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CAUSALITY - Discussion

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We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each element in one set is separated from every element in the other). Rule 1: x and y are d-connected if there is an unblocked path between them. By a "path" we mean any consecutive sequence of edges, disregarding their directionalities. By "unblocked path" we mean a path that can be traced without traversing a pair of arrows that collide "head-to-head". In other words, arrows that meet head-to-head do not constitute a connection for the purpose of passing information, such a meeting will be called a "collider". Example 1 This graph contains one collider, at t. The path x-r-s-t is unblocked, hence x and t are d-connected. So is also the path t-u-v-y, hence t and y are d-connected, as well as the pairs u and y, t and v, t and u, x and s etc.... However, x and y are not d-connected; there is no way of tracing a p

CAUSALITY - Discussion d -SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d- separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z

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