# Does ( P(B|A) - P(B|~A) ) / P(B|A) have a name?

Without going into the details, which are unnecessary here, this morning I found uses for the quantity

$$S = \dfrac{P(B|A) - P(B|\overline A)}{P(B|A)} ,$$

something like the amount of "probabilistic specialness due to a condition". Like likelihood ratio, it's a comparison of P(B|A) and P(B|not A), but perhaps in some ways even more natural.

For instance, if today has a 30% chance of rain given that it's spring, and would have a 20% chance of rain otherwise (given not spring), then the difference is 10%, which we then divide by 30%, giving 33%. So 33% of spring's rain chances are correlated with it being spring instead of another season.

If, instead, we imagine the spring chances were 1%, and non-spring were 0%, then S would be (.01 - 0) / .01, or 100%. In this way, values are normalized to 100%.

My current feeling is that this might be a natural way to associate conditions with events. If S is high, then we would be prone to saying, "It's raining, S(rain|spring) is close to 100%, therefore rain is characteristic of spring, and it's probably spring". Something like logical abduction.

I'd like to know if this has a name in probability theory (and just generally how it might appear).

• Jul 1, 2021 at 17:43
• I think that's it exactly. Please write a solution (copying text from Wiki instead of linking), and I'll mark it as the answer. Thank you
– dwn
Jul 1, 2021 at 18:39

the relative risk reduction (RRR) or efficacy is the relative decrease in the risk of an adverse event in the exposed group compared to an unexposed group. It is computed as $${\displaystyle (I_{u}-I_{e})/I_{u}}$$, where $$I_e$$ is the incidence in the exposed group, and $$I_{u}$$ is the incidence in the unexposed group. If the risk of an adverse event is increased by the exposure rather than decreased, term relative risk increase (RRI) is used, and computed as $${\displaystyle (I_{e}-I_{u})/I_{u}}$$. If the direction of risk change is not assumed, a term relative effect is used and computed as $${\displaystyle (I_{e}-I_{u})/I_{u}}$$