This paper studies the problem of {\em learning} the probability distribution
$P_X$ of a discrete random variable $X$ using indirect and sequential samples.
At each time step, we choose one of the possible $K$ functions, $g_1, \ldots,
g_K$ and observe the corresponding sample $g_i(X)$. The goal is to estimate the
probability distribution of $X$ by using a minimum number of such sequential
samples. This problem has several real-world applications including inference
under non-precise information and privacy-preserving statistical estimation. We
establish necessary and sufficient conditions on the functions $g_1, \ldots,
g_K$ under which asymptotically consistent estimation is possible. We also
derive lower bounds on the estimation error as a function of total samples and
show that it is order-wise achievable. Leveraging these results, we propose an
iterative algorithm that i) chooses the function to observe at each step based
on past observations; and ii) combines the obtained samples to estimate $p_X$.
The performance of this algorithm is investigated numerically under various
scenarios, and shown to outperform baseline approaches.

使用间接和序列样本学习离散随机变量 X 的概率分布 P_X 的问题已经在本文中得到研究，其中提出了一种基于几个条件限制的迭代算法，成功地在不同情况下实现了更好的效果比基线方法更好。