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In a graphical model, we say that set $A$ and $B$ are conditionally independent given set $C$ if all routes from $A$ to $B$ are blocked. There are multiple ways for the route to be blocked at a node. One instance for the route to be blocked at a node is when neither the descendants or the node itself is in $C$.

Given the following:

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let $A = \{x_1\}, \ B = \{x_3\}$ and $x_2 \notin C$. How then do we prove that

$$Pr(A,B|C) = Pr(A|C) Pr(B|C)$$

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There is only one route (or "path") from $A$ to $B$, namely $x_1\to x_2\to x_3$. Is this path blocked at any node? Yes, it is blocked at node $x_2$ since "neither the node $x_2$ nor its descendants" (which it does not have) are in $C$. So all the paths are blocked, and we have conditional independence.

In this particular example we can also argue directly, though it is a bit of a "degenerate case". The Bayes net modeling assumption is "each variable is conditionally independent of its non-descendants, given its parents”. Applying it to $x_1$ and $x_3$ we have $P(A,B)=P(A)P(B)$.

We now ask $$P(A,B|C) =?= P(A|C) P(B|C)$$

Since $x_2\notin C$, it is actually completely irrelevant that we have a Bayes net relating $x_2$ to $A$ and $B$. Two independent random variables are independent after conditioning on any subset of them. In detail:

If $C=\emptyset$ we have the original $P(A,B)=P(A)P(B)$.

If $C=A$ assuming for simplicity that the variables are discrete and $P(A=a)$ is never zero, and using notation $\delta(p,q)=\begin{cases}1 \text{ if } p=q\\ 0 \text{ else }\end{cases}$, we have

\begin{align*}P((A,B)=(a,b)|A=x) =& \delta(a,x) \frac{P((A,B)=(x,b))}{P(A=x)}\\&=\delta(a,x) P(B=b)=P(A=a|A=x)P(B=b|A=x)\end{align*}

where we used $P((A,B)=(x,b))=P(A=x)P(B=b)$ and $P(B=b)=P(B=b|A=x)$ by independence. This is conventionally written $P(A,B|A)=P(A|A)P(B|A)$.

If $C=B$ the argument is symmetric (assuming $P(B=b)$ is never zero).

Finally if $C=\{A, B\}$ we have under suitable assumptions

\begin{align*} P((A,B)=(a,b)|(A,B)=(x,y)))&=\delta((a,b), (x,y))=\delta(a,x)\delta(b,y)\\&=P(A=a|A=x, B=y)P(B=b|A=x, B=y) \end{align*}

or $P(A,B|A,B)=P(A|A,B)P(B|A,B)$.

All of this can be done with continuous random variables and their densities instead.

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  • $\begingroup$ Yes, but how do we use the rules of conditional probability to illustrate that this is indeed the case? $\endgroup$ – Sean Lee Apr 14 at 13:19
  • $\begingroup$ Are you interested in the general derivation of the D-separation rules or in a derivation of conditional independence for this particular case "from first principles"? $\endgroup$ – Max Apr 14 at 20:12
  • $\begingroup$ I'd like to see a proof from first principles for this case $\endgroup$ – Sean Lee Apr 14 at 21:49
  • $\begingroup$ I've added that. $\endgroup$ – Max Apr 15 at 9:07

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