# Agnostic Insurability of Model Classes

Narayana Santhanam, Venkat Anantharam.

Year: 2015, Volume: 16, Issue: 70, Pages: 2329−2355

#### Abstract

Motivated by problems in insurance, our task is to predict finite upper bounds on a future draw from an unknown distribution $p$ over natural numbers. We can only use past observations generated independently and identically distributed according to $p$. While $p$ is unknown, it is known to belong to a given collection $\mathcal{P}$ of probability distributions on the natural numbers.

The support of the distributions $p \in \mathcal{P}$ may be unbounded, and the prediction game goes on for *infinitely* many draws. We are allowed to make observations without predicting upper bounds for some time. But we must, with probability $1$, start and then continue to predict upper bounds after a finite time irrespective of which $p \in \mathcal{P}$ governs the data.

If it is possible, without knowledge of $p$ and for any prescribed confidence however close to $1$, to come up with a sequence of upper bounds that is never violated over an infinite time window with confidence at least as big as prescribed, we say the model class $\mathcal{P}$ is *insurable*. We completely characterize the insurability of any class $\mathcal{P}$ of distributions over natural numbers by means of a condition on how the neighborhoods of distributions in $\mathcal{P}$ should be, one that is both necessary and sufficient.