The multivariate hypergeometric distribution describes sampling without
replacement from a discrete population of elements divided into multiple
categories. Addressing a gap in the literature, we tackle the challenge of
estimating discrete distributions when both the total population size and the
sizes of its constituent categories are unknown. Here, we propose a novel
solution using the hypergeometric likelihood to solve this estimation
challenge, even in the presence of severe under-sampling. We develop our
approach to account for a data generating process where the ground-truth is a
mixture of distributions conditional on a continuous latent variable, such as
with collaborative filtering, using the variational autoencoder framework.
Empirical data simulation demonstrates that our method outperforms other
likelihood functions used to model count data, both in terms of accuracy of
population size estimate and in its ability to learn an informative latent
space. We demonstrate our method's versatility through applications in NLP, by
inferring and estimating the complexity of latent vocabularies in text
excerpts, and in biology, by accurately recovering the true number of gene
transcripts from sparse single-cell genomics data.

使用多元超几何分布以及变分自动编码器框架，本文提出了一种新的方法解决估计离散分布的挑战，并在极度欠采样的情况下进行估计。通过实证数据模拟和在自然语言处理和生物学领域的应用中的表现，我们证明了该方法的多样性和准确性。