Assume we have a rule {X} -> {Y} which means customers buy item X, they also buy item Y in a supermarket. I know the LIFT is used to measure the independence of X and Y which is equal $P(XY)/P(X)P(Y)$, the higher the better for the rules
Lets consider 2 cases:
1) We have the following probabilities for a rule:
$P(X)=0.52, P(Y)=0.77, P(XY)=0.5, P(X|Y)=0.65$ and $P(Y|X)=0.98$
The LIFT would be :
$$\frac{0.5}{0.52(0.77)}=1.25$$
2) We have the following probabilities for another rule {A}->{B}:
$P(A)=0.21, P(B)=0.4, P(AB)=0.18, P(A|B)=0.45$ and $P(B|A)=0.85$
The LIFT would be :
$$\frac{0.18}{0.21(0.4)}=2.14$$
A and B should be less dependent to each other compare to X and Y since they have less conditional probabilities ($P(A|B)<P(X|Y)$ and $P(B|A)<P(Y|X)$), but they still have higher lift.
Can it be said that lift does not linearly correlated with dependence and thus is not a good measure?
Best Answer
Random variables W and Z being independent implies:
$P(W)=P(W|Z)$
If we look at the probabilities you have there, the X and Y should have lower lift as the conditional probabilities are close to the unconditional ones, while for A and B, the conditional probabilities are farther away. This is part of what lift measures. The high conditional probability does not imply they are correlated.
So rewriting LIFT: $ \frac{P(Y|X)}{P(Y)}$ a weighted measure of conditional probability.