Simplicity bias, algorithmic probability, and the random logistic map

Abstract

Simplicity bias is an intriguing phenomenon prevalent in various input–output maps, characterized by a preference for simpler, more regular, or symmetric outputs. Notably, these maps typically feature high-probability outputs with simple patterns, whereas complex patterns are exponentially less probable. This bias has been extensively examined and attributed to principles derived from algorithmic information theory and algorithmic probability. In a significant advancement, it has been demonstrated that the renowned logistic map xk+1=μxk(1−xk), a staple in dynamical systems theory, and other one-dimensional maps exhibit simplicity bias when conceptualized as input–output systems. Building upon this work, our research delves into the manifestations of simplicity bias within the random logistic map, specifically focusing on scenarios involving additive noise. This investigation is driven by the overarching goal of formulating a comprehensive theory for the prediction and analysis of time series.Our primary contributions are multifaceted. We discover that simplicity bias is observable in the random logistic map for specific ranges of μ and noise magnitudes. Additionally, we find that this bias persists even with the introduction of small measurement noise, though it diminishes as noise levels increase. Our studies also revisit the phenomenon of noise-induced chaos, particularly when μ=3.83, revealing its characteristics through complexity-probability plots. Intriguingly, we employ the logistic map to illustrate a paradoxical aspect of data analysis: more data adhering to a consistent trend can occasionally lead to reduced confidence in extrapolation predictions, challenging conventional wisdom.We propose that adopting a probability-complexity perspective in analyzing dynamical systems could significantly enrich statistical learning theories related to series prediction and analysis. This approach not only facilitates a deeper understanding of simplicity bias and its implications but also paves the way for novel methodologies in forecasting complex systems behavior, especially in scenarios dominated by uncertainty and stochasticity.