Low complexity, low probability patterns and consequences for algorithmic probability applications

TL;DR

This paper investigates the occurrence of low complexity, low probability patterns across various systems and discusses their implications for algorithmic probability applications in science, biology, physics, and machine learning.

Contribution

It identifies mechanisms behind low complexity, low probability patterns and argues for their default consideration in algorithmic probability studies.

Findings

Low complexity, low probability patterns are common in real-world systems.

Certain mechanisms cause these patterns, affecting probability predictions.

Implications for physics, biology, and machine learning are discussed.

Abstract

Developing new ways to estimate probabilities can be valuable for science, statistics, and engineering. By considering the information content of different output patterns, recent work invoking algorithmic information theory has shown that a priori probability predictions based on pattern complexities can be made in a broad class of input-output maps. These algorithmic probability predictions do not depend on a detailed knowledge of how output patterns were produced, or historical statistical data. Although quantitatively fairly accurate, a main weakness of these predictions is that they are given as an upper bound on the probability of a pattern, but many low complexity, low probability patterns occur, for which the upper bound has little predictive value. Here we study this low complexity, low probability phenomenon by looking at example maps, namely a finite state transducer, natural time series data, RNA molecule structures, and polynomial curves. Some mechanisms causing low complexity, low probability behaviour are identified, and we argue this behaviour should be assumed as a default in the real world algorithmic probability studies. Additionally, we examine some applications of algorithmic probability and discuss some implications of low complexity, low probability patterns for several research areas including simplicity in physics and biology, a priori probability predictions, Solomonoff induction and Occam's razor, machine learning, and password guessing.