Differential Evolution Markov Chain with snooker updater and fewer chains

Differential Evolution Markov Chain (DE-MC) is an adaptive MCMC algorithm, in which multiple chains are run in parallel. Standard DE-MC requires at least N=2d chains to be run in parallel, where d is the dimensionality of the posterior. This paper extends DE-MC with a snooker updater and shows by simulation and real examples that DE-MC can work for d up to 50–100 with fewer parallel chains (e.g. N=3) by exploiting information from their past by generating jumps from differences of pairs of past states. This approach extends the practical applicability of DE-MC and is shown to be about 5–26 times more efficient than the optimal Normal random walk Metropolis sampler for the 97.5% point of a variable from a 25–50 dimensional Student t 3 distribution. In a nonlinear mixed effects model example the approach outperformed a block-updater geared to the specific features of the mode

Improving Simulation Efficiency of MCMC for Inverse Modeling of Hydrologic Systems with a Kalman-Inspired Proposal Distribution

Bayesian analysis is widely used in science and engineering for real-time forecasting, decision making, and to help unravel the processes that explain the observed data. These data are some deterministic and/or stochastic transformations of the underlying parameters. A key task is then to summarize the posterior distribution of these parameters. When models become too difficult to analyze analytically, Monte Carlo methods can be used to approximate the target distribution. Of these, Markov chain Monte Carlo (MCMC) methods are particularly powerful. Such methods generate a random walk through the parameter space and, under strict conditions of reversibility and ergodicity, will successively visit solutions with frequency proportional to the underlying target density. This requires a proposal distribution that generates candidate solutions starting from an arbitrary initial state. The speed of the sampled chains converging to the target distribution deteriorates rapidly, however, with increasing parameter dimensionality. In this paper, we introduce a new proposal distribution that enhances significantly the efficiency of MCMC simulation for highly parameterized models. This proposal distribution exploits the cross-covariance of model parameters, measurements and model outputs, and generates candidate states much alike the analysis step in the Kalman filter. We embed the Kalman-inspired proposal distribution in the DREAM algorithm during burn-in, and present several numerical experiments with complex, high-dimensional or multi-modal target distributions. Results demonstrate that this new proposal distribution can greatly improve simulation efficiency of MCMC. Specifically, we observe a speed-up on the order of 10-30 times for groundwater models with more than one-hundred parameters

Investigating prior parameter distributions in the inverse modelling of water distribution hydraulic models

PublishedJournal Article© 2014 Journal of Mechanical Engineering. All rights reserved. Inverse modelling concentrates on estimating water distribution system (WDS) model parameters that are not directly measurable, e.g. pipe roughness coefficients, which can, therefore, only be estimated by indirect approaches, i.e. inverse modelling. Estimation of the parameter and predictive uncertainty of WDS models is an essential part of the inverse modelling process. Recently, Markov Chain Monte Carlo (MCMC) simulations have gained in popularity in uncertainty analyses due to their effective and efficient exploration of posterior parameter probability density functions (pdf). A Bayesian framework is used to infer prior parameter information via a likelihood function to plausible ranges of posterior parameter pdf. Improved parameter and predictive uncertainty are achieved through the incorporation of prior pdf of parameter values and the use of a generalized likelihood function. We used three prior information sampling schemes to infer the pipe roughness coefficients of WDS models. A hypothetical case study and a real-world WDS case study were used to illustrate the strengths and weaknesses of a particular selection of a prior information pdf. The results obtained show that the level of parameter identifiability (i.e. sensitivity) is an important property for prior pdf selection.We are obliged to Jasper A. Vrugt and Cajo ter Braak for providing the code of the DREAM(ZS) algorithm and graphical post-processing software

A Bayesian multi-region radial composite reservoir model for deconvolution in well test analysis

In petroleum well test analysis, deconvolution is used to obtain information about the reservoir system. This information is contained in the response function, which can be estimated by solving an inverse problem in the pressure and flow rate measurements. Our Bayesian approach to this problem is based upon a parametric physical model of reservoir behaviour, derived from the solution for fluid flow in a general class of reservoirs. This permits joint parametric Bayesian inference for both the reservoir parameters and the true pressure and rate values, which is essential due to the typical levels of observation error. Using a set of flexible priors for the reservoir parameters to restrict the solution space to physical behaviours, samples from the posterior are generated using MCMC. Summaries and visualisations of the reservoir parameters' posterior, response, and true pressure and rate values can be produced, interpreted, and model selection can be performed. The method is validated through a synthetic application, and applied to a field data set. The results are comparable to the state of the art solution, but through our method we gain access to system parameters, we can incorporate prior knowledge that excludes non-physical results, and we can quantify parameter uncertainty

Multivariate Copula Analysis Toolbox (MvCAT): Describing Dependence and Underlying Uncertainty Using a Bayesian Framework

We present a newly developed Multivariate Copula Analysis Toolbox (MvCAT) which includes a wide range of copula families with different levels of complexity. MvCAT employs a Bayesian framework with a residual-based Gaussian likelihood function for inferring copula parameters and estimating the underlying uncertainties. The contribution of this paper is threefold: (a) providing a Bayesian framework to approximate the predictive uncertainties of fitted copulas, (b) introducing a hybrid-evolution Markov Chain Monte Carlo (MCMC) approach designed for numerical estimation of the posterior distribution of copula parameters, and (c) enabling the community to explore a wide range of copulas and evaluate them relative to the fitting uncertainties. We show that the commonly used local optimization methods for copula parameter estimation often get trapped in local minima. The proposed method, however, addresses this limitation and improves describing the dependence structure. MvCAT also enables evaluation of uncertainties relative to the length of record, which is fundamental to a wide range of applications such as multivariate frequency analysis