Find joint distribution from conditionals?

Question

I have the two conditionals $P(A|B)$ and $P(B|A)$ and would like to find the joint $P(A, B)$. I found

$$ P(A|B)P(B|A) = \frac{P(A, B)^2}{P(A)P(B)} $$

with Bayes' rule but this requires the marginals which I do not have access to. Is there no way to do this?

Answer 1

knowing both contitional, say

$$\mathbb{P}[A|B]=p_1$$

$$\mathbb{P}[B|A]=p_2$$

you know also that

$$\frac{\mathbb{P}[A]}{\mathbb{P}[B]}=\frac{p_1}{p_2}$$

and with this condition you should resolve your issue. If not, please provide the entire text with your attempts and you will surely be helped

Answer 2

There is not enough information to compute $P(A,B)$.

Consider the following two examples.

Example 1: $A = [0,2/3]$, $B = [1/3,1]$.

Example 2: $A = [1/3,5/9]$, $B = [4/9,2/3]$.

We have $P(A|B) = P(B|A) = 1/2$ in both cases, but $P(A,B)=1/3$ in the first example, and $P(A,B)=1/9$ in the second.