Chapter 4 Structure Learning

A Bayesian Network is a directed acyclic graph (DAG) due to the following constraints.
(TODO: list constraints)

Network structure learning from data is an NP-hard problem due to the acyclicity constraint (Zheng2018). Adapting the notation from Zheng et al. 2018, we assume \(G(W)\) is a d-node graph with the weighted adjacency matrix \(W\). We call the optimization score function \(F: \mathbb{R}^{dxd} \rightarrow \mathbb{R}\). The traditional optimization problem can be formulated as
\[ \begin{equation} \min_\underset{W \in \mathbb{R}^{dxd}} \; F(W) \\ G(W) \in DAGs \tag{4.1} \end{equation} \]

To reformulate @{eq:tradoptprob} in continuous form, we need to exactely characterize the acyclicity of the graphs with smooth function \(h: \mathbb{R}^{dxd} \rightarrow \mathbb{R}\) over real matrices with all levels set at zero. \[ \begin{equation} \min_\underset{W \in \mathbb{R}^{dxd}} \; F(W) \\ h(W) =0 \tag{4.2} \end{equation} \]