Why is this conditional dependency statement not correct? I am taking an online Bayesian statistics course and here is a question from the quiz:
"Is the following statement correct? $$p(a∣b,c)=p(a∣b)p(a∣c)$$ when $b$ and $c$ are independent."
I thought it was correct, but learned after submitting that it was incorrect. Why is that?
 A: If $b,c$ are independent you have $p(b,c)=p(b)p(c)$. If $b,c$ are conditionally independent on $a$, you have $p(b,c|a)=p(b|a)p(c|a)$. Also, former doesn't imply the latter in general and vice versa. Also none implies directly what you seek for. Given the information, you should be using the appropriate identity (i.e. the former here) to conclude the mentioned statement, and, you'll see that you won't be able to do it. 
Additionally, one intuitive example could be the following: $a:$ child has brown eyes, $b,c$: father/mother has brown eyes. Father and mother are probably independent, e.g. they come from very different families. But, genetically, child will depend on their joint.
Further Explanation: 
First of all, in the RHS, it's counter intuitive to have the probability of $a$ twice, which might give you a heads up to search for a contradiction. Here is one: If true, this identity should hold for all $a$, i.e. it should hold even if $a$ is totally (even conditionally) independent of the two or not. Then, LHS becomes $p(a|b,c)=p(a)$, and RHS becomes $p(a|b)p(a|c)=p(a)p(a)$, and the two are not equal, which is a contradiction. 
If you're going to show an identity is true, first you need to believe that it is true, and show it mathematically. If we have doubts, try to find a case that contradicts the statement, either numerically, or conceptually. Then, make your arguments rigorous. Trying to find cases where the identity may hold is a harder problem.  
