Bayesian Neural Tree Models for Nonparametric Regression

Frequentist and Bayesian methods differ in many aspects, but share some basic optimal properties. In real-life classification and regression problems, situations exist in which a model based on one of the methods is preferable based on some subjective criterion. Nonparametric classification and regression techniques, such as decision trees and neural networks, have frequentist (classification and regression trees (CART) and artificial neural networks) as well as Bayesian (Bayesian CART and Bayesian neural networks) approaches to learning from data. In this work, we present two hybrid models combining the Bayesian and frequentist versions of CART and neural networks, which we call the Bayesian neural tree (BNT) models. Both models exploit the architecture of decision trees and have lesser number of parameters to tune than advanced neural networks. Such models can simultaneously perform feature selection and prediction, are highly flexible, and generalize well in settings with a limited number of training observations. We study the consistency of the proposed models, and derive the optimal value of an important model parameter. We also provide illustrative examples using a wide variety of real-life regression data sets

Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes

In statistical practice, whether a Bayesian or frequentist approach is used in inference depends not only on the availability of prior information but also on the attitude taken toward partial prior information, with frequentists tending to be more cautious than Bayesians. The proposed framework defines that attitude in terms of a specified amount of caution, thereby enabling data analysis at the level of caution desired and on the basis of any prior information. The caution parameter represents the attitude toward partial prior information in much the same way as a loss function represents the attitude toward risk. When there is very little prior information and nonzero caution, the resulting inferences correspond to those of the candidate confidence intervals and p-values that are most similar to the credible intervals and hypothesis probabilities of the specified Bayesian posterior. On the other hand, in the presence of a known physical distribution of the parameter, inferences are based only on the corresponding physical posterior. In those extremes of either negligible prior information or complete prior information, inferences do not depend on the degree of caution. Partial prior information between those two extremes leads to intermediate inferences that are more frequentistic to the extent that the caution is high and more Bayesian to the extent that the caution is low

Resolving conflicts between statistical methods by probability combination: Application to empirical Bayes analyses of genomic data

In the typical analysis of a data set, a single method is selected for statistical reporting even when equally applicable methods yield very different results. Examples of equally applicable methods can correspond to those of different ancillary statistics in frequentist inference and of different prior distributions in Bayesian inference. More broadly, choices are made between parametric and nonparametric methods and between frequentist and Bayesian methods. Rather than choosing a single method, it can be safer, in a game-theoretic sense, to combine those that are equally appropriate in light of the available information. Since methods of combining subjectively assessed probability distributions are not objective enough for that purpose, this paper introduces a method of distribution combination that does not require any assignment of distribution weights. It does so by formalizing a hedging strategy in terms of a game between three players: nature, a statistician combining distributions, and a statistician refusing to combine distributions. The optimal move of the first statistician reduces to the solution of a simpler problem of selecting an estimating distribution that minimizes the Kullback-Leibler loss maximized over the plausible distributions to be combined. The resulting combined distribution is a linear combination of the most extreme of the distributions to be combined that are scientifically plausible. The optimal weights are close enough to each other that no extreme distribution dominates the others. The new methodology is illustrated by combining conflicting empirical Bayes methodologies in the context of gene expression data analysis

DIAMONDS: a new Bayesian Nested Sampling tool. Application to Peak Bagging of solar-like oscillations

To exploit the full potential of Kepler light curves, sophisticated and robust analysis tools are now required more than ever. Characterizing single stars with an unprecedented level of accuracy and subsequently analyzing stellar populations in detail are fundamental to further constrain stellar structure and evolutionary models. We developed a new code, termed Diamonds, for Bayesian parameter estimation and model comparison by means of the nested sampling Monte Carlo (NSMC) algorithm, an efficient and powerful method very suitable for high-dimensional and multi-modal problems. A detailed description of the features implemented in the code is given with a focus on the novelties and differences with respect to other existing methods based on NSMC. Diamonds is then tested on the bright F8 V star KIC~9139163, a challenging target for peak-bagging analysis due to its large number of oscillation peaks observed, which are coupled to the blending that occurs between $\ell=2,0$ peaks, and the strong stellar background signal. We further strain the performance of the approach by adopting a 1147.5 days-long Kepler light curve. The Diamonds code is able to provide robust results for the peak-bagging analysis of KIC~9139163. We test the detection of different astrophysical backgrounds in the star and provide a criterion based on the Bayesian evidence for assessing the peak significance of the detected oscillations in detail. We present results for 59 individual oscillation frequencies, amplitudes and linewidths and provide a detailed comparison to the existing values in the literature. Lastly, we successfully demonstrate an innovative approach to peak bagging that exploits the capability of Diamonds to sample multi-modal distributions, which is of great potential for possible future automatization of the analysis technique.Comment: 22 pages, 14 figures, 3 tables. Accepted for publication in A&