## Agnostic Insurability of Model Classes

** Narayana Santhanam, Venkat Anantharam**; 16(Nov):2329−2355, 2015.

### 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.