---
kind: concept
slug: imprecise-probability
title: Imprecise probability
summary: >-
  Replaces one probability per label with a family of compatible distributions (a credal set) built from split-CP p-values. Its spread gives uncertainty summaries that use the whole score profile, computed in linear time after sorting. Author-reported gains with LAC scores; probability baselines win under APS, RAPS and SAPS. One paper so far.
tags: [uncertainty, conformal, classification]
papers: ["2602.01667"]
related: [conformal-prediction, uncertainty-estimation, calibration, out-of-distribution-detection]
status: draft
updated_at: 2026-10-24
---

## Overview

A classifier that outputs one probability per label hides how much its
ranking of the labels is trusted. Imprecise probability keeps a family of
distributions instead of a single one. In the one paper here, that family is
built from split-CP p-values: each input gives a credal set, a set of
distributions that all agree with the p-value profile [arxiv:2602.01667 §3.1].
Its upper probability of an event is the largest normalised p-value inside
that event [arxiv:2602.01667 §3.1]. The conformal prediction regions are the
imprecise highest density regions of this set, so the threshold at each level
gives one cut of the set [arxiv:2602.01667 §3.2].

The point of the construction is to use the whole profile. A fixed-level set
size keeps only the labels above one cut and ignores how they are ordered
[arxiv:2602.01667 §1]. The summaries below use every p-value.

## Variants

All variants read the same sorted, normalised p-values. They differ in the
penalty applied to each cut or to each rank [arxiv:2602.01667 §3.3]. Each
card link points to the same paper.

### Generalised Hartley and entropy imprecision

Generalised Hartley weights each rank by the log of its size, and costs linear
time after sorting [arxiv:2602.01667 §3.3]. Entropy imprecision is the largest
entropy over the core, found by pooling adjacent masses until block averages
are nonincreasing [arxiv:2602.01667 §3.3]. Under LAC scores, the paper reports
U_1 (generalised Hartley) and U_2 above every tested fixed-level set size on
selective prediction, on all four datasets [arxiv:2602.01667 §5] [tab. 1].
Entropy imprecision scores close to U_1 on several conditions [tab. 1]. Card:
[Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](/papers/2602.01667).

### Tsallis-penalised summaries

A family of normalised weighted sums over the sorted p-values, indexed by q.
q = 0 gives fuzziness, q = 1 gives Hartley normalised by log K, q = 2 gives a
saturating form, and q going to infinity gives TV-MMI, which equals the
second-largest p-value [arxiv:2602.01667 §3.3] [app. B]. The paper does not
tune q on evaluation labels [arxiv:2602.01667 app. G]. Card:
[Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](/papers/2602.01667).

### Probability baselines

Entropy, one minus the top probability, and the second-largest probability are
the comparison points the paper uses [arxiv:2602.01667 §5] [app. D]. They are
not credal summaries. They appear here because the paper reports them against
the credal summaries and they sometimes win [arxiv:2602.01667 app. E].

## When to use which

| Situation | Use | Why |
|---|---|---|
| Classifier scored with LAC and you want one uncertainty number per input | U_1 (generalised Hartley) or U_2 | U_1 and U_2 beat every tested fixed-level set size on selective prediction, on four datasets [arxiv:2602.01667 §5] |
| Scores from APS, RAPS or SAPS | A probability baseline such as 1 - q_(1) | Under these scores the baseline beats all five credal summaries on every tested condition [arxiv:2602.01667 §5] [app. E] |
| Need a summary that compares across label counts | Normalised summaries only with a fixed K | The saturating normalisation depends on K [arxiv:2602.01667 §3.3] |
| Need to separate inputs for regression | Not the rank-grid summaries | Under a range condition on the score, every input gets the same rank core [arxiv:2602.01667 §3.4] |
| Need a validity claim after active labelling | Not from this paper | Acquired labels change the calibration set, so the paper makes no validity claim [arxiv:2602.01667 app. I] |
| Absolute atypicality matters | Report the raw maximum p-value next to the summary | Normalisation discards the raw maximum from the summary [arxiv:2602.01667 §3.1] |

## Implementation sketch

The common core is: sort the normalised p-values, take the gaps between
consecutive values as masses, then apply a rank penalty. The paper gives this
in words and formulas, not as code [arxiv:2602.01667 §3.3]. The card's
pseudocode is a sketch and has not been run against the paper's numbers
[arxiv:2602.01667 §3.3] [app. B].

```pseudocode
# Shared core of the credal summaries  [arxiv:2602.01667 §3.1] [§3.3]
# Input:  pvals: (K,) split-CP p-values for one input
# Output: masses and sorted values, from which each summary is a sum

def credal_core(pvals):
    # Sort largest first; this order is shared by every summary  [§3.3]
    alphas = sorted(pvals, reverse=True)
    # Raise the maximiser to one so the core is nonempty  [§3.1]
    alphas[0] = 1.0
    # Mass of rank j is alpha_j minus alpha_(j+1), with a zero pad  [§3.3]
    padded = alphas + [0.0]
    masses = [padded[j] - padded[j + 1] for j in range(len(alphas))]
    return alphas, masses
```

Variants differ only in the penalty applied after this step. The generalised
Hartley and integrated penalties weight each rank's mass by g(j)
[arxiv:2602.01667 §3.3]. Entropy imprecision needs an extra isotonic step on the
masses first [arxiv:2602.01667 §3.3]. The paper's Tsallis summaries are
normalised sums of the sorted values [arxiv:2602.01667 §3.3] [app. B].

## Contradictions

Only one paper cites this concept, so there are no disagreements between
papers yet. Within that paper, the headline result does not hold across score
families. With LAC scores, U_1 beats fixed-level set size everywhere and beats the
strongest probability baseline on Caltech-101 and ImageNetV2, but matches
entropy on CIFAR-100 and loses to probability baselines on Tiny ImageNet
[arxiv:2602.01667 §5] [app. E]. With
APS, RAPS and SAPS, the simple 1 - q_(1) baseline is above all five credal
summaries on selective prediction [arxiv:2602.01667 §5] [app. E]. The paper
reports both results, and so does this page.

## Papers

- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](/papers/2602.01667): defines credal sets from split-CP p-values and gives linear-time uncertainty summaries, with author-reported gains under LAC and a regression impossibility result.