How to think about Continuous Total Probability Theorem applied to a conditional?

Question

I'm trying to understand the following relationship intuitively.

We have the generic TPT which states:

$P(A)=\int_{R_X}P(A|X=t)f_X(t)dt$

Where $A$ is some event, and $X$ is a continuous random variable with possible values in $R_X$.

When we apply this to a conditional case, we now restrict the sample space to $B$ (for some event $B$), and we get:

$P(A|B)=\int_{X\in B}P(A|B|X=t)f_{X|B}(t)dt$

But apparently this is the same as:

$P(A|B)=\int_{X\in B}P(A|B,X=t)f_{X|B}(t)dt$

How do we get there? I was trying to think in terms of Venn diagrams but the fact that $X=t$ is a zero probability event is throwing me off.

Answer 1

It might help to look at what happens in the discrete case. Let $X$ be a discrete random variables taking values in a set $\mathcal X$. I will use $P(E,F,G)$ to denote $P(E\cap F\cap G)$. Then \begin{align} P(A|B) &=\frac{P(A,B)}{P(B)} \\&=\sum_{t\in \mathcal X}\frac{P(A, B, X=t)}{P(B)} \\&=\sum_{t\in \mathcal X}\frac{P(A, B, X=t)}{P(B, X=t)}\cdot \frac{P(B, X=t)}{P(B)} \\&=\sum_{t\in \mathcal X}P(A|B,X=t)P(X=t|B) \end{align} This is the discrete analogue of your integral formula.

Answer 2

All of it hidden in Conditional Expectation. In modern probability, conditional probability define as a special case of Conditional Expectation.

$p(A)= E(I_A)=E (E(I_A|X))$

by definition of conditional expectation $E(I_A|X)$ is a function of $X$ (measurable to $\sigma(X)$ that satisfy projection property ) $= \int E(I_A|X=x) f_X(x)dx=\int p(A|X=x) f_X(x)dx $

from now $E(I_A)=\int p(A|X=x) f_X(x)dx$ and so $E(I_{AB})=\int p(AB|X=x) f_X(x)dx$ $\hspace{.5cm}$ (1)

note in modern probability we have (this defination is same for continues and discrete variable, it does not depend to type of $Y$) :

$E(Y|B)=\frac{E(YI_B)}{E(I_B)} =\frac{E(YI_B)}{E(I_B)}$ $\hspace{.5cm}$ (2)

so

$p(A|B)=E(I_A|B)\overset{ (2)}{=}\frac{E(I_AI_B)}{E(I_B)}=\frac{E(I_{AB})}{E(I_B)}=\frac{p(AB)}{p(B)}\overset{ (1)}{=}\frac{\int p(AB|X=x) f_X(x)dx}{\int p(B|X=x) f_X(x)dx}$

it is over since you can calculate $p(AB)=E(I_{AB})=\int p(AB|X=x) f_X(x)dx$ and $p(B)$ .

but for more

$\frac{\int p(AB|X=x) f_X(x)dx}{\int p(B|X=x) f_X(x)dx}=\frac{\int p(AB|X=x) f_X(x)dx}{p(B)}= \int\frac{ p(AB|X=x)}{p(B)} f_X(x)dx= \int\frac{ p(AB|X=x)}{p(B)p(B|X=x)} p(B|X=x)f_X(x)dx= \int\frac{ p(AB|X=x)}{p(B|X=x)} \frac{p(B|X=x)f_X(x)}{p(B)}dx=\int p(A|B,X=x) \frac{p(B|X=x)f_X(x)}{p(B)}dx= \int p(A|B,X=x) f_X(x|B)dx $