[S-TT44] Seismic Big Data Analysis Based on the State-of-the-Art of Bayesian Statistics

[STT44-P01] A solution to the difficulty of posterior-mean and ABIC computations in fully Bayesian Monte-Carlo inference

*Daisuke Sato¹, Yukitoshi Fukahata¹ (1.Disaster Prevention Research Institute, Kyoto University)

The posterior distribution is constructed from the data-generating distribution and the prior of the model parameters in Bayesian inversions. In fully Bayesian inversions, priors are also introduced for hyperparameters, which specify the weights of the observation equation and the prior of the model parameters, and we evaluate the joint posterior of the model parameters and hyperparameters. The joint posterior is the formal solution in the fully Bayesian inferences that contains all the statistical information of the model parameters and hyperparameters; however, it is a challenging task to extract useful information about the model parameters from the joint posterior. Sato, Fukahata, & Nozue (2022) concluded that joint-posterior sampling requires an exponential number of samples (with respect to the number of model parameters) for generating sample means close enough to the population mean. They also showed that one effective solution is the joint-posterior integral (marginalization) over the model parameters, known as Akaike's Bayesian information criterion (Akaike, 1980), but ABIC is usually not analytically evaluable in nonlinear problems. It is also unrealistic to conduct numerical integration of the joint posterior over the model-parameter space in a brute-force manner.

In this study, we propose a new Monte Carlo technique for fast computing ABIC and posterior means in nonlinear problems, with testing its performance in a slip-inversion application. In the talk, we overview the present method and compare it to the conventional fully Bayesian Monte Carlo inferences. In linear inverse problems, the proposed method remarkably fastens the population-mean computations of the model parameters and hyperparameters. We also confirmed that the joint posterior is a unimodal distribution sharply peaked at the overfitting solution to the prior (an underfitting solution to the data), consistent with the observation of Sato, Fukahata, and Nozue (2022), which explains why the conventional fully Bayesian Monte Carlos often fail to estimate a certain number of model parameters. Furthermore, we benchmark the performance of the new method for nonlinear cases by solving the slip inversion with non-negative slip constraint, where the semianalytic form of ABIC is exceptionally known despite its non-Gaussianity.