Estimating Probabilities of Unobserved Species

Definitions and mathematical statement

Let $\mathcal S$ denote the species set in the supplied sampling model, and let $p=(p_t)_{t\in\mathcal S}$ be its unknown probability vector, with $p_t\geq 0$ and $\sum_{t\in\mathcal S}p_t=1$. Let $n$ be the database sample size and $X_t$ the number of observations of species $t$, so that the supplied model is \[ X=(X_t)_{t\in\mathcal S}\sim\operatorname{Multinom}(n,p),\qquad \sum_{t\in\mathcal S}X_t=n. \] The source describes $\mathcal S$ as very large but does not specify its cardinality or whether it is known. The usual multinomial notation applies to a finite species set; an intended infinite-species formulation would require clarification.

For a species $s\in\mathcal S$ whose observed count satisfies $X_s=0$, the stated research task is to construct an estimator $\widehat p_s=\widehat p_s(X,s)$ of $p_s$, or an estimator of \[ I_s=-\log_{10}p_s, \] and to quantify the uncertainty of the chosen estimate. The real-valued target $I_s$ is defined when $p_s>0$. The model does not exclude $p_s=0$; treating $I_s$ as $+\infty$ there would require an explicitly adopted extended-real convention. Observing $X_s=0$ alone does not specify which boundary convention is intended.

This is a research objective rather than a complete mathematical proposition. The source does not define statistical soundness, a loss or optimality criterion, an uncertainty procedure or its required guarantees, additional restrictions on $p$, or whether $s$ is fixed before sampling or selected after examining the database. Consequently, no universal performance quantifiers or precise theorem are asserted in this draft.

Short English statement

Estimate the population probability of a species absent from a multinomial database sample, or its negative base-ten logarithm, and describe the estimate's uncertainty. The supplied forensic-statistics research theme leaves the statistical guarantees, species-selection rule, model restrictions, and treatment of zero probability unresolved.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002673$, title Estimating Probabilities of Unobserved Species, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems, proposed year $2014$. No individual proposer is supplied.
• Supplied source citation: New Horizons in Statistical Decision Theory (2015), Oberwolfach Reports, pp. 2297--2352, DOI 10.4171/owr/2014/41; https://doi.org/10.4171/owr/2014/41. The background describes these reports as records of mathematical workshops and their open-problem discussions. This report-level reference supplies no exact passage or individual attribution for the task.
• Supplied literature reference: Oberwolfach Report 41/2014, https://doi.org/10.4171/owr/2014/41. Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified here. The supplied status is open; the accompanying assessment says that the extracted text identifies a forensic-statistics research theme rather than a determinate proposition, with no exact claim available to classify as proved or false. The source requests independent verification of time-sensitive status and cited claims.