Maximum Entropy Interval Aggregations

Given a probability distribution $\mathbf{p}=(p_{1},\ \ldots,p_{n})$ and an integer $1 \leq m < n$, we say that $\mathbf{q}=(q_{1},\ \ldots,\ q_{m})$ is a contiguous m-aggregation of p if there exist indices $0=i_{0} < i_{1} < \cdots < i_{m-1} < i_{m}=n$ such that for each $j=1, \ldots, m$ it holds that $q_{j}= \sum_{k=i_{i-1}+1}^{x_{j}}p_{k}$. In this paper, we consider the problem of efficiently finding the contiguous m-aggregation of maximum entropy. We design a dynamic programming algorithm that solves the problem exactly, and two more time-efficient greedy algorithms that provide slightly sub-optimal solutions. We also discuss a few scenarios where our problem matters.