Algorithms for Approximating Conditionally Optimal Bounds
George Bissias
TL;DR
The paper develops a framework for non-parametric confidence regions for samples from a univariate discrete distribution on a mesh with left-bounded support by extending Learned-Miller to monotone preorders. It proves that lexicographic low and high orders are extremal among monotone preorders, enabling tight bounds via conditional-optimality concepts and pessimal bounds $B_R^*$. It then provides concrete algorithms: closed-form bounds for the lexicographic orders, a polynomial-time PTAS for quantile orders, and Monte Carlo approaches for general mesh sizes, along with refinements that reduce computational burden without sacrificing correctness. The methods yield convergent, efficient approximations of both pessimal and pointwise-optimal bounds, scalable to mesh granularity through refinement strategies that focus computation on informative coordinates. This work broadens the scope of conditionally admissible bounds to preordered sample spaces, with practical implications for constructing confidence regions in discrete settings.
Abstract
This work develops algorithms for non-parametric confidence regions for samples from a univariate distribution whose support is a discrete mesh bounded on the left. We generalize the theory of Learned-Miller to preorders over the sample space. In this context, we show that the lexicographic low and lexicographic high orders are in some way extremal in the class of monotone preorders. From this theory we derive several approximation algorithms: 1) Closed form approximations for the lexicographic low and high orders with error tending to zero in the mesh size; 2) A polynomial-time approximation scheme for quantile orders with error tending to zero in the mesh size; 3) Monte Carlo methods for calculating quantile and lexicographic low orders applicable to any mesh size.