The Unseen Species Problem Revisited

The unseen species problem is a classical problem in statistics. It asks us to, given $n$ i.i.d. samples from an unknown discrete distribution over an unknown set, predict how many never before seen outcomes would be observed if $m$ additional samples were collected. For small $m$ we show the classical but poorly understood Good-Toulmin estimator to be minimax optimal to within a factor $2$ and resolve the open problem of constructing principled prediction intervals for it. For intermediate $m$ we propose a new estimator and, if one is willing to insist on respecting a certain problem symmetry, we get a pre-asymptotic optimality result up to an explicit multiplicative constant. This is in contrast to a pre-existing asymptotic result with an unknown constant in the exponent for the Smoothed Good-Toulmin estimator (SGTE). Our estimator vastly outperforms the standard SGTE in the worst case and performs substantially better on several real data sets, namely those with many rare species. For large $m$ we follow previous authors in assuming a power law tail and show that an older estimator without known rate guarantees actually achieves a marginally better rate than subsequent work. Moreover, we give a pre-asymptotic tail bound, in contrast with the purely asymptotic results in that subsequent work. We show in all three regimes that the same methods also achieve the same rate on incidence data, without further independence assumptions, provided that the sets are of bounded size. We establish, by means of bounded size biased couplings, concentration for some natural functionals of sequences of i.i.d. discrete-set-valued random variables which may be of independent interest.

💡 Research Summary

The paper revisits the classic “unseen species” problem, which asks how many new distinct outcomes will appear when additional samples are drawn from an unknown discrete distribution. The authors treat three prediction horizons—near (r ≤ 1), intermediate (1 < r ≲ log t), and distant (log t ≲ r)—and provide, for each regime, estimators that are provably close to minimax optimality while also offering practical prediction intervals.

In the near‑future regime the classical Good‑Toulmin estimator (GTE) is examined. The authors prove that for any r∈