Bayesian Network - unclear homework example

Question

I am not sure if it is me or the example:

A doctor gives a patient a drug dependent on their age and gender. The patient has a probability to recover depending on whether s/he receives the drug, how old s/he is and which gender the patient has. Additionally it is known that age and gender are conditional independent if nothing else is known from the patient.

i) Draw the Bayesian Network which describes the situation.

ii) How does the factorized probability distribution look like?

iii) Write down the formula to compute the probability that a patient recovers given that you know if s/he gets the drug. Write down the formula using only probabilities which are part of the factorized probability distribution.

I think I could handle task ii) and iii) if someone would show me how the network looks like.

What is not clear to me is the give fact that "A doctor gives a patient a drug dependent on their age and gender" and "patient has a probability to recover depending on whether s/he receives the drug, how old s/he is and which gender the patient has".

Could someone help me with that?

Answer 1

i) Let $R$ represent the patient recovering, $D$ represent the drug being administered, $A$ represent the patient's age, and $G$ represent the patient's gender. Lastly $X\rightarrow Y$ means that there's an arrow from $X$ to $Y$ in the Bayesian network. Now let's break down the problem.

A doctor gives a patient a drug dependent on their age and gender.

This translates to $A\rightarrow D$ and $G\rightarrow D$.

The patient has a probability to recover depending on whether s/he receives the drug, how old s/he is and which gender the patient has.

This translates to $D\rightarrow R$, $A\rightarrow R$, and $G\rightarrow R$.

Additionally it is known that age and gender are conditional independent if nothing else is known from the patient.

This doesn't add any new arrows and translates to $P(A,G)=P(A)P(G)$. However, if we knew $D$ and let's say $A$, we might be able to infer insight on the value of $G$ (and maybe infer $A$ if we knew $G$ and $D$). Same story is we replace $D$ with $R$.

ii) This isn't the only way to factorize the joint distribution (there's a more elegant way that uses the conditional independence of $A$ and $G$), but this way will help us with part iii), \begin{align*} P(A,G,R,D) &=P(A|G,R,D)P(G,R,D) \\\ &=P(A|G,R,D)P(G|R,D)P(R,D) \\\ &=P(A|G,R,D)P(G|R,D)P(R|D)P(D) \\\ \end{align*} On the other hand, they may be looking for a form which conforms more clearly to the arrow relations like, \begin{align*} P(R,D,A,G) &=P(R|D,A,G)P(D,A,G) \\\ &=P(R|D,A,G)P(D|A,G)P(A,G) \\\ &=P(R|D,A,G)P(D|A,G)P(A)P(G) \\\ \end{align*}

iii) Using the fully factored form in the first part of ii) we can solve for $P(R|D)$, $$ P(R|D) = \frac{P(A,G,R,D)}{P(A|G,R,D)P(G|R,D)P(D)} $$