[Paper Review] Computable Randomness is Inherently Imprecise

Gert de Cooman, Jasper De Bock|arXiv (Cornell University)|Mar 2, 2017

Computability, Logic, AI Algorithms14 references3 citations

TL;DR

This paper introduces a game-theoretic framework for computable randomness using interval forecasts instead of precise probabilities, showing that randomness can be inherently imprecise. It proves that every infinite binary sequence is computably random with respect to a filter of interval forecasts, and that this filter may lack a smallest interval—demonstrating that randomness cannot always be captured by precise models without loss of precision.

ABSTRACT

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise, forecasting systems, and study its properties. The richer mathematical structure that thus arises lets us better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every infinite sequence of zeroes and ones has an associated filter of intervals with respect to which it is computably random. It may happen that none of these intervals is precise, which justifies the title of this paper. We illustrate this by showing that computable randomness associated with non-stationary precise forecasting systems can be captured by a stationary interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision.

Motivation & Objective

• To extend the concept of computable randomness to imprecise forecasting systems using interval probabilities.
• To investigate whether computable randomness can be defined without relying on precise probability forecasts.
• To understand the mathematical structure of randomness when forecasts are intervals rather than point values.
• To explore the trade-off between model simplicity (stationarity) and precision in forecasting systems.
• To demonstrate that some sequences cannot be computably random under any precise stationary forecast, but are random under an interval forecast.

Proposed method

• Uses the game-theoretic probability framework, modeling forecasts as interval bounds for buying and selling prices of gambles.
• Defines a supermartingale multiplier for interval forecasts, where the expected value of the multiplier does not increase under the interval forecast.
• Applies computability constraints to ensure that all processes (forecasts, multipliers) are effectively computable.
• Constructs specific computable supermartingale multipliers to test randomness under interval forecasts, including for non-stationary and stationary cases.
• Uses a filtering approach to identify all interval forecasts with respect to which a given sequence is computably random.
• Employs a proof by contradiction to show that certain interval forecasts must be part of the randomness filter, even when precise forecasts are not.

Experimental results

Research questions

• RQ1 Can computable randomness be meaningfully defined when forecasts are intervals rather than precise probabilities?
• RQ2 Is there always a set of interval forecasts with respect to which a given infinite binary sequence is computably random?
• RQ3 Can a stationary interval forecast capture the randomness of a sequence that is not computably random under any precise stationary forecast?
• RQ4 Can the filter of interval forecasts for which a sequence is computably random fail to have a smallest element?
• RQ5 Is it possible for the smallest interval in such a filter to be non-degenerate (i.e., not a single point), implying inherent imprecision in randomness?

Key findings

• Every infinite binary sequence has a non-empty filter of interval forecasts with respect to which it is computably random.
• There exist sequences that are not computably random under any precise stationary forecasting system, but are computably random under a stationary interval forecast.
• The filter of interval forecasts for which a sequence is computably random may not have a smallest element, indicating that randomness is inherently imprecise.
• Even when a smallest interval exists in the filter, it may be non-degenerate (i.e., have positive length), confirming that randomness cannot always be captured precisely.
• A computable supermartingale multiplier for a non-stationary precise forecast can be simulated by a computable supermartingale multiplier for a stationary interval forecast, which must then be less precise.
• The proof shows that if a sequence were not computably random under a symmetric interval forecast $[1/2 - ho, 1/2 + ho]$, a contradiction arises with the known computable randomness under a precise $1/2$ forecast.