Bayesian learning of Gaussian mixture model for calculating debris flow exceedance probability

Probabilistic modelling of debris flow data provides useful information for quantitative risk assessment, such as exceedance probabilities (EPs) of debris flow quantities. This task can defy many classical statistical models because debris flow data are frequently collected over years or even decades, and the nonuniformity of the nature and the complex physical mechanism of debris flows lead to multimodal distribution characteristics of observational data. This paper proposes a Bayesian framework for learning Gaussian mixture model (GMM) of debris flow quantities (e.g. total discharge Qtotal and maximum impact pressure Pmax) and calculating their EPs for risk-informed decision making. GMM provides great flexibility to fit observation data, but are intrinsically unidentifiable due to the label switching. These computational difficulties are addressed using Random Gibbs Sampling and Bridge Sampling in the proposed framework, allowing incorporating the statistical uncertainty in GMM parameters into EP estimation. Equations are derived for the proposed approach and are illustrated using Qtotal and Pmax data at Jiangjia Ravine, China. Results show that the proposed approach identifies a bivariate GMM of Qtotal and Pmax reflecting the multimodal characteristics of the observed data and quantifies the statistical uncertainty of GMM parameters. Incorporating the statistical uncertainty into EP estimation provides robust estimates.