Suppose we have 2 events $C_1$ and $C_2$ so that $C_1$, $C_2$, $C_1'$ and $C_2'$ partition the probability space. Let the event I'm interested in be denoted by $X$. $C_1$ and $C_2$ and their complements are conditionally independent given $X$. The probability of $X$ is given by, \begin{equation*} P(X)=P(X|C_1,C_2)P(C_1,C_2)+P(X|C_1',C_2)P(C_1',C_2)+P(X|C_1,C_2')P(C_1',C_2)+P(X|C_1',C_2')P(C_1',C_2') \end{equation*} Using conditional independence, we evaluate this as follows, \begin{equation*} P(X) =P(C_1|X)P(C_2|X)P(X)+P(C_1'|X)P(C_2|X)P(X)+P(C_1|X)P(C_2'|X)P(X)+P(C_1'|X)P(C_2'|X)P(X) \end{equation*} But then I have $P(X)$ on both side of the equality in the second equation. Is there a difference between these two (are the ones on the RHS the prior probability and the $P(X)$ on the LHS the posterior)? Any clarification would be appreciated.
In my case $P(X)$ is unknown, and I'm trying to solve for it. Would I go about this by guessing values of $P(X)$ for those on the RHS and evaluate this for the value of $P(X)$ on the LHS?