Abstract
The heuristic identification of peaks from noisy complex spectra often leads to misunderstanding of the physical and chemical properties of matter. In this paper, we propose a framework based on Bayesian inference, which enables us to separate multipeak spectra into single peaks statistically and consists of two steps. The first step is estimating both the noise variance and the number of peaks as hyperparameters based on Bayes free energy, which generally is not analytically tractable. The second step is fitting the parameters of each peak function to the given spectrum by calculating the posterior density, which has a problem of local minima and saddles since multipeak models are nonlinear and hierarchical. Our framework enables the escape from local minima or saddles by using the exchange Monte Carlo method and calculates Bayes free energy via the multiple histogram method. We discuss a simulation demonstrating how efficient our framework is and show that estimating both the noise variance and the number of peaks prevents overfitting, overpenalizing, and misunderstanding the precision of parameter estimation.
Highlights
Spectroscopy is at the heart of all sciences concerned with matter and energy
We provide a straightforward and efficient scheme that calculates this bivariate function by using the exchange Monte Carlo method and the multiple histogram method.19,20) We demonstrated our framework through simulation
We constructed a framework that enables the dual estimation of the noise variance and the number of peaks and demonstrated the effectiveness of our framework through simulation
Summary
Spectroscopy is at the heart of all sciences concerned with matter and energy. An electromagnetic spectrum indicates the electronic states and the kinetics of atoms. We constructed a Bayesian framework for estimating both the noise variance and the number of peaks from spectra with white Gaussian noise by expanding the previous framework by Nagata et al.6) The noise variance and the number of peaks are respectively estimated by hyperparameter optimization and model selection These estimations are carried out by maximizing a function called the marginal likelihood,16–18) which is a conditional probability of observed data given the noise variance and the number of peaks in our framework. These prior density models can be replaced with any other model without loss of generality in our framework
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