I have 4 variables and I know the following probabilities:
1) $P(A|B_1)$; $P(A|B_2)$; $P(A|B_3)$;
2) $P(B_1)$; $P(B_2)$; $P(B_3)$.
3) $P(A)$
Of course, I have:
4) $P(A,B_1)$; $P(A,B_2)$; $P(A,B_3)$;
And I need:
$P(A|B_1,B_2,B_3)$
Is it possible calculate this probability using some rule o theorem of probability?
I was thinking to use Chain's rule but for this case, It is enough data.
What I have is a table like following:
It contains the nationality for the name "Bernhard" of people who live in Spain. The name represents the variable $B_1$. Same for $B_2$ and $B_3$ (surname 1 and surname 2).
What I need is given name, surname 1 and surname 2 ($B_1$, $B_2$ and $B_3$) what is the probability of being from Germany (for example).
Two extremes of dependency here. If the $B$ terms are both independent so $P(B_1,B_2)=P(B_1)P(B_2)$ and conditionally independent on $A$ so $P(B_1,B_2|A)=P(B_1|A)P(B_2|A)$ then
$$ P(A|B_1,B_2,B_3)=\frac{P(B_1,B_2,B_3|A)P(A)}{P(B_1,B_2,B_3)} $$ $$ =\frac{P(B_1|A)P(B_2|A)P(B_3|A)}{P(B_1)P(B_2)P(B_3)}P(A) $$ $$ =\frac{P(A|B_1)P(B_1)P(A|B_2)P(B_2)P(A|B_3)P(B_3)}{P(A)P(A)P(A)P(B_1)P(B_2)P(B_3)}P(A) $$ $$ =\frac{P(A|B_1)P(A|B_2)P(A|B_3)}{P(A)^2} $$ Similarly, if the terms are completely dependent and conditionally dependent $$ P(A|B_1,B_2,B_3)=\frac{P(B_1,B_2,B_3|A)P(A)}{P(B_1,B_2,B_3)} $$ $$ =\frac{P(B_1|A)}{P(B_1)}P(A) $$ $$ =P(A|B_1) $$ These should form (loose? tight?) bounds.