# Towards Testing Monotonicity of Distributions Over General Posets

## TL;DR

This paper investigates the sample complexity of testing distribution monotonicity over general posets, introducing a new property called bigness, and establishing lower bounds and tools for upper bounds in various poset structures.

## Contribution

It introduces the concept of bigness for distributions, derives lower bounds for testing monotonicity, and provides tools for analyzing upper bounds in general posets.

## Key findings

- Lower bound of Ω(n/log n) for testing bigness.
- Lower bounds for testing monotonicity over specific posets.
- Sublinear sample complexity bounds for certain cases.

## Abstract

In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution $p$ over a poset is monotone if, for any pair of domain elements $x$ and $y$ such that $x \preceq y$, $p(x) \leq p(y)$. To understand the sample complexity of this problem, we introduce a new property called bigness over a finite domain, where the distribution is $T$-big if the minimum probability for any domain element is at least $T$. We establish a lower bound of $\Omega(n/\log n)$ for testing bigness of distributions on domains of size $n$. We then build on these lower bounds to give $\Omega(n/\log{n})$ lower bounds for testing monotonicity over a matching poset of size $n$ and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem.