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In former articles, we have been looking at many different concepts related to Bayesian Networks: how to represent them in a compact manner, their reasoning patterns, and I-Map. In this article, we will take a look at D-separation and try to understand it both intuitively and mathematically.
Understanding D-separation
So, what exactly is D-separation in context to Bayesian Networks and what can it be used for? Simply said, it is a more formal procedure for determining independence. In other words, if two variables are d-separated relative to a set of variables, Z, in a directed graph, then they are independent conditional on Z in all probability distributions such a graph can represent. What does this mean? It means that two variables X and Y are independent conditional on Z if knowledge about X gives you no extra information about Y once you have knowledge of Z.
To completely understand how it is done, we will first need to talk about active and inactive trails. Intuitively, we can say that a path is active if it implies dependence. Two variables, X and Y, might be connected by multiple paths in a graph, where some or none are active. If none of the paths are active then X and Y are d-separated. Let us take a look at four different cases and determine whether the parts are active or not:
