The mathematical concept of set diversity is a somewhat neglected topic in current applied decision sciences and optimization. We take this opportunity to discuss the issue.

## The problem

Consider the following problem: for a number of items `U = {x_1`

, … `x_n}`

pick a small set of them `X = {x_i1, x_i2, ..., x_ik}`

such that there is a high probability one of the `x in X`

is a “success.” By success I mean some standard business outcome such as making a sale (in the sense of any of: propensity, appetency, up selling, and uplift modeling), clicking an advertisement, adding an account, finding a new medicine, or learning something useful.

This is common in:

- Search engines. The user is presented with a page consisting of “top results” with the hope that one of the results is what the user wanted.
- Online advertising. The user is presented with a number of advertisements in enticements in the hope that one of them matches user taste.
- Science. A number of molecules are simultaneously presented to biological assay hoping that at least one of them is a new drug candidate, or that the simultaneous set of measurements shows us where to experiment further.
- Sensor/guard placement. Overlapping areas of coverage don’t make up for uncovered areas.
- Machine learning method design. The random forest algorithm requires diversity among its sub-trees to work well. It tries to ensure by both per-tree variable selections and re-sampling (some of these issues discussed here).

In this note we will touch on key applications and some of the theory involved. While our group specializes in practical data science implementations, applications, and training, our researchers experience great joy when they can re-formulate a common problem using known theory/math and the reformulation is game changing (as it is in the case of set-scoring).

Minimal spanning trees, the basis of one set diversity metric.

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