---
kind: concept
slug: conformal-prediction
title: Conformal prediction
summary: >-
  Turns model scores or errors into sets, alarm thresholds or safety levels at a chosen error level, from labelled calibration data and one quantile. Shown for LLM planners (a singleton set means act), VLA failure alarms, classifier credal summaries and a quadruped value-function filter. Assumes exchangeable calibration data. Four author-reported papers.
tags: [uncertainty, calibration, robotics, llm-planning, failure-detection, safety]
papers: ["2307.01928", "2602.01667", "2606.21386", "2412.09989"]
related: [uncertainty-estimation, calibration, anomaly-detection, out-of-distribution-detection, llm-planning, safety-filter]
status: draft
updated_at: 2026-10-25
---

## Overview

Conformal prediction wraps a model with a rule that returns a set of candidate
answers instead of one answer. The set is sized so that the true answer is in
it with at least a chosen probability, 1−ε, on new inputs drawn from the same
distribution as the calibration data [arxiv:2307.01928 §3.1] [eq. 1]. The
method needs a score for each candidate and a labelled calibration set. Each
paper here plugs in a different score: the LLM's confidence in each option
[arxiv:2307.01928 §3.1], a classifier's probability for each label
[arxiv:2602.01667 app. D], a failure detector's score per control step
[arxiv:2606.21386 §3.3], or the error of a learned value function on held-out
states [arxiv:2412.09989 §V-A.4]. The last one returns a level, not a set.

The split form used here works in three steps. Score each calibration example
by how unlike its correct answer the model found it. Take the empirical quantile
of those scores at a level set by N and ε, giving one threshold q̂. At test
time, keep every candidate whose score clears that threshold [arxiv:2307.01928
§3.1] [eq. 2].

Four papers apply this so far. The first applies it to language-model planners:
a planner lists candidate next steps, the candidates become a multiple-choice
question, and the set size decides whether a robot acts or asks for help
[arxiv:2307.01928 §2] [arxiv:2307.01928 §3.2]. The paper reports this with
author-reported results on tabletop and mobile-manipulation hardware
[arxiv:2307.01928 §4]. The second uses the same kind of calibration differently.
It does not form sets. It sets alarm thresholds on the scores of a failure
detector for finetuned robot policies [arxiv:2606.21386 §3.3]. The third uses
the split-CP p-values of each input to build uncertainty summaries. It does not
make a decision set either. It ranks inputs by how uncertain they are, and
compares its summaries with set size [arxiv:2602.01667 §3.3] [§5]. The fourth
calibrates a learned reachability value function for a quadruped. It takes a
quantile of prediction errors on simulated states, and uses the result as the
level at which a safety filter starts to act [arxiv:2412.09989 §V-A.4]
[§V-B.2]. This is the only paper here where the output is a level that gates a
controller, not a set or an alarm.

The evidence is narrow. The planner paper's results come from its own tabletop
and mobile-manipulation runs with PaLM-2L [arxiv:2307.01928 §4]. The detector
paper's come from six real-world tasks and Libero-Plus, with two VLA models
[arxiv:2606.21386 §4]. The credal paper's come from image classifiers on four
datasets [arxiv:2602.01667 §5]. The quadruped paper's come from simulation with
100 trials per row and ten hardware trials per row [arxiv:2412.09989 §VI-D]
[§VII].

## Variants

### Single-step set over multiple-choice options

Options A to D come from the LLM, and a fifth option E ("an option not listed")
is appended so that the right action can fall outside the list. The score is
one minus the normalised confidence of the correct option. The threshold is set
from N calibration scenarios and the chosen ε [arxiv:2307.01928 §2]
[arxiv:2307.01928 §3.1]. A singleton set means the robot acts; a larger set
means it asks a human to pick [arxiv:2307.01928 §3.2]. Card: [Robots That Ask
For Help](/papers/2307.01928).

### Dataset-conditional bound

The marginal guarantee holds on average over draws of the calibration set. The
dataset-conditional form instead holds with probability 1−δ over the calibration
set itself. It needs an adjusted ε̂ taken from an inverse Beta quantile, with
v = ⌊(N+1)ε̂⌋ [arxiv:2307.01928 §3.1] [eq. 2]. The paper uses δ = 0.01 and
N = 400 [arxiv:2307.01928 §3.1] [app. A.7].

