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We consider estimation of generalized additive models using basis expansions with Bayesian model selection. Although Bayesian model selection is an intuitively appealing tool for regression splines by virtue of the flexible knot placement and model-averaged function estimates, its use has traditionally been limited to Gaussian additive regression, as posterior search of the model space requires a tractable form of the marginal model likelihood. We introduce an extension of the method to the exponential family of distributions using the Laplace approximation to the likelihood. Although the Laplace approximation is successful with all Gaussian-type prior distributions in providing a closed-form expression of the marginal likelihood, there is no broad consensus on the best prior distribution to be used for nonparametric regression via model selection. We observe that the classical unit information prior distribution for variable selection may not be suitable for nonparametric regression using basis expansions. Instead, our study reveals that mixtures of g-priors are more suitable. A large family of mixtures of g-priors is considered for a detailed examination of how various mixture priors perform in estimating generalized additive models. Furthermore, we compare several priors of knots for model selection-based spline approaches to determine the most practically effective scheme. The model selection-based estimation methods are also compared with other Bayesian approaches to function estimation. Extensive simulation studies demonstrate the validity of the model selection-based approaches. We provide an R package for the proposed method.

In this paper, we study the identifiability and the estimation of the parameters of a copula-based multivariate model when the margins are unknown and are arbitrary, meaning that they can be continuous, discrete, or mixtures of continuous and discrete. When at least one margin is not continuous, the range of values determining the copula is not the entire unit square and this situation could lead to identifiability issues that are discussed here. Next, we propose estimation methods when the margins are unknown and arbitrary, using pseudo log-likelihood adapted to the case of discontinuities. In view of applications to large data sets, we also propose a pairwise composite pseudo log-likelihood. These methodologies can also be easily modified to cover the case of parametric margins. One of the main theoretical result is an extension to arbitrary distributions of known convergence results of rank-based statistics when the margins are continuous. As a by-product, under smoothness assumptions, we obtain that the asymptotic distribution of the estimation errors of our estimators are Gaussian. Finally, numerical experiments are presented to assess the finite sample performance of the estimators, and the usefulness of the proposed methodologies is illustrated with a copula-based regression model for hydrological data.

The analysis of data stored in multiple sites has become more popular, raising new concerns about the security of data storage and communication. Federated learning, which does not require centralizing data, is a common approach to preventing heavy data transportation, securing valued data, and protecting personal information protection. Therefore, determining how to aggregate the information obtained from the analysis of data in separate local sites has become an important statistical issue. The commonly used averaging methods may not be suitable due to data nonhomogeneity and incomparable results among individual sites, and applying them may result in the loss of information obtained from the individual analyses. Using a sequential method in federated learning with distributed computing can facilitate the integration and accelerate the analysis process. We develop a data-driven method for efficiently and effectively aggregating valued information by analyzing local data without encountering potential issues such as information security and heavy transportation due to data communication. In addition, the proposed method can preserve the properties of classical sequential adaptive design, such as data-driven sample size and estimation precision when applied to generalized linear models. We use numerical studies of simulated data and an application to COVID-19 data collected from 32 hospitals in Mexico, to illustrate the proposed method.

This paper considers the problem of estimating the distribution of a response variable conditioned on observing some factors. Existing approaches are often deficient in one of the qualities of flexibility, interpretability and tractability. We propose a model that possesses these desirable properties. The proposed model, analogous to classic mixture regression models, models the conditional quantile function as a mixture (weighted sum) of basis quantile functions, with the weight of each basis quantile function being a function of the factors. The model can approximate any bounded conditional quantile model. It has a factor model structure with a closed-form expression. The calibration problem is formulated as convex optimization, which can be viewed as conducting quantile regressions of all confidence levels simultaneously and does not suffer from quantile crossing by design. The calibration is equivalent to minimization of Continuous Probability Ranked Score (CRPS). We prove the asymptotic normality of the estimator. Additionally, based on risk quadrangle framework, we generalize the proposed approach to conditional distributions defined by Conditional Value-at-Risk (CVaR), expectile and other functions of uncertainty measures. Based on CP decomposition of tensors, we propose a dimensionality reduction method by reducing the rank of the parameter tensor and propose an alternating algorithm for estimating the parameter tensor. Our numerical experiments demonstrate the efficiency of the approach.

Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.

Probabilistic graphical models (PGMs) provide a compact and flexible framework to model very complex real-life phenomena. They combine the probability theory which deals with uncertainty and logical structure represented by a graph which allows one to cope with the computational complexity and also interpret and communicate the obtained knowledge. In the thesis, we consider two different types of PGMs: Bayesian networks (BNs) which are static, and continuous time Bayesian networks which, as the name suggests, have a temporal component. We are interested in recovering their true structure, which is the first step in learning any PGM. This is a challenging task, which is interesting in itself from the causal point of view, for the purposes of interpretation of the model and the decision-making process. All approaches for structure learning in the thesis are united by the same idea of maximum likelihood estimation with the LASSO penalty. The problem of structure learning is reduced to the problem of finding non-zero coefficients in the LASSO estimator for a generalized linear model. In the case of CTBNs, we consider the problem both for complete and incomplete data. We support the theoretical results with experiments.

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging.

In the usual Bayesian setting, a full probabilistic model is required to link the data and parameters, and the form of this model and the inference and prediction mechanisms are specified via de Finetti's representation. In general, such a formulation is not robust to model mis-specification of its component parts. An alternative approach is to draw inference based on loss functions, where the quantity of interest is defined as a minimizer of some expected loss, and to construct posterior distributions based on the loss-based formulation; this strategy underpins the construction of the Gibbs posterior. We develop a Bayesian non-parametric approach; specifically, we generalize the Bayesian bootstrap, and specify a Dirichlet process model for the distribution of the observables. We implement this using direct prior-to-posterior calculations, but also using predictive sampling. We also study the assessment of posterior validity for non-standard Bayesian calculations. We provide a computationally efficient way to calibrate the scaling parameter in the Gibbs posterior so that it can achieve the desired coverage rate. We show that the developed non-standard Bayesian updating procedures yield valid posterior distributions in terms of consistency and asymptotic normality under model mis-specification. Simulation studies show that the proposed methods can recover the true value of the parameter efficiently and achieve frequentist coverage even when the sample size is small. Finally, we apply our methods to evaluate the causal impact of speed cameras on traffic collisions in England.

High-dimensional feature selection is a central problem in a variety of application domains such as machine learning, image analysis, and genomics. In this paper, we propose graph-based tests as a useful basis for feature selection. We describe an algorithm for selecting informative features in high-dimensional data, where each observation comes from one of $K$ different distributions. Our algorithm can be applied in a completely nonparametric setup without any distributional assumptions on the data, and it aims at outputting those features in the data, that contribute the most to the overall distributional variation. At the heart of our method is the recursive application of distribution-free graph-based tests on subsets of the feature set, located at different depths of a hierarchical clustering tree constructed from the data. Our algorithm recovers all truly contributing features with high probability, while ensuring optimal control on false-discovery. Finally, we show the superior performance of our method over other existing ones through synthetic data, and also demonstrate the utility of the method on two real-life datasets from the domains of climate change and single cell transcriptomics.

Brain tumors are one of the life-threatening forms of cancer. Previous studies have classified brain tumors using deep neural networks. In this paper, we perform the later task using a collaborative deep learning technique, more specifically split learning. Split learning allows collaborative learning via neural networks splitting into two (or more) parts, a client-side network and a server-side network. The client-side is trained to a certain layer called the cut layer. Then, the rest of the training is resumed on the server-side network. Vertical distribution, a method for distributing data among organizations, was implemented where several hospitals hold different attributes of information for the same set of patients. To the best of our knowledge this paper will be the first paper to implement both split learning and vertical distribution for brain tumor classification. Using both techniques, we were able to achieve train and test accuracy greater than 90\% and 70\%, respectively.