# Draw the graphical model for the distribution

I would like to draw the graphical model for the distribution in this question and capture as many independence assumption as possible. The distribution is:

Let $$A = \{\text{flipping coin A}\}$$ and $$B\{\text{flipping coin B}\}$$ and $$C = \begin{cases} 1 & \text{both heads or both tails} \\ 0 & \text{otherwise}\end{cases}$$. So this means that $$A \perp\kern-5pt\perp B ,B \perp\kern-5pt\perp C, C \perp\kern-5pt\perp A$$ but $$A$$ and $$B$$ are not conditionally independent when given $$C$$

So, would the following graphical model (belief network) be an accurate representation?

Yes this is correct. Notice that conditioning on $$C$$ provides some information about the result of $$A$$ and $$B$$. In fact, if $$C=1$$, then $$A=B$$, otherwise $$A \neq B$$, so of course $$A$$ and $$B$$ are not independent conditioned on $$C$$.
The information you're capturing in this graphical model is: $$A$$ and $$B$$ are independent and $$C$$ is constant when conditioning on $$A$$ and $$B$$.
• can i also state that $A \perp\kern-5pt\perp C$ and $B \perp\kern-5pt\perp C$?