Measures of classification bias derived from sample size analysis

TL;DR

The paper tackles the problem of measuring classifier bias across demographics by reframing bias as the detectability of performance differences, quantified through the sample size required to statistically distinguish two demographic error rates. It introduces a chi-squared based bias metric $N_{\alpha,\beta}(\varepsilon_1,\varepsilon_2)$, derived via an $\arcsin(\sqrt{\cdot})$ transformation, and shows that larger $N$ corresponds to lower bias, with key properties such as invariance to $\alpha$ and $\beta$ up to a constant and $\lim_{(\varepsilon_1,\varepsilon_2)\to(0^+)} N_{\alpha,\beta}=\infty$ when no bias exists. The authors compare this metric to the conventional difference and ratio of error rates, demonstrating that it can yield different bias rankings and emphasizing its stability near zero error rates. They extend the discussion to multiple demographics, discuss test-agnostic desirable properties, and outline future directions, including Bayesian and information-theoretic perspectives and extensions to more complex demographic structures. Overall, the work provides a simple, interpretable, and principled ranking tool for algorithmic bias grounded in fundamental statistical detectability.

Abstract

We propose the use of a simple intuitive principle for measuring algorithmic classification bias: the significance of the differences in a classifier's error rates across the various demographics is inversely commensurate with the sample size required to statistically detect them. That is, if large sample sizes are required to statistically establish biased behavior, the algorithm is less biased, and vice versa. In a simple setting, we assume two distinct demographics, and non-parametric estimates of the error rates on them, e1 and e2, respectively. We use a well-known approximate formula for the sample size of the chi-squared test, and verify some basic desirable properties of the proposed measure. Next, we compare the proposed measure with two other commonly used statistics, the difference e2-e1 and the ratio e2/e1 of the error rates. We establish that the proposed measure is essentially different in that it can rank algorithms for bias differently, and we discuss some of its advantages over the other two measures. Finally, we briefly discuss how some of the desirable properties of the proposed measure emanate from fundamental characteristics of the method, rather than the approximate sample size formula we used, and thus, are expected to hold in more complex settings with more than two demographics.

Figures (3)

• Figure 1: A colormap of the values of $\log N_{\alpha,\beta} (\varepsilon_1,\varepsilon_2)$.
• Figure 2: From left to right: the contours of the functions $D_{err}(\varepsilon_1, \varepsilon_2) = \varepsilon_2 - \varepsilon_1$, $R_{err}(\varepsilon_1, \varepsilon_2) = \varepsilon_2 / \varepsilon_1$, and $N_{\alpha,\beta} (\varepsilon_1,\varepsilon_2)$ in Eq. \ref{['eq:twoproportions-arcSin']}.
• Figure 3: In each image, in a $30 \times 30$ grid, a black dot corresponds to a misclassified sample, and a white dot to a correctly classified sample. Top:$\varepsilon = 0.00$ (left) against $\varepsilon = 0.01$ (right). Bottom:$\varepsilon = 0.49$ (left) against $\varepsilon = 0.50$ (right).

Theorems & Definitions (16)

• Proposition 3.1
• proof
• Remark 3.1
• Remark 3.2
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 4.1
• proof