Consider a setting with $N$ independent individuals, each with an unknown
parameter, $p_i \in [0, 1]$ drawn from some unknown distribution $P^\star$.
After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i
\sim \text{Binomial}(t, p_i)$ per individual, our objective is to accurately
estimate $P^\star$. This problem arises in numerous domains, including the
social sciences, psychology, health-care, and biology, where the size of the
population under study is usually large while the number of observations per
individual is often limited. Our main result shows that, in the regime where $t
\ll N$, the maximum likelihood estimator (MLE) is both statistically minimax
optimal and efficiently computable. Precisely, for sufficiently large $N$, the
MLE achieves the information theoretic optimal error bound of
$\mathcal{O}(\frac{1}{t})$ for $t < c\log{N}$, with regards to the earth
mover's distance (between the estimated and true distributions). More
generally, in an exponentially large interval of $t$ beyond $c \log{N}$, the
MLE achieves the minimax error bound of $\mathcal{O}(\frac{1}{\sqrt{t\log
N}})$. In contrast, regardless of how large $N$ is, the naive "plug-in"
estimator for this problem only achieves the sub-optimal error of
$\Theta(\frac{1}{\sqrt{t}})$.

研究了在观测了 Bernoulli 试验结果的基础上，针对具有未知分布的参数，利用最大似然估计方法准确地估计总体分布，提出了一种在样本容量不大于总体个体个数的情况下，MLE 能达到统计最优和高效计算的方法。

学习参数人口的最大似然估计

Maximum Likelihood Estimation for Learning Populations of Parameters

This paper develops theoretical results regarding noisy 1-bit compressed
sensing and sparse binomial regression. We show that a single convex program
gives an accurate estimate of the signal, or coefficient vector, for both of
these models. We demonstrate that an s-sparse signal in R^n can be accurately
estimated from m = O(slog(n/s)) single-bit measurements using a simple convex
program. This remains true even if each measurement bit is flipped with
probability nearly 1/2. Worst-case (adversarial) noise can also be accounted
for, and uniform results that hold for all sparse inputs are derived as well.
In the terminology of sparse logistic regression, we show that O(slog(n/s))
Bernoulli trials are sufficient to estimate a coefficient vector in R^n which
is approximately s-sparse. Moreover, the same convex program works for
virtually all generalized linear models, in which the link function may be
unknown. To our knowledge, these are the first results that tie together the
theory of sparse logistic regression to 1-bit compressed sensing. Our results
apply to general signal structures aside from sparsity; one only needs to know
the size of the set K where signals reside. The size is given by the mean width
of K, a computable quantity whose square serves as a robust extension of the
dimension.

论文使用单一的凸优化程序分别对于噪声的一位非压缩感知和稀疏二项回归进行精确估计。作者展示，使用简单的凸优化程序，可以从单位比特量度中精确地估计在 R ^ n 中的 s 稀疏信号。