Reference no: EM133040303
Question: Generate data from a graph with 4 nodes X = {x1, x2, x3, x4}. If the joint distribution P(X) can be written as a product of conditionally independent distributions
4
P(X|G) = Π P(xk|PA(xk), G)
k=1
then the graph is said to be a directed acyclic graph. Here the notation PA(xk), G means the set of nodes which are parents of xk, given graph G.
Assume further that this conditionally independent distribution P(xk|PA(xk), G) is gaussian and define the set Ik = {j; xj≠k ∈ PA(xk)} to be the set of indices which denote if a RV xj≠k is a parent of xk.
We assume that P(xk|PA(xk), G)~N(αk+βkIk XIk, σk2)
We wish to compute the marginal likelihood
4
P(X|G) = ∏ P(xk|xIk, G)
k=1
P(xk|xIk, G) = P(xk|PA(xk), G) = ∫ P(xk|PA(xk), G, Θk) P(Θ|G) dΘk
where Θk = {αk, βklk , σk2} and βkIk is coefficient of the impact of parent node (Ik) on node k.
Task 01: Consider the following priors
Let βk∗ = (αk, βk)
βk ∗|PA(xk)~N [0, gσk2(Xik*T X1k*)-1
XIk∗ = [1, Xk] and σk2~IG(a, b)
Write down an expression for
P(xk|PA(xk)) = ∫∫ P (xk|PA(xk), βk∗, σk2) P(βk∗|σk2)P(σk2) dβk∗dσk2
And hence compute
4
P(X|G) = Π P(xk|PA(xk))
k=1
Task 02: The joint distribution always equal to
P(X) = P(x1) P(x2|x1) P(x3|x2, x1)P(x4|x1, x2, x3)
Or any other ordering of nodes.
Write down an expression for P(x1) P(x2|x1) P(x3|x2, x1)P(x4|x1, x2, x3) if
βj<k,k*|σk2, Xj<k ∗~N [0, gσk 2(Xj<k∗T Xj<k*)-1]
Write down an expression for P(X|G). Note it will be very similar to the task 01 only X's will change.
Task 03: Generate data from the models in task 01 and task 02. For each model compute
I. P(X|G1) = P(x1) P(x2|x1) P(x3|x1)P(x4|x2, x3)
II. P(X|G2) = P(x1) P(x2|x1) P(x3|x2, x1)P(x4|x1, x2, x3)