We study the consistency and optimality of the maximum marginal likelihood estimate (MMLE) in the hyperparameter inference for large-degree-of-freedom models. We perform main analyses within the exponential family, where the natural parameters are hyperparameters. First, we prove the consistency of the MMLE for the general linear models when estimating the scales of variance in the likelihood and prior. The proof is independent of the number ratio of data to model parameters and excepts the ill-posedness of the associated regularized least-square model-parameter estimate that is shown asymptotically unbiased. Second, we generalize the proof to other models with a finite number of hyperparameters. We find that the extensive properties of cost functions in the exponential family generally yield the consistency of the MMLE for the likelihood hyperparameters. Besides, we show the MMLE asymptotically almost surely minimizes the Kullback-Leibler divergence between the prior and true predictive distributions even if the true data distribution is outside the model space under the hypothetical asymptotic normality of the predictive distributions applicable to non-exponential model families. Our proof validates the empirical Bayes method using the hyperparameter MMLE in the asymptotics of many model parameters, ensuring the same qualification for the empirical-cross-entropy cross-validation.

Considering the purpose of the session relating early engineering developments in site response and soil-structure interaction, this paper focuses on the development of studies regarding site-city interaction following the striking site response observations obtained in Mexico City during the 1985 Guerrero-Michoacan event, The first part presents an overview of the investigations on multiple structure-soil-structure interaction, starting with Mexico-city like environments with dense urbanization on soft soils, which later evolved with the concept of metamaterials. Up to now, such investigations have been largely relying on numerical simulations in 2D and 3D media, coupling soft surface soil layers and simplified building models, including also some theoretical developments using various mechanical concepts. They also relied on a number of laboratory experiments on reduced-scale mock-ups with diverse vibratory sources (shaking table, acoustic devices). The latest studies coupled full-scale experiments on mechanical analogs such as forests or wind turbine farms involving sets of resonators with similar frequencies, and numerical simulation to investigate their impact on the propagation of surface (Rayleigh) waves. Almost all such studies converge in predicting lower ground motion amplitude for sites located within the ''urbanized'' area, but none of them can be considered a ''groundtruth'' proof for a real earthquake in a real city. The second part thus takes advantage of the long duration of strong motion observations in the Kanto area thanks to the KiK-net, K-NET and JMA (Shin-dokei) networks, to investigate the possible changes in site response with time. The first results obtained with the event-specific site terms derived from Generalized Inversion Techniques (Nakano et al., 2015) indicate a systematic reduction of the low frequency (0.2 -1 Hz) site amplification, in the central-south Tokyo area. As this frequency band corresponds both to the site frequency (very thick deposits) and to the high-rise buildings, the discussion focuses on the possible relation with the extensive construction in some areas of downtown Tokyo over the last 2 decades.