D-Separation (D-Separation) is a graphical method for judging whether variables are conditionally independent. Compared with non-graphical methods, D-Separation is more intuitive and simple to calculate. For a DAG ( directed acyclic graph ) E, the D-Separation method can quickly determine whether two nodes are conditionally independent.
Concept: Many machine learning models can be explained from the perspective of probability. One of the important models is the probabilistic graphical model, but the soul of the probabilistic graphical model is the conditional independence between model variables . Because of independence, there are various probabilistic graphical models, such as LDA, HMM and other models. In the probability map, how is the independence between variables reflected? The D-separation criterion is a simple technique for judging independence in a probability graph.
Simply put, if in the probabilistic graphical model, for example, there are no edges between X and Y nodes, there must be some independence between X and Y. This independence can be made under the condition of a certain subset Z. They are independent, or the two of them may be independent. At this time, we call X and Y separated by D-. Now give the criterion of D-separation first:
Definition: When the path p is separated (or blocked) by the node set Z, d-, if and only if the following conditions hold:
① If p contains the following chain i->m->j or i<-m->j, then the node m is in the set Z.
② If p contains collider (collision point) i->m<-j, then node m is not in Z and m's children are not in Z.
More specifically, if Z d- separates X and Y if and only if Z blocks every path between X and Y.
Next, we will introduce the above guidelines step by step. We can split them into three rules for consideration:
First of all, let's explain the path here. When we talk about the path between two nodes, we don't care about the direction of the edge between them.
Independence without condition set
Rule : If any path (path) from x to y passes through a collider (collision point) , then x and y are independent. Note that the path here ignores the direction of the edge, and the collision point refers to its node with multiple arrows pointing to it, that is, similar to A->B<-C in the figure below
head-to-head: A, C independent
If the model is seen in it, they are considered to be connected, and they are considered to be independent, but
if their central node B is part of the condition, it is contrary to their original attributes, and they are not connected.
become independent.
Rule 2 : When any path between x and y passes through the nodes in Z, and Z does not contain the collision point or the children of the collision point, then x and y are independent.
tail-to-tail: A<-B->C: A, C are not independent
head-to-tail: A->B->C: A and C are not independent
Note that if the two models are seen there, they are considered connected, that is, they are not independent, but
if their central node B is part of the condition, it is the opposite of their original properties , they are disconnected
and become independent.
Example:
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