### Sequence-level calibration for multi-step plans

Calibration uses whole correct episodes. Each episode gets a score equal to the
lowest per-step confidence of its correct options, and one threshold covers the
episode [arxiv:2307.01928 §3.3] [eq. 4]. At test time the set is rebuilt at
each step from the context reached so far, using the same threshold. The robot
acts on singletons and asks otherwise [arxiv:2307.01928 §3.3] [alg. 1]. The
paper's hardware tabletop run reached task success 0.74 against a target of
0.75, with the multi-step sets [arxiv:2307.01928 §4.2] [tab. 1].

### Several acceptable options

When more than one option is correct, the calibration label is the correct
option that the LLM ranks highest. The set then has to contain that option, and
the human is asked to choose the highest-ranked correct one [arxiv:2307.01928
§3.3] [app. A.3] [app. A.4]. The appendix also reports a bimanual run on an
all-ambiguous distribution [arxiv:2307.01928 app. A.5].

### Constant band for a failure-detector threshold

A detector gives a score per control step, and one threshold is set per score
from calibration data. The alarm fires when either score is at or above its
threshold [arxiv:2606.21386 §3.3]. The paper calibrates each threshold with a
constant conformal band at significance level 0.05, using 20 successful
calibration rollouts, which are removed from the test set [arxiv:2606.21386
app. D]. It does not use a time-dependent band, because episode lengths vary,
for example in its Drawer task [arxiv:2606.21386 §3.3] [app. D]. The band's
formula is not on the card, so read the paper's appendix D before implementing
it. Card: [VLA-FAIL](/papers/2606.21386).

### Calibrated level for a learned value function

A learned value function estimates how close a robot state is to failure. Its
prediction error is measured on held-out states whose true value comes from a
reachability solver [arxiv:2412.09989 §V-A.2]. The paper picks a confidence
level β and a violation rate ε, finds the largest k that a binomial tail bound
allows, and takes the (N − k)/N quantile of the errors as a level δ. With
probability at least 1 − β, the fraction of samples whose predicted value
overestimates the true value by more than δ is at most ε [arxiv:2412.09989
§V-A.4] [eq. 11] [eq. 12]. The paper calls the proof a split-conformal argument
with a Beta distribution [arxiv:2412.09989 app. A] [eq. 17]. At run time, the
safety filter passes the nominal command through while the calibrated value is
above δ, and solves a QP below it [arxiv:2412.09989 §V-B.2] [eq. 15]. With
β = 10^-12 and N = 10^7 samples, the paper tabulates δ = 0.22, 0.49 and 0.99 m
for ε = 10^-1, 10^-2 and 10^-3, and uses 0.49 m [arxiv:2412.09989 tab. II]
[§V-A.4]. Card: [One Filter to Deploy Them All](/papers/2412.09989).

### Credal summaries of the p-value profile

Keep all the split-CP p-values for an input, not only the labels above one
cut-off. The p-values define a family of distributions that are all compatible
with them, called a credal set. Its spread gives uncertainty numbers that
take every label into account, and these come out in linear time once the
p-values are sorted [arxiv:2602.01667 §3.1] [§3.3]. Before the summaries are
computed, the largest p-value is raised to one. The raw maximum is kept apart
as conformal credibility [arxiv:2602.01667 §3.1].

