Markov Chain Confidence Intervals and Biases

We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

Approximations of geometrically ergodic reversible markov chains

A common tool in the practice of Markov chain Monte Carlo (MCMC) is to use approximating transition kernels to speed up computation when the desired kernel is slow to evaluate or is intractable. A limited set of quantitative tools exists to assess the relative accuracy and efficiency of such approximations. We derive a set of tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, $L_2(\pi)$. Our results apply to approximations of reversible chains which are geometrically ergodic, as is typically the case for applications to MCMC. The focus of our work is on determining whether the approximating kernel will preserve the geometric ergodicity of the exact chain, and whether the approximating stationary distribution will be close to the original stationary distribution. For reversible chains, our results extend the results of Johndrow et al. (2015) from the uniformly ergodic case to the geometrically ergodic case, under some additional regularity conditions. We then apply our results to a number of approximate MCMC algorithms.

Quantification of uncertainties in the assessment of an atmospheric release source applied to the autumn 2017 ¹⁰⁶Ru event

Using a Bayesian framework in the inverse problem of estimating the source of an atmospheric release of a pollutant has proven fruitful in recent years. Through Markov chain Monte Carlo (MCMC) algorithms, the statistical distribution of the release parameters such as the location, the duration, and the magnitude as well as error covariances can be sampled so as to get a complete characterisation of the source. In this study, several approaches are described and applied to better quantify these distributions, and therefore to get a better representation of the uncertainties. First, we propose a method based on ensemble forecasting: physical parameters of both the meteorological fields and the transport model are perturbed to create an enhanced ensemble. In order to account for physical model errors, the importance of ensemble members are represented by weights and sampled together with the other variables of the source. Second, once the choice of the statistical likelihood is shown to alter the nuclear source assessment, we suggest several suitable distributions for the errors. Finally, we propose two specific designs of the covariance matrix associated with the observation error. These methods are applied to the source term reconstruction of the 106Ru of unknown origin in Europe in autumn 2017. A posteriori distributions meant to identify the origin of the release, to assess the source term, and to quantify the uncertainties associated with the observations and the model, as well as densities of the weights of the perturbed ensemble, are presented.

Estimating drift and minorization coefficients for Gibbs sampling algorithms

Gibbs samplers are common Markov chain Monte Carlo (MCMC) algorithms that are used to sample from intractable probability distributions when sampling directly from full conditional distributions is possible. These types of MCMC algorithms come up frequently in many applications, and because of their popularity it is important to have a sense of how long it takes for the Gibbs sampler to become close to its stationary distribution. To this end, it is common to rely on the values of drift and minorization coefficients to bound the mixing time of the Gibbs sampler. This manuscript provides a computational method for estimating these coefficients. Herein, we detail the several advantages of the proposed methods, as well as the limitations of this approach. These limitations are primarily related to the “curse of dimensionality”, which for these methods is caused by necessary increases in the numbers of initial states from which chains need be run and the need for an exponentially increasing number of grid points for estimation of minorization coefficients.

Cosmological Parameter Inference with Bayesian Statistics

Bayesian statistics and Markov Chain Monte Carlo (MCMC) algorithms have found their place in the field of Cosmology. They have become important mathematical and numerical tools, especially in parameter estimation and model comparison. In this paper, we review some fundamental concepts to understand Bayesian statistics and then introduce MCMC algorithms and samplers that allow us to perform the parameter inference procedure. We also introduce a general description of the standard cosmological model, known as the ΛCDM model, along with several alternatives, and current datasets coming from astrophysical and cosmological observations. Finally, with the tools acquired, we use an MCMC algorithm implemented in python to test several cosmological models and find out the combination of parameters that best describes the Universe.

Quantification of the modelling uncertainties in atmospheric release source assessment and application to the reconstruction of the autumn 2017 Ruthenium 106 source

Using a Bayesian framework in the inverse problem of estimating the source of an atmospheric release of a pollutant has proven fruitful in recent years. Through Markov chain Monte Carlo (MCMC) algorithms, the statistical distribution of the release parameters such as the location, the duration, and the magnitude as well as the likelihood covariances can be sampled so as to get a complete characterisation of the source. In this study, several approaches are described and applied to improve on these distributions, and therefore to get a better representation of the uncertainties. First, a method based on ensemble forecasting is proposed: physical parameters of both the meteorological fields and the transport model are perturbed to create an enhanced ensemble. In order to account for model errors, the importance of ensemble members are represented by weights and sampled together with the other variables of the source. Secondly, the choice of the statistical likelihood is shown to alter the nuclear source assessment, and several suited distributions for the errors are advised. Finally, two advanced designs of the covariance matrix associated to the observation error are proposed. These methods are applied to the case of the detection of Ruthenium 106 of unknown origin in Europe in autumn 2017. A posteriori distributions meant to identify the origin of the release, to assess the source term, to quantify the uncertainties associated to the observations and the model, as well as densities of the weights of the perturbed ensemble, are presented.