Deterministic CPD's independencies in PGM

Question

I am studying Probabilistic graphical model based on this course and book by same professor. I am reading deterministic CPD's independency. Seeing the image below from the lecture, the lecture says if C is a deterministic function of A and B, then D and E are independent to each other given A and B. And it only applies when C is "deterministic" function.

I don't understand this deterministic part. I understand that without the deterministic condition, D and E are independent to each other given C. But why it is not possible in case when C is stochastic function of A and B? Anyway C is depending on A and B regardless of its function type (stochastic and deterministic function).

In the book and lecture, this part is explained by only one line. For me, it is not enough.

Answer 1

If C is observed, then D and E are independent of each other, regardless of whether C is a deterministic function of A and B or not. I believe the case the book/course was trying to illustrate was that if A and B are observed but C isn't, then the independency between D and E depends on the deterministic nature of C's CPD: if C is a deterministic function of A and B, then observing them implies than you know C, and therefore D and E are not independent anymore. However, if C is a non-deterministic function of A and B then you cannot make such a conclusion. This case is also explained on slide 6 here: https://cedar.buffalo.edu/~srihari/CSE674/Chap5/5.2-DeterministicCPD.pdf

In case there is still some fuziness in your head around deterministic and non-deterministic CPDs, here is an illustration that might help: We choose to represent “It is raining today” as a Non-deterministic CPD because even though we can model the influence of "it was raining yesterday" and the "average rainfall for the season", we think it also depends on a lot of other unmodeled factors: humidity, air pressure, ...

According to this model, if someone knows:

• whether it was raining yesterday
• the average rainfall for the season

but not:

• whether it is raining today (imagine he/she is on a trip and we have been exchanging messages)

Then learning that I just went to the movies will give he/she additional clues on whether I have worn a raincoat when I went out today. Because it will give them clues on whether it was raining today. We choose to represent “I’ll wear a raincoat when I go out” as a Deterministic CPD equal to "( It is raining = “yes” ) AND ( I have a raincoat = “yes” )".

According to this model, if someone knows:

• That it is raining
• That I have a raincoat

but not:

• That I was wearing a raincoat when I went out

Then learning that people in the street found my coat ugly will not give he/she additional clues on whether I found my lost keys in my raincoat's pocket. Because they already inferred that I was wearing a raincoat when I went out, they already know the chances that I found my keys.