The summaries include generalised Hartley, entropy imprecision, TV-MMI and
Tsallis-penalised forms [arxiv:2602.01667 §3.3] [app. B]. TV-MMI equals the
second-largest p-value [arxiv:2602.01667 §3.3]. Under LAC scores, it is only a
relabelling of the second-highest probability, so it adds no new ranking
[arxiv:2602.01667 app. D] [app. F]. The paper reports these on image
classifiers. With LAC, the Tsallis summaries U_1 and U_2 beat every fixed-level
set size on selective prediction, with paired gains of 0.50 to 1.86 points
[arxiv:2602.01667 §5] [tab. 1]. With APS, RAPS and SAPS, the simple baseline
1 − q_(1) has a higher mean than all five summaries on every tested condition
[arxiv:2602.01667 §5] [app. E]. Card: [Beyond Prediction-Set
Size](/papers/2602.01667).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Model gives option probabilities and you can label about 400 i.i.d. scenarios | Single-step set, marginal guarantee | Gives the target success level on new scenarios from the same distribution [arxiv:2307.01928 §3.1] [app. A.7] |
| You need the guarantee to hold for the calibration draw itself | Dataset-conditional bound (eq. 2) | Stated with δ = 0.01 [arxiv:2307.01928 §3.1]. The released notebooks do not compute it (see Contradictions) |
| Plans have several steps and calibration data are whole episodes | Sequence-level score and causal sets | One threshold for the episode [arxiv:2307.01928 §3.3]. Only valid while earlier actions are correct [arxiv:2307.01928 app. A.2] |
| Several different steps would be acceptable in one scene | Label with the highest-ranked correct option | The paper defines the label this way [arxiv:2307.01928 app. A.3] |
| The model's token probabilities are not exposed | Not covered by this paper | The authors did not use GPT-4 for this reason [arxiv:2307.01928 §4.3] |
| You are choosing between set-based and free-form confidence scores | Multiple-choice options with conformal thresholds | Raw sequence probabilities gave larger sets and more help in the paper's simulation [arxiv:2307.01928 app. A.9] |
| You want an alarm on a policy's per-step score and have no failure data, only successful rollouts | Constant band on the detector score | This is the paper's setup, with 20 successful calibration rollouts [arxiv:2606.21386 app. D] |
| Episode lengths differ a lot across tasks | Constant band, not time-dependent | The paper gives episode-length variation as its reason [arxiv:2606.21386 §3.3] |
| Calibration data are few | Expect unstable thresholds | With 20 rollouts and a low significance level, one outlier can raise the threshold and lower recall [arxiv:2606.21386 app. D] |
| You need an uncertainty ranking from conformal p-values, not a decision set, on a classifier with LAC scores | Credal summaries U_1 or U_2 | Beat fixed-level set size on selective prediction on four image datasets, author-reported [arxiv:2602.01667 §5] [tab. 1] |
| Scores are APS, RAPS or SAPS | Simple baseline 1 − q_(1), not the credal summaries | It had the higher mean on every tested condition [arxiv:2602.01667 app. E] |
| Calibration has fewer than 500 examples, randomised APS, α = 0.01 | Raw set size | TV-MMI overtook it only from 500 calibration examples [arxiv:2602.01667 app. F] |
| Regression with absolute-residual-type scores | Expect no per-input ranking from the rank-core summaries | Under a range condition, every input gets the same rank core [arxiv:2602.01667 §3.4] |
| A learned value function gates a safety override, and you can compute ground-truth values offline for held-out states | Calibrated level δ from the binomial rule and an error quantile | Gives an overestimation bound at chosen β and ε on the validation distribution [arxiv:2412.09989 §V-A.4] [tab. II] |
| Deployment differs from the calibration distribution | Do not treat δ as a bound on the real robot | The authors say the guarantee does not carry over under distribution shift [arxiv:2412.09989 §VIII] |

The paper reports 0.67 help for the conformal method against 0.81 for the
Simple Set baseline at the same plan success of 0.87 on a mobile manipulator
[arxiv:2307.01928 §4.3] [tab. 2]. The baseline is cumulative thresholding on
LLM confidence, not conformal, and its ε was tuned to match plan success
[arxiv:2307.01928 §4.2] [arxiv:2307.01928 §4.3]. The comparison is therefore not
at an equal target level.

For the failure detector, the paper's threshold-dependent tables use the
constant band, and the authors say these tables are less reliable for comparing
methods than their threshold-free tables [arxiv:2606.21386 app. D]. Cite the
threshold-free results when comparing detectors. The paper claims no formal
safety guarantee, and says its method is empirical [arxiv:2606.21386 §6]. Do
not read the band as the 1−ε guarantee from the planner paper. The two papers
use different calibration data and different score types, so their results are
not directly comparable.

## Implementation sketch

The core is split conformal calibration plus a singleton test. Notation follows
the paper: N calibration scenarios, five option labels, and ε set by the user
[arxiv:2307.01928 §2] [arxiv:2307.01928 §3.1].

```pseudocode
# Calibrate a threshold from labelled scenarios  [§3.1] [eq. 1]
# Input:  probs (N, 5) normalised option confidences
# Input:  labels (N,) index of the correct option
# Input:  eps, the user-chosen error level

def calibrate(probs, labels, eps):
    # Nonconformity: one minus confidence of the correct option  [§3.1]
    scores = 1 - probs[range(N), labels]
    # Quantile level ceil((N+1)(1-eps))/N, capped at 1  [eq. 1]
    level = min(1.0, ceil((N + 1) * (1 - eps)) / N)
    # Take the upper empirical quantile of the scores  [§3.1]
    return quantile(scores, level, method="higher")

def prediction_set(probs, qhat):
    # Keep each option whose confidence clears 1 - qhat  [§3.1]
    return [y for y in options if probs[y] >= 1 - qhat]

def act_or_ask(probs, qhat):
    # Singleton: act. Otherwise ask a human to choose  [§3.2]
    kept = prediction_set(probs, qhat)
    if len(kept) == 1:
        return act(kept[0])
    return ask_human(kept)
```

