Einstein from Noise: Statistical Analysis

Abstract “Einstein from noise” (EfN) is a prominent example of the model bias phenomenon, where systematic errors in the statistical model lead to spurious but consistent estimates. In the EfN experiment, one falsely believes that a set of observations contains noisy, shifted copies of a template signal (e.g., an Einstein image), whereas in reality, it contains only pure noise observations. To estimate the signal, the observations are first aligned with the template using cross-correlation and then averaged. Although the observations contain nothing but noise, it was recognized early on that this process produces a signal that resembles the template signal! This model bias pitfall was at the heart of a central scientific controversy about validation techniques in structural biology. This paper provides a comprehensive statistical analysis of the EfN phenomenon above. We show that the Fourier phases of the EfN estimator (namely, the average of the aligned noise observations) converge to the Fourier phases of the template signal, thereby explaining the observed structural similarity. Additionally, we prove that the convergence rate of Fourier phases is inversely proportional to the number of noise observations and, in the high-dimensional regime, to the Fourier magnitudes of the template signal. Moreover, in the high-dimensional regime, the EfN estimator converges to a scaled version of the template signal. This work not only deepens the theoretical understanding of the EfN phenomenon but also highlights potential pitfalls in template matching techniques and emphasizes the need for careful interpretation of noisy observations across disciplines in engineering, statistics, physics, and biology. Competing Interest Statement The authors have declared no competing interest. Footnotes We have made small adjustments and rearranged the existing proofs.