Learning coefficients for hierarchical learning models in Bayesian Estimation

Recently, singular learning theory has been analyzed using algebraic geometry as its basis. It is essential to determine the normal crossing divisors of learning machine singularities through a blowing-up process to observe the behaviors of state probability functions in learning theory. In this paper, we investigate learning coefficients for multi-layered neural networks with linear units, especially when dealing with a large number of layers in Bayesian estimation. We make use of the valuable results obtained by Aoyagi(2023), which provide the main terms for Bayesian generalization error and the average stochastic complexity (free energy). These terms are widely employed in numerical experiments, such as in information criteria

Statistical Learning Theory of Quasi-Regular Cases

Many learning machines such as normal mixtures and layered neural networks are not regular but singular statistical models, because the map from a parameter to a probability distribution is not one-to-one. The conventional statistical asymptotic theory can not be applied to such learning machines because the likelihood function can not be approximated by any normal distribution. Recently, new statistical theory has been established based on algebraic geometry and it was clarified that the generalization and training errors are determined by two birational invariants, the real log canonical threshold and the singular fluctuation. However, their concrete values are left unknown. In the present paper, we propose a new concept, a quasi-regular case in statistical learning theory. A quasi-regular case is not a regular case but a singular case, however, it has the same property as a regular case. In fact, we prove that, in a quasi-regular case, two birational invariants are equal to each other, resulting that the symmetry of the generalization and training errors holds. Moreover, the concrete values of two birational invariants are explicitly obtained, the quasi-regular case is useful to study statistical learning theory

Estimating LOCP cancer mortality rates in small domains in Spain using its relationship with lung cancer

The distribution of lip, oral cavity, and pharynx (LOCP) cancer mortality rates in small domains (defined as the combination of province, age group, and gender) remains unknown in Spain. As many of the LOCP risk factors are preventable, specific prevention programmes could be implemented but this requires a clear specification of the target population. This paper provides an in-depth description of LOCP mortality rates by province, age group and gender, giving a complete overview of the disease. This study also presents a methodological challenge. As the number of LOCP cancer cases in small domains (province, age groups and gender) is scarce, univariate spatial models do not provide reliable results or are even impossible to fit. In view of the close link between LOCP and lung cancer, we consider analyzing them jointly by using shared component models. These models allow information-borrowing among diseases, ultimately providing the analysis of cancer sites with few cases at a very disaggregated level. Results show that males have higher mortality rates than females and these rates increase with age. Regions located in the north of Spain show the highest LOCP cancer mortality rates.The work was supported by Project MTM2017-82553-R (AEI, UE), Project PID2020-113125RB-I00/MCIN/ AEI/10.13039/501100011033 and Proyecto Jóvenes Investigadores PJUPNA2018-11

Posterior Covariance Information Criterion

We introduce an information criterion, PCIC, for predictive evaluation based on quasi-posterior distributions. It is regarded as a natural generalisation of the widely applicable information criterion (WAIC) and can be computed via a single Markov chain Monte Carlo run. PCIC is useful in a variety of predictive settings that are not well dealt with in WAIC, including weighted likelihood inference and quasi-Bayesian predictio