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Journal of the Royal Statistical Society. Series C Applied Statistics. Bibcode : ITSP Journal of the American Statistical Association. Computational Statistics. International Journal of Coal Geology. Mean field simulation for Monte Carlo integration. Feynman-Kac formulae. Genealogical and interacting particle approximations.

Lecture Notes in Mathematics. Del Moral - A. Doucet - A. Series B Statistical Methodology. Annals of Statistics. Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences Diss. Stanford University.

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Mathematics and Computers in Simulation. Operations Research. Statistical Science. Bibcode : StaSc Retrieved Nucleic Acids Research. Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability. Atzberger, P. Berg, Bernd A. World Scientific. Bolstad, William M. Understanding Computational Bayesian Statistics. Casella, George; George, Edward I. The American Statistician. Gelfand, A. Gelman, Andrew ; Carlin, John B. Bayesian Data Analysis 1st ed. They are also a foundational tool in formulating many machine learning problems.

This course is the second in a sequence of three. Following the first course, which focused on representation, this course addresses the question of probabilistic inference: how a PGM can be used to answer questions. Even though a PGM generally describes a very high dimensional distribution, its structure is designed so as to allow questions to be answered efficiently. The course presents both exact and approximate algorithms for different types of inference tasks, and discusses where each could best be applied.

The highly recommended honors track contains two hands-on programming assignments, in which key routines of the most commonly used exact and approximate algorithms are implemented and applied to a real-world problem.


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  • Markov Chain Monte Carlo in Practice?

Thanks a lot for professor D. Really a very good starting point for PGM model and preparation for learning part. I learned pretty much from this course. The acceptance rate is the ratio of accepted to generated proposals and is typically updated batch-wise. In general, by decreasing the proposal standard deviation the acceptance rate increases and vice versa.

Theoretically, for single component updating schemes like in this work , the optimal target acceptance rate is 0. This strategy, which we refer to as the FSL strategy, tunes the acceptance rate to 0. It works by multiplying the proposal variance by the ratio of accepted to rejected samples, i. Since this method never ceases the adaptation of the standard deviations, it theoretically loses ergodicity of the chain Roberts and Rosenthal, , Since this method features diminishing adaptation, the chain remains ergodic Roberts and Rosenthal, We compared all three strategies and the default, with no adaptation, on the number of effective samples they generated see below and on accuracy and precision, using ground truth simulation data.

We sampled all models with 20, samples, without thinning and using the point optimized Maximum Likelihood Estimator MLE as a starting point. We reported statistics over the first 10, samples in the article, considering it is the common number of samples in MCMC sampling, and report estimates over all 20, samples as Supplementary Figures. Burn-in is the process of discarding the first z samples from the chain and using only the remaining samples in subsequent analysis.

The idea is that if the starting point had a low probability then the limited number of early samples may over sample low probability regions. By discarding the first z samples as a burn-in, the hope is that, by then, the chain has converged to its stationary distribution and that all further samples are directly from the stationary distribution Robert, Theoretically, burn-in is unnecessary since any empirical average. Additionally, since it can not be predicted how long it will take for the chain to reach convergence, the required burn-in can only be estimated post-hoc.

In practice, discarding the first few thousand samples as a burn-in often works and is less time-consuming than generating a lot of samples to average out the effects of a low probability starting position. An alternative to burn-in, or, to reduce the need for burn-in, is to use a Maximum Likelihood Estimator as starting point for the MCMC sampling van Ravenzwaaij et al. If the optimization routine did its work well, the MLE should be part of the stationary distribution of the Markov chain, removing the need for burn-in altogether.

To evaluate the effect of burn-in and initialization single-slice datas was sampled using the NODDI model with the default starting point and with MLE. We compare these starting points on moving mean and moving standard deviation, as well as on autocorrelation the correlation of a chain with itself given by:. Thinning is the process of using only every k th step of the chain for analysis, while all other steps are discarded, with as goal reducing autocorrelation and obtaining relatively independent samples.

Several authors have recommended against the use of thinning, stating that it is often unnecessary, always inefficient and reduces the precision of the posterior estimates Geyer, ; MacEachern and Berliner, ; Jackman, ; Christensen et al. The only valid reason for thinning is to avoid bias in the standard error estimate of posterior mean, when that mean estimate was computed over all non-thinned samples MacEachern and Berliner, ; Link and Eaton, In general, thinning is only considered worthwhile if there are storage limitations, or when the cost of processing the output outweighs the benefits of reduced variance of the estimator Geyer, ; MacEachern and Berliner, ; Link and Eaton, For example, 1, samples with an ESS of have a higher information content than 2, samples with an ESS of The ESS can be defined as the minimum size of a set of posterior samples taken directly from the posterior , which have the same efficiency measure of quality in the posterior density estimation as a given chain of samples obtained from MCMC sampling Martino et al.

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Conversely, ESS theory can quantify how many samples should be taken in a chain to reach a given quality of posterior estimates. We use the ESS theory to comparing proposal adaptation strategies and to estimating the minimum number of samples necessary for adequate sampling of diffusion microstructure models. Multivariate ESS theory Vats et al.

Figure 2. Since online monitoring of the ESS during MCMC sampling is an expensive operation, and terminating on ESS will yield different sample sizes for different voxels, we instead use the ESS theory to estimate a fixed minimum number of samples needed to reach a desired ESS when averaged over a white matter mask. For this study we used two groups of ten subjects coming from two studies, each whith a different acquisition protocol. These datasets were acquired at a resolution of 1. These four-shell, high number of directions, and very high maximum b- value datasets allow a wide range of models to be fitted.

These datasets had a resolution of 2.

Markov Chain Monte Carlo and the Metropolis Alogorithm

Additional b0 volumes were acquired with a reversed phase encoding direction which were used to correct susceptibility related distortion in addition to bulk subject motion with the topup and eddy tools in FSL version 5. We refer to these datasets as RLS-pilot — 2 mm - dir - b3k and to the multi-shell direction table as the RLS-pilot table.

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These three-shell datasets represent a relatively short time acquisition protocol that still allows many models to be fitted. All other models are estimated on all data volumes. The whole brain mask is used during sampling, whereas averages over the WM mask are used in model or data comparisons. The WM mask was calculated by applying a lower threshold of 0.

We performed ground truth simulations to illustrate the effects of the adaptive proposals on effective sample sizes and on accuracy and precision of parameter estimation. To ensure Gaussianity of the sampled parameter distributions, we generate the parameters with a smaller range than the support of the sampling priors Table 4. Table 4. The simulation ranges per model parameters. We generate uniformly distributed parameter values using the upper and lower bounds presented. Analogous to Harms et al. We compute a measure of accuracy as the inverse of the mean of the average estimate error over ten thousand random repeats and a measure of precision as the inverse of the standard deviation of the average estimates.

Finally, we aggregate these results per model and per experiment over 10 independent ground truth simulation trials into a mean and standard error of the mean SEM for both accuracy and precision.

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We then present burn-in and thinning given an effective proposal strategy, and end with ESS estimates on the minimum number of samples needed for adequate characterization of the posterior distribution. Comparisons are based on multivariate Effective Sample Size, and accuracy and precision using ground truth simulations. The illustration clearly shows that without adaptive proposals the chain can get stuck in the same position for quite some time, while all adaptive proposal methods can adapt the standard deviations to better cover the support of the posterior distribution. Figure 3. Results were computed with an initial proposal standard deviation of 0.

A Gaussian distribution function was fitted to the samples, superimposed in blue on the sample histograms, with its mean indicated by the blue dot.