If I have a conditional probability matrix for binary variables $A,B,C$ with entries of the form
$$
P(A | B \cap C) =
\left(
\begin{matrix}
P(a_1 | b_1 \cap c_1) & P(a_1| b_1 \cap c_2) & P(a_1|b_2 \cap c_1) & P(a_1 | b_2 \cap c_2) \\
P(a_2 | b_1 \cap c_1) & P(a_2| b_1 \cap c_2) & P(a_2|b_2 \cap c_1) & P(a_2 | b_2 \cap c_2)
\end{matrix}
\right)
$$
how can I use this matrix to obtain the matrix
$$
P(A | C) =
\left(
\begin{matrix}
P(a_1 | c_1) & P(a_1| c_2) \\
P(a_2 | c_1) & P(a_2| c_2)
\end{matrix}
\right)
$$
where $A$ is assumed to be independent of $B$?
Clearly in general $P(X | Y \cap Z) = P(X|Z)$ under the assumption that $X$ and $Y$ are independent, but I am confused as to how the entries of the matrices above are related.
Best Answer
One needs the values of the conditional probabilities $P(b_1\mid c_j)$ and $P(b_2\mid c_j)=1-P(b_1\mid c_j)$, since $$ P(a_i\mid c_j)=P(a_i\mid b_1\cap c_j)P(b_1\mid c_j)+P(a_i\mid b_2\cap c_j)P(b_2\mid c_j). $$