What is this Bayesian equations left hand side

Question

What is the left hand side of this equation $$ X =\frac{p(a,b\mid c,d)}{p(a\mid c,d)} $$

Answer 1

Hint:

Start with writing $P(X\mid Y)=P(X,Y)/P(Y)$ for numerator and denominator.

Then you can simplify and recognize another conditional probability.

Answer 2

$$ X =\frac{p(a,b\mid c,d)}{p(a\mid c,d)} $$

Let $A = a \cap b, B = c \cap d$.

We are going to use the conditional probability formula $$P(A\mid B) = \frac{P(A \cap B) }{P(B)}$$

$p(a,b|c,d) = p(A|B) = \frac{p(A\cap B)}{p(B)} $

$p(a|c,d) = p(a|B) = \frac{p(a \cap B)}{p(B)}$

Therefore, $\frac{p(a,b\mid c,d)}{p(a\mid c,d)} = \frac{\frac{p(A\cap B)}{p(B)}}{\frac{p(a \cap B)}{p(B)}}= \frac{p(A\cap B)}{p(a \cap B)} = \frac{p(a\cap b\cap c\cap d)}{p(a \cap c\cap d)}$

If $a,b,c,d$ are independent(although not specified in the question), then $X=p(a) $