# Conditional independence of dependent variables (D-Separation)

I have given the following Bayesian network and need to evaluate whether q is independent of s given r.

I know that the path over the node r is blocked because r is observed. But what about the other direct path? Probabilistically it states P(s|q), therefore q is directly dependent on s. But does this also apply to conditional independence?

I do not have any probabilities given and need to solve the task using D-Separation. The problem is when we want to check if two variables are d-separated, we need to check whether the path is blocked. In the case of the lower path that includes r it is, because r is observed and we have a head-to-tail connection. But what case do we have for the other path? Here we do not have any nodes in between.

• you probably have to check whether P(q|c)*P(s|c) = P(q and s|c)
– Alberto Sinigaglia
Dec 17, 2021 at 11:08
• @AlbertoSinigaglia Only the model was given. I do not have the exact probabilities and need to solve it with D-separation. (I will make the question more clear)
– Marcello Zago
Dec 17, 2021 at 11:21

\eqalign{ \textrm{Conditional independence} &\Leftrightarrow p(q,s|r) = p(q|r)p(s|r)\cr p(q,s|r) &= p(s|q,r)p(q|r)\cr p(s|q,r) &\ne p(s|r)\cr } (because observing $$q$$ and $$r$$ would tell us more about the distribution of $$s$$ than observing $$r$$ alone) \eqalign{ \Rightarrow p(s|q,r)p(q|r) &\ne p(q|r)p(s|r) \cr \Rightarrow p(q,s|r) &\ne p(q|r)p(s|r) } $$\Rightarrow$$ $$q$$ and $$s$$ are not conditionally independent given $$r$$, I believe.