The causal bandit problem aims to sequentially learn the intervention that maximizes the expectation of a reward variable within a system governed by a causal graph. Most existing approaches assume prior knowledge of the graph structure, or impose unrealistically restrictive conditions on the graph. In this paper, we assume a Gaussian linear directed acyclic graph (DAG) over arms and the reward variable, and study the causal bandit problem when the graph structure is unknown. We identify backdoor adjustment sets for each arm using sequentially generated experimental and observational data during the decision process, which allows us to estimate causal effects and construct upper confidence bounds. By integrating estimates from both data sources, we develop a novel bandit algorithm, based on modified upper confidence bounds, to sequentially determine the optimal intervention. We establish both case-dependent and case-independent upper bounds on the cumulative regret for our algorithm, which improve upon the bounds of the standard multi-armed bandit algorithms. Our empirical study demonstrates its advantage over existing methods with respect to cumulative regret and computation time.

Directed Acyclic Graphs (DAGs) provide a powerful framework to model causal relationships among variables in multivariate settings; in addition, through the do-calculus theory, they allow for the identification and estimation of causal effects between variables also from pure observational data. In this setting, the process of inferring the DAG structure from the data is referred to as causal structure learning or causal discovery. We introduce BCDAG, an R package for Bayesian causal discovery and causal effect estimation from Gaussian observational data, implementing the Markov chain Monte Carlo (MCMC) scheme proposed by Castelletti & Mascaro (2021). Our implementation scales efficiently with the number of observations and, whenever the DAGs are sufficiently sparse, with the number of variables in the dataset. The package also provides functions for convergence diagnostics and for visualizing and summarizing posterior inference. In this paper, we present the key features of the underlying methodology along with its implementation in BCDAG. We then illustrate the main functions and algorithms on both real and simulated datasets.

The traditional directed acyclic graph (DAG) model assumes data are generated independently from the underlying joint distribution defined by the DAG. In many applications, however, individuals are linked via a network and thus the independence assumption does not hold. We propose a novel Gaussian DAG model for network data, where the dependence among individual data points (row covariance) is modeled by an undirected graph. Under this model, we develop a maximum penalized likelihood method to estimate the DAG structure and the row correlation matrix. The algorithm iterates between a decoupled lasso regression step and a graphical lasso step. We show with extensive simulated and real network data, that our algorithm improves the accuracy of DAG structure learning by leveraging the information from the estimated row correlations. Moreover, we demonstrate that the performance of existing DAG learning methods can be substantially improved via de-correlation of network data with the estimated row correlation matrix from our algorithm.