Reversing a conditional probability

Question

If I'm given a P(X|Y) table and P(Y), how can I find P(Y|X)?

I understand that $P(Y|X)=\frac{P(X|Y)P(Y)}{P(X)}$ but how do i find P(X)?

Furthermore, If i'm told that random variable X is given a value, does this affect P(Y)?

Answer 1

Assume that you are given a decomposition $\mathscr D$ of your probability space $\Omega$, $\mathscr D=\{D_1,\ldots,D_k\}$, i.e., $D_i\cap D_j=\emptyset$ for $i\neq j$ and $D_1\cup\ldots\cup D_k=\Omega$. Then the formula for total probability for the event $A$ is given by $$ {\rm P}\{A\}=\sum_i{\rm P}\{A\mid D_i\}{\rm P}\{D_i\}. $$ To find ${\rm P}\{D_i\mid A\}$ you just go $$ {\rm P}\{D_i\mid A\}=\frac{{\rm P}\{A\mid D_i\}{\rm P}\{D_i\}}{{\rm P}\{A\}}=\frac{{\rm P}\{A\mid D_i\}{\rm P}\{D_i\}}{\sum_j{\rm P}\{A\mid D_j\}{\rm P}\{D_j\}}, $$ which is called the Bayes formula.

The first is often referred as a priory probability (before the experiment), the second as a posteriori probability (after the experiment, you fix your probabilities of $D_j$ based on the information of the event $A$).