---
kind: concept
slug: discriminability-index
title: Discriminability index (d')
summary: >-
  A d' value measures how well two classes can be told apart, from their
  distributions alone. It covers the Bayes index d'_b, which is the optimal
  error rate for two normals with equal priors, and two cheaper indices, Simpson
  and Fitter's d'_a and Egan and Clarke's d'_e. The cheaper indices can
  understate separation when variances differ. Bayes error needs integration.
tags: [classification, evaluation, signal-detection]
papers: ["2012.14331"]
related: []
status: draft
updated_at: 2026-10-19
---

## Overview

A discriminability index turns two class distributions into one number that
says how separable they are. The shared idea is that a larger value means the
classes overlap less. For two normals with equal covariance, the true d' is the
Mahalanobis distance between the means [arxiv:2012.14331 §5]. With unequal
covariances, the indices differ, and only some of them match the Bayes-optimal
error [arxiv:2012.14331 §3.1.4].

## Variants

### Bayes index d'_b

For equal priors, the Bayes index is defined from the optimal error $p(e)$ as
$d'_b = -2\,Z(p(e))$, where $Z$ is the standard normal quantile. It can also be
written with the percent correct $p(c)$ as $2\,Z(p(c))$
[arxiv:2012.14331 §3.1.4]. It is the index that matches the Bayes-optimal
classifier. Computing $p(e)$ requires integrating a normal over the decision
region. Card: [Methods to integrate multinormals and compute classification measures](/papers/2012.14331).

The paper also reads $d'_b$ as the accuracy at the point on the ROC curve that
maximises hit rate minus false-alarm rate [arxiv:2012.14331 §3.1.5].

### Simpson and Fitter d'_a

Let $\mathbf{S}_{\text{rms}} = [(\bm{\Sigma}_a + \bm{\Sigma}_b)/2]^{1/2}$ and
$\bm{d} = \bm{\mu}_a - \bm{\mu}_b$. Then $d'_a = \lVert \mathbf{S}_{\text{rms}}^{-1}
\bm{d} \rVert$ [arxiv:2012.14331 §3.1.4]. It needs no integration. In 1D, the
paper says it underestimates discriminability by up to about 30% at an extreme
ratio of standard deviations [arxiv:2012.14331 §3.1.4] [fig. 4a]. Card: the same
[paper card](/papers/2012.14331).

### Egan and Clarke d'_e

Let $\mathbf{S}_{\text{avg}} = (\mathbf{S}_a + \mathbf{S}_b)/2$, where each
$\mathbf{S}$ is a sd matrix. Then $d'_e = \lVert \mathbf{S}_{\text{avg}}^{-1}
\bm{d} \rVert$ [arxiv:2012.14331 §3.1.4]. It needs no integration. In 1D, the
paper reports that $d'_e$ approaches $d'_b$ at large separation
[arxiv:2012.14331 §3.1.4] [fig. 4c], and the card notes that $d'_e$ is the
fallback when $d'_b$ becomes infinite in 1D [arxiv:2012.14331 §5].

### Two bivariate normals with correlation

For two bivariate normals with correlation $\rho$, the paper gives a closed form
in the per-axis differences $d'_x$ and $d'_y$:
$d'^2 = (d_x'^2 + d_y'^2 - 2\rho\, d'_x d'_y) / (1 - \rho^2)$
[arxiv:2012.14331 §3.1.4]. The card does not say which of the indices above
this equals.

## When to use which

| Situation | Use | Why |
|---|---|---|
| Equal priors and you need the Bayes-optimal error | $d'_b$, from an integral of $p(e)$ | It is defined from that optimum [arxiv:2012.14331 §3.1.4] |
| Equal covariances, a closed form is enough | $d'_a$ or $d'_e$ | Both reduce to the Mahalanobis distance when $\bm{\Sigma}_a = \bm{\Sigma}_b$ [arxiv:2012.14331 §3.1.4] |
| Unequal variances, 1D, large separation | $d'_e$ | It approaches $d'_b$ [arxiv:2012.14331 §3.1.4] [fig. 4c] |
| Unequal variances, any dimension | Integrate $p(e)$ and report $d'_b$ | $d'_a$ can understate by about 30% at an extreme sd ratio, so it is not a good Bayes index [arxiv:2012.14331 §3.1.4] [fig. 4a] |
| Very high separation (d' above about 10) | Ray or quadratic integration, not Monte Carlo | Sampling is inaccurate beyond $d' \approx 10$; the integrals stay accurate to about 75 [arxiv:2012.14331 §5] [fig. 8a] |
| Custom decision boundary | Treat the value as that boundary's error | The paper defines $d'_b$ for the Bayes boundary with equal priors [arxiv:2012.14331 §3.1.4] [arxiv:2012.14331 §3.1.6] |

## Implementation sketch

Language-agnostic pseudocode of the closed-form indices. $d'_b$ needs $p(e)$
from a Bayes-error integral, which the card covers in its Method section
[arxiv:2012.14331 §2] [arxiv:2012.14331 §3.1].

```pseudocode
# Closed-form discriminability indices for two normals  [arxiv:2012.14331 §3.1.4]
# Input:  mu_a, mu_b (k,); cov_a, cov_b (k, k) covariance matrices
# Output: d_a (Simpson and Fitter), d_e (Egan and Clarke)

def closed_form_indices(mu_a, cov_a, mu_b, cov_b):
    # Mean difference, the d vector  [§3.1.4]
    d = mu_a - mu_b
    # Simpson and Fitter: whiten by the root of the averaged covariance  [§3.1.4]
    s_rms = matrix_sqrt((cov_a + cov_b) / 2)
    d_a = norm(solve(s_rms, d))
    # Egan and Clarke: whiten by the average of the sd matrices  [§3.1.4]
    s_avg = (matrix_sqrt(cov_a) + matrix_sqrt(cov_b)) / 2
    d_e = norm(solve(s_avg, d))
    return d_a, d_e


# Bayes index from the optimal error, equal priors  [§3.1.4]
def bayes_index(p_error):
    # d'_b = -2 Z(p(e)), with Z the standard normal quantile
    return -2 * normal_quantile(p_error)
```

## Pitfalls

- **Which index you report.** The Bayes index $d'_b$ is not a metric. It is
  symmetric and positive, but it breaks the triangle inequality
  [arxiv:2012.14331 §3.1.4].
- **Ordering in 1D.** The card states $d'_a \le d'_e \le d'_b$ in 1D
  [arxiv:2012.14331 §3.1.4].
- **Infinite values.** Beyond about 75, $p(e)$ underflows and the paper's
  $d'_b$ is infinite. The paper says $d'_e$ can stand in for it in 1D only
  [arxiv:2012.14331 §3.1.4] [arxiv:2012.14331 §5].
- **Sample-based fits.** Estimating means and covariances from samples uses
  the sample means and covariances, and the optimal boundary follows from them
  [arxiv:2012.14331 §3.2]. The paper leaves to the investigator whether a
  deviation from normality is small enough for the problem
  [arxiv:2012.14331 §3.6] [fig. 7b].

## Contradictions

None known among the cited papers. Only one card covers this concept so far.

## Papers

- [Methods to integrate multinormals and compute classification measures](/papers/2012.14331): defines $d'_b$, $d'_a$ and $d'_e$, and gives the integration methods that compute $p(e)$ for non-quadratic domains.