Question

This task explores the distinction between d-connection and dependence. Consider the distribution class

p(a, b, c) p(c I b)p(b I a)p(a) ,
One might expect that this means that a and c are dependent, aℿc. Our interest is to show that there are non-trivial for which a is d-connected to c distributions for which a ℿ c.
Consider dom(a) = dom(c)= { l, 2} and dom(b)={1,2,3}

p(a)=(3/2 2/5) p(b|a)={1/4 1/12 2/3, 15/40 1/8 1/2}
p(c I b) ={⅓ ½ 15/40, ⅔ ½ 5/3}

Answer 1

Exploring the distinction between d-connection and dependence in a distribution class.

Step-by-step explanation:

In this task, we are exploring the distinction between d-connection and dependence in the context of a distribution class. We are given the distribution class p(a, b, c) p(c | b)p(b | a)p(a) and we want to determine whether a and c are dependent, aℿc, or not. We can do this by analyzing the distributions and checking for d-connection.

D-connection refers to a directed path between two variables that does not contain any colliders. On the other hand, dependence implies a statistical relationship between two variables.

To determine whether a and c are dependent or not, we need to check if there is a d-connection path between them in the given distribution class. If there is a d-connection path, it means that a and c are dependent even if they are not statistically related.