(Probability) Association rules - LIFT is useless? 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?
 A: 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. 
A: In you example, A and B had lower probabilities than X and Y. Therefore the denominator of lift is small and the lift may easily get high. In an extreme case, if C and D occurred on just one row both, but together, the lift would be maximal possible, N (the data size). So the pattern would appear very strong but it could well be just noise. However, if you measured the strength with leverage (P(CD)-P(C)P(D)), you would get quite a different result, leverage<1/N (when maximum leverage is 0.25). 
In general, lift measures dependence between two variable-value combinations (events), while leverage measures dependence between the corresponding binary variables. It depends on your modelling purposes which one you want to measure. Leverage tends to be more robust in the sense that the patterns with highest leverage tend to have approximately the same leverage in future data, while lift values may vary a lot (a pattern with highest possible lift may actually express negative dependence in future data). On the other hand, leverage favours patterns where marginal probabilities P(C) and P(D) are near 0.5 and you may miss interesting but less frequent patterns. 
If you use lift to measure the strength, I suggest to evaluate also statistical significance of the pattern. Fisher's exact test is reliable even for infrequent patterns.
