What's the name of the quantity $\mathbb{P}(A\cap B)/(\mathbb{P}(A)\mathbb{P}(B))\;$?

Question

In a physics book, I've come across the quantity

$$ \frac{\def\P{\mathbb{P}}\P(A\cap B)}{\P(A)\P(B)}\,, $$

where $A$ and $B$ are events.

The author calls this quantity the correlation of $A$ and $B$, but the expression above does not agree with the definition of "correlation" that I'm familiar with. (For one thing, when $A$ and $B$ are independent, the quantity above equals $1$, yet the correlation I'm familiar with is $0 \neq 1$ when the events are independent.)

I must conclude that the author's word choice reflects the usage within physics.

Does this quantity have a standard name in mathematics?

Answer 1

There are a number of dependence measures which are defined in similar way. For example, given two $\sigma$-fields $F$ and $G$:

$\alpha(F,G)=sup_{(A\in F,B\in G)}|P(A\cap B)-P(A)P(B)|$

$\phi(F,G)=sup_{(A\in F,B\in G)}|P(A|B)-P(A)|$

Your dependence measure defined using similar logic:

$\delta(A,B)=\frac{P(A\cap B)}{P(A)P(B)}=\frac{P(A|B)}{P(A)}$

• If events $A$ and $B$ are independent $\delta(A,B)=1$

• If events $A$ and $B$ are dependent than $P(A|B)>(<)P(A)$ and $\delta(A,B)>(<)1$

Answer 2

I think the book is simply saying that it is a measure of how correlated the two variables are. If they are totally independent (un-correlated) then $P(A\cap B)=P(A)P(B)$ and the quantity is $1$. If they are mutually exclusive the quantity is $0$. In general, $$\frac{P(A\cap B)}{P(A)P(B)}=\frac{P(A|B)}{P(A)}=\frac{P(B|A)}{P(B)}$$ So it measures what proportion of $A$ occurs "in" $B$ or vice versa.

But to answer the question, I think it's fair to say that 1) it almost certainly has some name and 2) that name is almost certainly not standard. So no - it has no name.

Answer 3

I think "pointwise mutual information" is what comes close. PMI is defined as

$\log{\frac{p(x,y)}{p(x)p(y)}}$