Confusion about independence and conditional independence

Question

I have a question based on an exercise from Grimmett's & Strizaker's book, "Probability and Random Processes".

I don't understand the fact that

$$ \mathbb{P}(A \cap B | T) = \mathbb{P}(A | T)\mathbb{P}(B | T) \qquad \& \qquad \mathbb{P}(A \cap B | T^c) = \mathbb{P}(A | T^c)\mathbb{P}(B | T^c) $$

The conditional independence of $A$ and $B$, given $T$ or $T^c$, is being attributed to the independence of $A$ and $B$. However, I've seen that independence does not imply conditional independence.

So, my questions are :

Why are the above equations correct ? (can we show their validity in a more mathematical strict way ?)

Generally, when can independence imply conditional independence, as in this case ?

Thank you in advance

Answer 1

There seems to be an implicit assumption that the reliability of witness A, the reliability of witness B and whether T occurred are all independent, i.e. both unconditional and conditional independence. With that assumption, the book's argument is valid.

Without that implicit assumption, it is possible to construct a counter-example, as you suspected. Consider the following case with $\alpha=\beta=0.9$ and $\mathbb P(T)=0.001$

T occurs   A reliable   B reliable  A says T  B says T Probability
--------   -----------  ----------  --------  -------- ----------- 
    1          1            1          1        1         0.001    
    0          1            1          0        0         0.809    
    0          1            0          0        1         0.090
    0          0            1          1        0         0.090 
    0          0            0          1        1         0.010

The two reliabilities are unconditionally independent but here $\mathbb P(T \mid A \cap B) =\frac{1}{11} \not= \frac{81}{1080}$, i.e. $0.0909\ldots \not= 0.075$.

Since $\frac{1}{11}$ is still small, the book's conclusion "somewhat small for a judicial conclusion" is still reasonable and this is in fact the highest possible value for $\mathbb P(T \mid A \cap B)$ with unconditional independence of the two reliabilities.