Ramblings for [Relevant Search](http://manning.com/turnbull). 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:

## Additive Boosting

Primary | Secondary | Sum
--------|-----------|-----
10      | 0.9       | 10.9
10.1    | 0.7       | 10.8
10      | 0.8       | 10.8


## Multiplicative Boosting

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. 

## Additive & Multiplicative Boosts Are The same

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:

<pre>
X + A = A + X
      = Y * X
    Y = (X + A) / X
</pre>

So the additive boost can be restated multiplicative as:

<pre>
X * (X + A) / X 
</pre>

Similarly multiplicative boost (M), restated as additive:

<pre>
X * M = MX
X + (M - 1)X = MX
</pre>

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.