Learning Bayesian Networks: A Copula Approach for Mixed-Type Data.

Estimating dependence relationships between variables is a crucial issue in many applied domains and in particular psychology. When several variables are entertained, these can be organized into a network which encodes their set of conditional dependence relations. Typically however, the underlying network structure is completely unknown or can be partially drawn only; accordingly it should be learned from the available data, a process known as structure learning. In addition, data arising from social and psychological studies are often of different types, as they can include categorical, discrete and continuous measurements. In this paper, we develop a novel Bayesian methodology for structure learning of directed networks which applies to mixed data, i.e., possibly containing continuous, discrete, ordinal and binary variables simultaneously. Whenever available, our method can easily incorporate known dependence structures among variables represented by paths or edge directions that can be postulated in advance based on the specific problem under consideration. We evaluate the proposed method through extensive simulation studies, with appreciable performances in comparison with current state-of-the-art alternative methods. Finally, we apply our methodology to well-being data from a social survey promoted by the United Nations, and mental health data collected from a cohort of medical students. R code implementing the proposed methodology is available at https://github.com/FedeCastelletti/bayes_networks_mixed_data .

Bayesian graphical modeling for heterogeneous causal effects.

There is a growing interest in current medical research to develop personalized treatments using a molecular-based approach. The broad goal is to implement a more precise and targeted decision-making process, relative to traditional treatments based primarily on clinical diagnoses. Specifically, we consider patients affected by Acute Myeloid Leukemia (AML), an hematological cancer characterized by uncontrolled proliferation of hematopoietic stem cells in the bone marrow. Because AML responds poorly to chemotherapeutic treatments, the development of targeted therapies is essential to improve patients' prospects. In particular, the dataset we analyze contains the levels of proteins involved in cell cycle regulation and linked to the progression of the disease. We evaluate treatment effects within a causal framework represented by a Directed Acyclic Graph (DAG) model, whose vertices are the protein levels in the network. A major obstacle in implementing the above program is represented by individual heterogeneity. We address this issue through a Dirichlet Process (DP) mixture of Gaussian DAG-models where both the graphical structure as well as the allied model parameters are regarded as uncertain. Our procedure determines a clustering structure of the units reflecting the underlying heterogeneity, and produces subject-specific estimates of causal effects based on Bayesian Model Averaging (BMA). With reference to the AML dataset, we identify different effects of protein regulation among individuals; moreover, our method clusters patients into groups that exhibit only mild similarities with traditional categories based on morphological features.

Bayesian inference of causal effects from observational data in Gaussian graphical models.

We assume that multivariate observational data are generated from a distribution whose conditional independencies are encoded in a Directed Acyclic Graph (DAG). For any given DAG, the causal effect of a variable onto another one can be evaluated through intervention calculus. A DAG is typically not identifiable from observational data alone. However, its Markov equivalence class (a collection of DAGs) can be estimated from the data. As a consequence, for the same intervention a set of causal effects, one for each DAG in the equivalence class, can be evaluated. In this paper, we propose a fully Bayesian methodology to make inference on the causal effects of any intervention in the system. Main features of our method are: (a) both uncertainty on the equivalence class and the causal effects are jointly modeled; (b) priors on the parameters of the modified Cholesky decomposition of the precision matrices across all DAG models are constructively assigned starting from a unique prior on the complete (unrestricted) DAG; (c) an efficient algorithm to sample from the posterior distribution on graph space is adopted; (d) an objective Bayes approach, requiring virtually no user specification, is used throughout. We demonstrate the merits of our methodology in simulation studies, wherein comparisons with current state-of-the-art procedures turn out to be highly satisfactory. Finally we examine a real data set of gene expressions for Arabidopsis thaliana.

Bayesian learning of multiple directed networks from observational data.

Graphical modeling represents an established methodology for identifying complex dependencies in biological networks, as exemplified in the study of co-expression, gene regulatory, and protein interaction networks. The available observations often exhibit an intrinsic heterogeneity, which impacts on the network structure through the modification of specific pathways for distinct groups, such as disease subtypes. We propose to infer the resulting multiple graphs jointly in order to benefit from potential similarities across groups; on the other hand our modeling framework is able to accommodate group idiosyncrasies. We consider directed acyclic graphs (DAGs) as network structures, and develop a Bayesian method for structural learning of multiple DAGs. We explicitly account for Markov equivalence of DAGs, and propose a suitable prior on the collection of graph spaces that induces selective borrowing strength across groups. The resulting inference allows in particular to compute the posterior probability of edge inclusion, a useful summary for representing flow directions within the network. Finally, we detail a simulation study addressing the comparative performance of our method, and present an analysis of two protein networks together with a substantive interpretation of our findings.