[S-TT40] Seismic Big Data Analysis Based on the State-of-the-Art of Bayesian Statistics

[STT40-P08] Problems in the reduction of the posterior distribution reduction in fully Bayesian inversion and their geophysical implication

*Daisuke Sato¹, Yukitoshi Fukahata¹, Yohei Nozue² (1.Disaster Prevention Research Institute, Kyoto University, 2.Graduate School of Science, Kyoto University)

Keywords:Inversion analysis, Fully Bayesian inference, Geophysical applications

In Bayesian inversions, the posterior distribution is constructed from the data generating distribution and the prior distribution of the model parameters. In fully Bayesian inversions, prior distributions are also introduced for hyperparameters, which represent the narrowness of these distribution functions, and we evaluate the joint posterior distributions of the model parameters and hyperparameters. The fully Bayesian inversion has the advantage in objective probabilistic evaluation of both the model parameters and hyperparameters (Fukuda and Johnson, 2008), whereas its formal solution, the joint posterior is difficult to interpret, and extracting useful information about the model parameters from the joint posterior (reduction of the posterior) earns essential importance in the result interpretation (Matsu'ura, 1990). For instance, Sato and Fukahata (2019; 2020, SSJ) reported that the joint posterior can become pathological as the number of model parameters increases for the fixed number of data; specifically, the global probability maximum of the joint posterior outputs an overly smoothed solution, and the appropriate estimate is no longer close to either the global maximum or the local maximum (then having almost zero probability in the joint posterior).

In this study, we apply the above results to crustal deformation data to demonstrate how the reduction of the posterior in the fully Bayesian inversions affects the result interpretation in the inversion analysis. Focusing on the marginalization of the joint posterior, Sato and Fukahata (2021) classify the reduction into the following three: (1) the joint posterior of the model parameters and hyperparameters, maximized with respect to the hyperparameters (profile likelihood method; Murphy and Van der Vaart 2000), (2) the marginal posterior of the model parameters, obtained by integrating the joint posterior with respect to the hyperparameters (marginal likelihood method; Carlin and Louis 2008; Fukuda and Johnson, 2008), and (3) the marginal posterior of the hyperparameters, obtained by integrating the joint posterior with respect to the model parameters (the Akaike Bayesian information criterion; Akaike, 1980; Yabuki and Matsu'ura, 1992). In the linear inverse problem, as the number of model parameters increases, the solutions that emerge with finite probability from these distribution functions are represented by the two solutions: (a; the MAP estimate) the probability maximum of the joint posterior and (b; the ABIC estimate) the probability maximum of the joint posterior when using the probability maximum of the marginal posterior of the hyperparameters. In the presentation, we address the problem of estimating the displacement velocity field from spatially discrete GNSS data with the Laplacian smoothing constraint. We set the joint posterior using the basis function expansion methods (e.g. Okazaki et al., 2021) and examine how the MAP and ABIC estimates vary with the number of basis functions (the number of model parameters). This comparison will show that as we increase the number of basis functions, the ABIC estimates exhibit a more detailed structure, with the plateau of the resolution indicating the resolution upper bounds specified by the data, while the MAP solution generates an excessively smooth field. High-resolution estimation, which aims to make the best use of the observed data by using a large number of basis functions, requires special attention to the reduction.