Randomness is inherently imprecise

TL;DR

This paper explores how imprecision in forecasting systems affects the concept of randomness, showing that sequences can be random for imprecise models even when not for precise ones, highlighting the complexity of randomness.

Contribution

It introduces a martingale-theoretic framework for randomness with interval forecasts, revealing new properties and distinctions from precise forecasting systems.

Findings

Sequences can be random for interval forecasts without being random for any precise system.

Imprecise randomness cannot always be reduced to oversimplified precise models.

The set of sequences random for non-vacuous interval forecasts is meagre, similar to precise systems.

Abstract

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise, forecasting systems, and study their properties. The richer mathematical structure that thus arises lets us, amongst other things, better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every sequence of binary outcomes has an associated filter of intervals it is random for. It may happen that none of these intervals is precise -- a single real number -- which justifies the title of this paper. We illustrate this by showing that randomness associated with non-stationary precise forecasting systems can be captured by a constant interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision. But imprecise randomness can't always be explained away as a result of oversimplification: we show that there are sequences that are random for a constant interval forecast, but never random for any {\comp} (more) precise forecasting system. We also show that the set of sequences that are random for a non-vacuous interval forecasting system is meagre, as it is for precise forecasting systems.