Information gains from Monte Carlo Markov Chains
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Information gains from Monte Carlo Markov Chains Ahmad Mehrabia , A. Ahmadi Department of Physics, Bu-Ali Sina University, Hamedan, Iran Received: 2 October 2019 / Accepted: 7 April 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we present a novel method to compute the relative entropy as well as the expected relative entropy using an MCMC chain. The relative entropy from information theory can be used to quantify differences in posterior distributions of a pair of experiments. In cosmology, the relative entropy has been proposed as an interesting tool for model selection, experiment design, forecasting and measuring information gain from subsequent experiments. In contrast to Gaussian distributions, these quantities are not available analytically and one needs to use numerical methods to estimate them which are computationally very expensive. We propose a method and provide its python package to estimate the relative entropy as well as expected relative entropy from an MCMC sample. We consider the linear Gaussian model to check the accuracy of our code. Our results indicate that the relative error is below 0.2% for sample size larger than 105 in the linear Gaussian model. In addition, we study the robustness of our code in estimating the expected relative entropy in the Gaussian case.
1 Introduction In contrast to a few decades ago, there are a large number of probes in cosmology, which provide us remarkable information about content and evolution of the Universe. These datasets have been extensively used to study and constrain free model parameters in literature (see [1– 4]and references therein). Bayesian inference provides a common and widely used method to constrain free model parameters. In this case, a prior probability density in parameter space is updated to obtain the posterior distribution using an observational data. Since an analytic solution in the Bayesian inference is very limited, one has to develop a numerical method to find the posterior. Among these, the Monte Carlo Markov Chain (MCMC) techniques are widely accepted and used in different problems. The purpose of an MCMC algorithm is to construct a sample of points in parameter space which is called a chain and then obtain posterior probability density from it. The simplest and widely used MCMC algorithm is Metropolis-Hasting [5], but considering different situations other algorithms like Gibbs sampling [6,7] and Hamilton Monte-Carlo [8] have been used to obtain the posterior distributions. To quantify the difference between probability distributions from different surveys, a robust framework is needed. Initially motivated from information theory, the relative entropy or Kullback–Leibler divergence has been proposed to measure differences in two probability densities [9] . In
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addition, this method has been used for experiment design and fore
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