Can I update an expectation $\mathbb{E}[X | A, B, C]$ with a conditional probability $P(X | Y)$?
Hello everyone, I am tackling a problem where I have a conditional Expectation $\mathbb{E}[X | A, B, C]$, for some unknown Random Variables $A, B, C, \dots$ and two known probabilities, $P(X)$ and $P(X | Y)$. Where $ Y \not\in {A, B, C, \dots}$.
Now I would like to use those two probabilities to update the expectation. I thought about using multiplying $\mathbb{E}[X | A, B, C, \dots]$ ratio $\frac{P(X|Y = y)}{P(X)}$ but I am not quite sure how to interpret the new quantity.
Is it possible to get $\mathbb{E}[X | A, B, C, \dots, Y]$ with the new probabilities? I've been trying but don't get there without knowing $A, B, C$ since the only way I can think of doing it would be Total Expectation.
Many thanks for any ideas and insights you can provide!
Two events $A_1, A_2$ are independent if we can write $$ \mathbb{P}(A_1 \cap A_2)=\mathbb{P}(A_1)\mathbb{P}(A_2) $$
Two random variables $X_1, X_2$ are independent if we can write $$ \mathbb{P}(X_1=x_1, X_2=x_2)=\mathbb{P}(X_1=x_1|X_2=x_2)\mathbb{P}(X_2=x_2)=\mathbb{P}(X_1=x_1)\mathbb{P}(X_2=x_2) $$ i.e. the joint probability as the product of the marginal probabilities. Intuitively, you can ignore the information on $X_2$ to know the probability of $X_1=x_1$.
For ease of exposition take the random variables $X,A,B,Y$ such that $Y$ is independent of $X,A,B$. This condition is the one interesting for you, according to your question and comments.
The conditional expectation $\mathbb{E}[X |A = a,B = b, Y = y]$, is: $$ \sum_{x} xP(X = x | A = a,B = b, Y = y ) = \sum_{x} \frac{xP(X=x,A = a,B = b, Y = y)}{P(A = a,B = b, Y = y)} $$
Because of the independence of $Y$, $$ \mathbb{E}[X |A = a,B = b, Y = y]=\sum_{x} \frac{xP(X=x,A = a,B = b)}{P(A = a,B = b)} $$ That is, since $Y$ is independent, you cannot have a value for $ \mathbb{E}[X |A = a,B = b, Y = y]$ without information on $P(A = a,B = b)$.