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Register a new userUmbrella sampling: a powerful method to sample tails of distributions
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We present the umbrella sampling (US) technique and show that it can be used to sample extremely low probability areas of the posterior distribution that may be required in statistical analyses of data. In this approach sampling of the target likelihood is split into sampling of multiple biased likelihoods confined within individual umbrella windows. We show that the US algorithm is efficient and highly parallel and that it can be easily used with other existing MCMC samplers. The method allows the user to capitalize on their intuition and define umbrella windows and increase sampling accuracy along specific directions in the parameter space. Alternatively, one can define umbrella windows using an approach similar to parallel tempering. We provide a public code that implements umbrella sampling as a standalone python package. We present a number of tests illustrating the power of the US method in sampling low probability areas of the posterior and show that this ability allows a considerably more robust sampling of multi-modal distributions compared to the standard sampling methods. We also present an application of the method in a real world example of deriving cosmological constraints using the supernova type Ia data. We show that umbrella sampling can sample the posterior accurately down to the $approx 15sigma$ credible region in the $Omega_{rm m}-Omega_Lambda$ plane, while for the same computational work the affine-invariant MCMC sampling implemented in the {tt emcee} code samples the posterior reliably only to $approx 3sigma$.
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The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed to efficiently sample a multimodal probability measure. We show that this method can be seen as a variant of the well-known Wang-Landau algorithm. Adapting results on the convergence of the Wang-Landau algorithm, we prove the convergence of the SHUS algorithm. We also compare the two methods in terms of efficiency. We finally propose a modification of the SHUS algorithm in order to increase its efficiency, and exhibit some similarities of SHUS with the well-tempered metadynamics method.
In performing a Bayesian analysis, two difficult problems often emerge. First, in estimating the parameters of some model for the data, the resulting posterior distribution may be multi-modal or exhibit pronounced (curving) degeneracies. Secondly, in selecting between a set of competing models, calculation of the Bayesian evidence for each model is computationally expensive using existing methods such as thermodynamic integration. Nested Sampling is a Monte Carlo method targeted at the efficient calculation of the evidence, but also produces posterior inferences as a by-product and therefore provides means to carry out parameter estimation as well as model selection. The main challenge in implementing Nested Sampling is to sample from a constrained probability distribution. One possible solution to this problem is provided by the Galilean Monte Carlo (GMC) algorithm. We show results of applying Nested Sampling with GMC to some problems which have proven very difficult for standard Markov Chain Monte Carlo (MCMC) and down-hill methods, due to the presence of large number of local minima and/or pronounced (curving) degeneracies between the parameters. We also discuss the use of Nested Sampling with GMC in Bayesian object detection problems, which are inherently multi-modal and require the evaluation of Bayesian evidence for distinguishing between true and spurious detections.
The speed of many one-line transformation methods for the production of, for example, Levy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, for the class of decreasing probability densities fast rejection implementations like the Ziggurat by Marsaglia and Tsang promise a significant speed-up if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transform maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.
We describe a simple method of umbrella trajectory sampling for Markov chains. The method allows the estimation of large-deviation rate functions, for path-extensive dynamic observables, for an arbitrary number of models within a certain family. The general relationship between probability distributions of dynamic observables of members of this family is an extended fluctuation relation. When the dynamic observable is chosen to be entropy production, members of this family include the forward Markov chain and its time reverse, whose probability distributions are related by the expected simple fluctuation relation.
The goal of Project GAUSS is to return samples from the dwarf planet Ceres. Ceres is the most accessible ocean world candidate and the largest reservoir of water in the inner solar system. It shows active cryovolcanism and hydrothermal activities in recent history that resulted in minerals not found in any other planets to date except for Earths upper crust. The possible occurrence of recent subsurface ocean on Ceres and the complex geochemistry suggest possible past habitability and even the potential for ongoing habitability. Aiming to answer a broad spectrum of questions about the origin and evolution of Ceres and its potential habitability, GAUSS will return samples from this possible ocean world for the first time. The project will address the following top-level scientific questions: 1) What is the origin of Ceres and the origin and transfer of water and other volatiles in the inner solar system? 2) What are the physical properties and internal structure of Ceres? What do they tell us about the evolutionary and aqueous alteration history of icy dwarf planets? 3) What are the astrobiological implications of Ceres? Was it habitable in the past and is it still today? 4) What are the mineralogical connections between Ceres and our current collections of primitive meteorites? GAUSS will first perform a high-resolution global remote sensing investigation, characterizing the geophysical and geochemical properties of Ceres. Candidate sampling sites will then be identified, and observation campaigns will be run for an in-depth assessment of the candidate sites. Once the sampling site is selected, a lander will be deployed on the surface to collect samples and return them to Earth in cryogenic conditions that preserves the volatile and organic composition as well as the original physical status as much as possible.
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