The Variational Bayesian Approach to Fitting Mixture Models to Circular Wave Direction Data

Abstract

The emerging variational Bayesian (VB) technique for approximate Bayesian statistical inference is a nonsimulation-based and time-efficient approach. It provides a useful, practical alternative to other Bayesian statistical approaches such as Markov chain Monte Carlo-based techniques, particularly for applications involving large datasets. This article reviews the increasingly popular VB statistical approach and illustrates how it can be used to fit Gaussian mixture models to circular wave direction data. This is done by taking the straightforward approach of padding the data; this method involves adding a repeat of a complete cycle of the data to the existing dataset to obtain a dataset on the real line. The padded dataset can then be analyzed using the standard VB technique. This results in a practical, efficient approach that is also appropriate for modeling other types of circular, or directional, data such as wind direction.

Efficient Interrogation of the Kinetic Barriers Demarcating Catalytic States of a Tyrosine Kinase with Optimal Physical Descriptors and Mixture Models

“…For periodic CVs, we use EM algorithm to fit von‐Mises Fisher (vMF) mixtures in lieu of Gaussians because of their better performance shown in earlier works [101,102] . In this case, an optimal number of vMF functions is obtained using the minimum Bayesian Information Criterion (BIC) score [103,104] owing to the intractability of analytical solutions while using variational approach with vMF mixtures [105–107] . For a model M the BIC is defined as $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr BIC_M = - 2{\rm{ln}}\left( {{\rm{{\cal L}}}_M } \right) + k{\rm{ln}}\left( N \right)\hfill\cr}}$$$ …”

“…[101,102] In this case, an optimal number of vMF functions is obtained using the minimum Bayesian Information Criterion (BIC) score [103,104] owing to the intractability of analytical solutions while using variational approach with vMF mixtures. [105][106][107] For a model M the BIC is defined as…”

Simulated Tempering Markov Chain Monte Carlo for Full Waveform Analysis

“…The purpose of the full waveform analysis is to solve the parameters with y , which can be seen as the estimate of the prior information according to the posterior information. During the processing, the update is not always right, a probability density function of the posterior distribution [7] can be used to assess the parameters after each update. i y can be considered as a sample of Poisson distribution with intensity   i F , and each i y is independent.…”