Where variants differ:

- **Dataset-conditional.** Replace the level in `calibrate` with the adjusted
  ε̂ from the Beta quantile in eq. 2 [arxiv:2307.01928 §3.1] [eq. 2].
- **Multi-step.** Score each correct episode by the minimum step confidence
  [eq. 4], set one threshold on episodes, and call `act_or_ask` at each step
  with the current context [alg. 1] [arxiv:2307.01928 §3.3].
- **Several acceptable options.** Set the calibration label to the
  highest-ranked correct option [arxiv:2307.01928 app. A.3].
- **Constant band on a detector score.** Calibrate one threshold per score on
  successful rollouts at significance 0.05, then alarm when
  `score >= threshold` for either score [arxiv:2606.21386 §3.3] [app. D]. The
  band's exact formula is not on the card, so read the paper's appendix D
  before implementing it.
- **Credal summaries.** Take the same (K,) vector of split-CP p-values. Sort it
  in decreasing order and raise the largest to one. The summaries are then
  weighted sums over the sorted values [arxiv:2602.01667 §3.1] [§3.3] [app. B].
  The entropy term needs an isotonic step (pool adjacent violators) on the
  masses. The paper describes that step in words, not as code
  [arxiv:2602.01667 §3.3].
- **Calibrated value level.** Compute the errors s_i = V_pred − V_true on
  held-out states with ground truth. Choose β and ε, then take δ from the
  binomial rule in eq. 11 and the error quantile in §V-A.4. The paper gives the
  k-rule in eq. 11 and app. A, and this card gives only its outline, so read
  them before coding. Switch the filter when the calibrated value is at or
  below δ [arxiv:2412.09989 §V-B.2] [eq. 15].

## Contradictions

- **Dataset-conditional guarantee: paper vs released code.** The paper states
  the eq. 2 guarantee with δ = 0.01 and an adjusted ε̂ [arxiv:2307.01928 §3.1].
  The released notebooks use the requested ε directly, which gives only the
  marginal guarantee. The card's Paper vs code section records the search that
  found no Beta-quantile adjustment ([card](/papers/2307.01928)).
- **Calibration size: paper vs released code.** The paper uses N = 400
  [arxiv:2307.01928 app. A.7]. The card records 100 in the tabletop notebook
  and 200 in the mobile one ([card](/papers/2307.01928)).
- **LLM: paper vs released code.** The paper's main results use PaLM-2L
  [arxiv:2307.01928 §4]. The released notebooks call GPT-3.5, which the README
  discloses ([card](/papers/2307.01928)). Results from one model do not carry
  over to the other without a check.
- **Set size as an uncertainty signal depends on the score.** Under LAC, the
  credal summaries beat fixed-level set size on classifiers. Under APS, RAPS and
  SAPS, the raw baseline 1 − q_(1) beats the summaries [arxiv:2602.01667 §5]
  [app. E]. The planner paper uses set size as its act-or-ask rule and does not
  compare summaries of the set [arxiv:2307.01928 §3.2]. These are different
  tasks and scores, so the results do not conflict. The choice of score still
  changes which signal ranks best.
- **Value-function level: paper vs released config.** The paper switches the
  filter at δ = 0.49 m on the uncalibrated output [arxiv:2412.09989 §V-B.2]
  [tab. II]. The released hardware config adds a calibration offset and switches
  at a different level, which the card puts at about 0.85 m in raw network units
  ([card](/papers/2412.09989)). The paper tabulates ε = 10^-3, which the released
  calibration grid does not cover ([card](/papers/2412.09989)).

## Papers

- [Robots That Ask For Help](/papers/2307.01928): applies split conformal prediction to LLM next-step choices, so a robot acts on singleton sets and asks for help otherwise.
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](/papers/2602.01667): turns split-CP p-values into credal uncertainty summaries that, with LAC scores, beat fixed-level set size on selective prediction (author-reported).
- [VLA-FAIL](/papers/2606.21386): sets alarm thresholds for a failure detector on robot policies with a constant conformal band fitted on successful rollouts.
- [One Filter to Deploy Them All](/papers/2412.09989): calibrates a learned reachability value network with a binomial-tail quantile rule and uses the resulting level to switch a quadruped safety filter (author-reported).