teach me about D-separation in Bayesian graphical models

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D-separation is a fundamental concept in Bayesian networks that provides a graphical method for determining conditional independence between variables. In this simple network with three nodes A, B, and C, D-separation helps us understand when variables are independent given evidence about other variables. This concept is essential for probabilistic inference, understanding causal relationships, and efficient computation in complex networks. Information flow in Bayesian networks is determined by three fundamental connection patterns. First, serial connections form chains where information flows from X through Y to Z. Second, diverging connections create forks where Y is a common cause of both X and Z. Third, converging connections form colliders where both X and Z influence Y. Understanding these patterns is essential for applying D-separation rules to determine conditional independence.