setting, using the MetropolisHastings algorithm. The models and algorithms will be illustrated using various artificial examples. The full posterior estimation procedures using homogeneous and inhomogeneous models will be compared. For the examples considered the fully Bayesian estimation for inhomogeneous models performs very favourably when compared to methods using homogeneous models, allowing differential smo...
Markov Chain Monte Carlo and Spatial Point Processes
, 1999
"... this paper) reversibility holds, that is f P(x, A)(,x) = f PC, B A for all A, B , whereby r is clearly invariant ..."
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Cited by 15 (5 self)
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this paper) reversibility holds, that is f P(x, A)(,x) = f PC, B A for all A, B , whereby r is clearly invariant
Asymptotic Variance and Convergence Rates of NearlyPeriodic MCMC Algorithms
, 2001
"... We consider nearlyperiodic chains, which may have excellent functionalestimation properties but poor distributional convergence rate. We show how simple modications of the chain (involving using a random number of iterations) can greatly improve the distributional convergence of the chain. We prov ..."
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Cited by 11 (4 self)
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We consider nearlyperiodic chains, which may have excellent functionalestimation properties but poor distributional convergence rate. We show how simple modications of the chain (involving using a random number of iterations) can greatly improve the distributional convergence of the chain. We prove various theoretical results about convergence rates of the modied chains. We also consider a number of examples. 1.