Abstract
Abstract. Formal and informal Bayesian approaches have found widespread implementation and use in environmental modeling to summarize parameter and predictive uncertainty. Successful implementation of these methods relies heavily on the availability of efficient sampling methods that approximate, as closely and consistently as possible the (evolving) posterior target distribution. Much of this work has focused on continuous variables that can take on any value within their prior defined ranges. Here, we introduce theory and concepts of a discrete sampling method that resolves the parameter space at fixed points. This new code, entitled DREAM(D) uses the recently developed DREAM algorithm (Vrugt et al., 2008, 2009a, b) as its main building block but implements two novel proposal distributions to help solve discrete and combinatorial optimization problems. This novel MCMC sampler maintains detailed balance and ergodicity, and is especially designed to resolve the emerging class of optimal experimental design problems. Three different case studies involving a Sudoku puzzle, soil water retention curve, and rainfall – runoff model calibration problem are used to benchmark the performance of DREAM(D). The theory and concepts developed herein can be easily integrated into other (adaptive) MCMC algorithms.
Highlights
Formal and informal Bayesian methods have found widespread application and use to summarize parameter and model predictive uncertainty in hydrologic modeling
Monte Carlo methods are admirably suited to generate samples from the posterior parameter distribution, but generally inefficient when confronted with complex, multimodal, and high-dimensional model-data synthesis problems. This has stimulated the development of Markov Chain Monte Carlo (MCMC) methods that generate a random walk through the search space and iteratively visit solutions with stable frequencies stemming from an invariant probability distribution
In this paper we present a discrete implementation of Differential Evolution Adaptive Metropolis (DREAM), that is especially designed to efficiently retrieve the posterior distribution of noncontinuous and combinatorial search and optimization problems
Summary
Formal and informal Bayesian methods have found widespread application and use to summarize parameter and model predictive uncertainty in hydrologic modeling. This deficiency is frequently caused by an inappropriate selection of q(·) used to generate trial moves in the Markov Chain This inspired Vrugt et al (2008, 2009a,b) to develop a simple adaptive RWM algorithm called Differential Evolution Adaptive Metropolis (DREAM) that runs multiple chains simultaneously for global exploration, and automatically tunes the scale and orientation of the proposal distribution during the evolution to the posterior distribution. This new code, hereafter referred to as DREAM(D) uses DREAM as its main building block, and implements three novel proposal distributions to explicitly recognize differences in topology between discrete and Euclidean search spaces This sampling method maintains detailed balance and ergodicity, and provides explicit information about the posterior uncertainty of the optimal solution.
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