Connecting PM and MAP in Bayesian spectral deconvolution by extending exchange Monte Carlo method and using multiple data sets

Abstract

Nagata et al. proposed a parameter estimation method using Markov chain Monte Carlo (MCMC) for the spectral deconvolution of observed data. However, a systematic error occurs when the parameters to be estimated are close. In this paper, we first clarify that the exchange symmetry of parameters, which is essentially included in the spectral deconvolution problem, causes the systematic error. In particular, we show that estimation from a single data set is inherently difficult because the posterior distribution becomes unimodal or multimodal depending on the data set when the parameters to be estimated are close. Second, we alleviate the problem to the case of using multiple data sets and propose an extension of the exchange Monte Carlo method to low temperatures. This extension corresponds to bridging the gap between posterior mean (PM) estimation and maximum a posteriori (MAP) estimation. The above alleviation and bridging achieve a good estimation even when the parameters are close.