On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach

TL;DR

This paper reviews methods for estimating confidence intervals for binomial proportions in astronomy, highlighting the superiority of Bayesian approaches over traditional methods like normal approximation and Clopper-Pearson, especially in small samples.

Contribution

It demonstrates the advantages of Bayesian binomial confidence intervals using the beta distribution and criticizes the common but flawed traditional methods.

Findings

Bayesian intervals are more accurate in small samples.

Normal approximation often underestimates uncertainty.

Clopper & Pearson intervals can be overly conservative.

Abstract

I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small-to-intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, AGN fraction, SMBH fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals--the 'normal approximation' and the Clopper & Pearson approach--are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (i) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. R, matlab, mathematica, IDL, python); and (ii) to demonstrate convincingly the major flaws of both the 'normal approximation' and the Clopper & Pearson approach for error estimation.