The Markov Condition (plato.stanford.edu)

The Markov Condition (MC) is a foundational concept in probabilistic causation, crucial for understanding how complex joint probability distributions factorize in graphical models. It states that the joint probability of a set of variables \(V = \{X_1, X_2, \ldots, X_n\}\), structured as a directed acyclic graph (DAG), can be factorized into the product of the conditional probabilities of each variable given its parents: \(P(X_1, \ldots, X_n) = \prod_i P(X_i | \text{PA}(X_i))\). This factorization simplifies the representation and computation of joint distributions by leveraging the causal ordering and dependencies encoded in the graph. Beyond factorization, the MC implies specific conditional independence relations between variables, characterized by the graphical notion of d-separation. D-separation provides a necessary and sufficient criterion to determine when two variables are independent given a conditioning set, based solely on the structure of the DAG. It accounts for colliders—nodes where causal paths converge—and their descendants, offering a nuanced method to read off independence properties from the graph without direct probability calculations. This connection between graph structure and probabilistic independence is pivotal for model interpretation, causal inference, and efficient learning in AI and machine learning, enabling algorithms to exploit these conditional independencies for better scalability and causal discovery.