Nonergodicity in Psychology and Neuroscience

by

Madhur Mangalam

• LAST REVIEWED: 27 October 2022
• LAST MODIFIED: 27 October 2022
• DOI: 10.1093/obo/9780199828340-0295

Introduction

Inference concerning cause-and-effect linkages in modern psychology and neuroscience follows the ergodic premise, which states that the mean response of representative samples allows predictions about the characteristics of specific sample members. However, emerging evidence suggests that empirical data in these fields often violate ergodic assumptions. In these cases, theoretical results for ensemble averages cannot be used to understand and interpret data acquired from time averages. As a result, a large corpus of research in psychology and neuroscience fails to meet the fundamental requirements of scientific inquiry, from the inability to replicate study findings to the inability to test hypotheses concerning nonlinear far-from-equilibrium dynamics that characterize the emergence and creativity of biological and psychological behavior. Ongoing work attempts to identify relevant ergodic observables for studying nonergodic processes.

General Overview

The success of psychology and neuroscience depends on making causal inferences about a phenomenon that generalizes to new individuals in the long run. Typically, studies infer the cause from statistical tests conducted on aggregated data from a random sample to infer the cause. This policy assumes that the study phenomenon is “ergodic.” Ergodicity has its roots in statistical physics, as discussed in Lebowitz and Penrose 1973 and Moore 2015. Ergodicity holds when individual-level variability is (i) homogeneous, resembling variability at the group level, and (ii) stationary, exhibiting homogeneous mean and variance over time. That is, when the time-average variance of any single trajectory resembles the ensemble-average variance of those multiple trajectories and when the time-average variance of a subset within a single trajectory resembles the time-average variance for the entire single trajectory. In short, a stochastic process is ergodic if any collection of samples represents the entire process’s average statistical properties. Conversely, a stochastic process is nonergodic when its statistics change with time. For example, a gamble can be modeled as in Peters 2019: toss a coin, and for heads you win 50 percent of your current wealth of $1, for tails you lose 40 percent. The expected wealth across infinite such gambles simply does not reflect what happens over time. As a more intuitive example of a coin toss, let us randomly choose one of the two fair coins and then toss the selected coin n times, and let the outcome be 1 for heads and 0 for tails. Then the ensemble average is ½(½ + ½) = ½, which is equal to the long-term average of ½ for either coin. Hence, this stochastic process is ergodic. Now let us randomly choose one of the two coins: one fair and the other has two heads, and then toss the selected coin n times. Although the ensemble average for this case is ½(½ + 1) = ¾, the long-term average is ½ for the fair coin and 1 for the two-headed coin. Hence, this stochastic process is nonergodic. Unfortunately, the form and behavior that distinguish life as an evolving, innovating, and adaptive process distinct from nonliving processes violate these assumptions, inevitably making them nonergodic. Mangalam and Kelty-Stephen 2021, Molenaar 2008, Peters 2019 nicely highlight the challenges posed by nonergodicity in social sciences. Specifically, group-level statistical modeling that assumes ergodicity—when it is not (which is usually the case with psychology and neuroscience studies)—cloaks artifacts of nonergodicity and conceals any genuine individual differences.

• Lebowitz, J. L., and O. Penrose. 1973. Modern ergodic theory. Physics Today 26:23–29.

  DOI: 10.1063/1.3127948

  Provides a historical perspective on the evolution of the modern ergodic theory in statistical mechanics.

• Mangalam, M., and D. G. Kelty-Stephen. 2021. Point estimates, Simpson’s paradox, and nonergodicity in biological sciences. Neuroscience & Biobehavioral Reviews 125:98–107.

  DOI: 10.1016/j.neubiorev.2021.02.017

  An excellent review attributing incidences of Simpson’s paradox observed in biomedical, behavioral, and psychological data to nonergodicity in these data. Advocates using statistical measures that encode the degree and type of nonergodicity in measurements to avoid Simpson’s paradox.

• Molenaar, P. C. M. 2008. On the implications of the classical ergodic theorems: Analysis of developmental processes has to focus on intra-individual variation. Developmental Psychobiology 50.1: 60–69.

  DOI: 10.1002/dev.20262

  Makes a convincing case that developmental processes must be analyzed at the individual level using time series data, as standard statistical techniques that capitalize variation across individuals are insensitive to large degrees of heterogeneity within individuals.

• Moore, C. C. 2015. Ergodic theorem, ergodic theory, and statistical mechanics. Proceedings of the National Academy of Sciences of the United States of America 112.7: 1907–1911.

  DOI: 10.1073/pnas.1421798112

  Discusses the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, and their influence on the fields of mathematics and statistical physics.

• Peters, O. 2019. The ergodicity problem in economics. Nature Physics 15.12: 1216–1221.

  DOI: 10.1038/s41567-019-0732-0

  Highlights that prevailing formulations of economic theory—e.g., expected utility theory—indiscriminately assume ergodicity mainly because of faulty historical precedence. Argues that carefully addressing ergodic assumptions resolves many puzzles besetting the current economic formalism.