Ramblings for Relevant Search. Primary and Secondary ranking factors in relevance. Problem: you want to rank with two factors. Simply multiplying scores ignores the relative significance of the primary/secondary scores. For example, in the tables below, if the first column is much more important to ranking than the second factor. The second factor is more of a tie breaker for ties in the primary criteria. Additive boosting seems to do a better job at this:
| Primary | Secondary | Sum |
|---|---|---|
| 10 | 0.9 | 10.9 |
| 10.1 | 0.7 | 10.8 |
| 10 | 0.8 | 10.8 |
| Primary | Secondary | Product |
|---|---|---|
| 10 | 0.9 | 9.0 |
| 10.1 | 0.7 | 7.06 |
| 10 | 0.8 | 8 |
Notice how on the second row of multiplicative boosting table, you see a step down from 0.8 -> 0.7 in the secondary ranking factor has caused a significant step down in the relevance score. This secondary impact for additive boosting has had a minor impact on the score, acting more like a nudge or tie breaker. Multiplying by a difference of 0.1 is a rather significant change in the final product.
You could solve this with multiplicative boosting by translating the secondary boost to a much lighter scalar. Instead of multiplying to scores together, you could make multiplicative boosting have a much lighter touch by computing a different factor. This requires you to really know the ranges of the secondary factor
| Primary | Secondary | Scalar | Product |
|---|---|---|---|
| 10 | 0.9 | 0.99 | 9.9 |
| 10.1 | 0.7 | 0.98 | 9.89 |
| 10 | 0.8 | 0.97 | 9.7 |
Would computing a scalar from a relevance score be difficult?
In some ways, when there's no clear primary vs secondary relationship, and instead you just have a bunch of ranking factors of amorphous prioritization, trying to narrow things down to an appropriate scalar might be most appropriate.
Ultimately, you can restate an additive boost as a multiplicative boost and vice-versa.
Given base score X and additive boost value A, A can be restated as multiplicative boost as follows:
X + A = A + X
= Y * X
Y = (X + A) / X
So the additive boost can be restated multiplicative as:
X * (X + A) / X
Similarly multiplicative boost (M), restated as additive:
X * M = MX X + (M - 1)X = MX
What's not satisfying is now the translated additive/multiplicative boosts are now functions of the base score X. Instead of a simple addition of A to the base score, you now multiply (X + A) / X.
Fundamentally this equivalence doesn't satisfy the question when you add vs when you multiply.