Uncertainty Quantification, Calibration, and Conformal Prediction

Visual: reliability diagram plus conformal prediction-set construction showing calibration data, coverage guarantee, uncertainty type, and AV decision use.

Uncertainty quantification is the practice of making model uncertainty explicit enough to support decisions. Calibration asks whether predicted probabilities match empirical frequencies. Conformal prediction wraps a predictor with finite-sample coverage guarantees under exchangeability assumptions.

Related docs

• Gaussian Noise, Covariance, Information, Whitening, and Uncertainty Ellipses
• Mahalanobis and Chi-Square Gating
• Mixture Models and Multimodal Beliefs
• Information Theory for Perception and Machine Learning
• Evaluation, Calibration, and Data Leakage: First Principles
• Sensor Likelihoods, Noise, and Error Budgets
• World Model Evaluation and Planning Objectives

Why it matters for AV, perception, SLAM, and mapping

Autonomy decisions depend on uncertainty:

• A tracker gates detections using predicted innovation covariance.
• A planner slows down when occupancy or localization uncertainty grows.
• A perception model threshold turns scores into object reports.
• A world model exposes multiple future modes for interactive agents.
• A safety monitor needs coverage claims that survive distribution shift audits.

Raw neural confidence is not enough. Guo et al. showed that modern neural networks can be poorly calibrated and that temperature scaling is a practical post-hoc fix for many classification settings. Deep ensembles often provide strong predictive uncertainty and OOD sensitivity. MC dropout provides one approximate Bayesian route. Conformal prediction provides distribution-free prediction sets with explicit coverage when the calibration and test examples are exchangeable.

Core definitions

Aleatoric and epistemic uncertainty

Aleatoric uncertainty is irreducible randomness in observations or outcomes:

wet pavement returns, photon noise, occlusion, future human choice

Epistemic uncertainty is model ignorance:

unseen airport layout, novel vehicle type, insufficient training data

In practice they interact. A far pedestrian under rain is both noisy and less represented in training.

Predictive distribution

A probabilistic predictor returns:

p_theta(y | x)

For regression, a common Gaussian output is:

y | x ~ N(mu_theta(x), Sigma_theta(x))

For classification:

p_theta(y = k | x) = softmax_k(z_theta(x))

Uncertainty quality is about the distribution, not just the point estimate.

Calibration

A classifier is calibrated if:

P(Y = y_hat | confidence = p) = p

For binary or multiclass confidence:

confidence = max_k p_theta(k | x)

If examples assigned 0.9 confidence are correct about 90 percent of the time, the model is calibrated on that slice.

Sharpness

Calibration alone is not enough. A model that always predicts base rates may be calibrated but useless. Sharpness measures concentration of predictive distributions. The goal is calibrated and sharp.

Prediction set

A prediction set returns a set C(x) rather than one label:

P(Y in C(X)) >= 1 - alpha

The set should be small when the model is confident and large when the input is ambiguous.

First-principles math

Negative log-likelihood

For data {(x_i, y_i)}:

NLL = - (1 / n) sum_i log p_theta(y_i | x_i)

NLL rewards probability mass on the realized outcome. It penalizes confident wrong predictions heavily, which is useful for safety-critical perception.

Brier score

For class probabilities p_i and one-hot target e_{y_i}:

Brier = (1 / n) sum_i ||p_i - e_{y_i}||_2^2

It is a proper scoring rule and often easier to decompose visually than NLL.

Expected calibration error

Partition examples into confidence bins B_m:

acc(B_m)  = mean_i in B_m 1[y_i = y_hat_i]
conf(B_m) = mean_i in B_m confidence_i

Then:

ECE = sum_m |B_m| / n * |acc(B_m) - conf(B_m)|

ECE is easy to report but depends on binning and can hide class, range, weather, and scene-specific miscalibration.

Temperature scaling

Given logits z, calibrated probabilities are:

p_k = softmax(z_k / T)

T is fitted on a held-out calibration set by minimizing NLL. If T > 1, probabilities are softened. Temperature scaling does not change the predicted class ranking; it changes confidence.

Ensembles

For M models:

p(y | x) = (1 / M) sum_m p_m(y | x)

Regression with mean and variance can decompose uncertainty:

E[y] = (1 / M) sum_m mu_m
Var[y] = (1 / M) sum_m (sigma_m^2 + mu_m^2) - E[y]^2

The average predicted variance captures aleatoric uncertainty; disagreement between means is a proxy for epistemic uncertainty.

MC dropout

At test time, keep dropout active and sample predictions:

y_hat_s = f_{theta, dropout_s}(x)

The empirical mean and variance approximate a Bayesian predictive distribution under assumptions that connect dropout to variational inference.

Split conformal prediction

Given a trained model and calibration examples (x_i, y_i), define a nonconformity score s_i where larger means worse fit.

For classification, one simple score is:

s_i = 1 - p_theta(y_i | x_i)

Let q be the empirical quantile:

q = ceil((n + 1) * (1 - alpha)) / n quantile of {s_i}

For a new example:

C(x) = {y: 1 - p_theta(y | x) <= q}

Under exchangeability of calibration and test examples:

P(Y_new in C(X_new)) >= 1 - alpha

For regression, if s_i = |y_i - mu(x_i)|, the conformal interval is:

[mu(x) - q, mu(x) + q]

For heteroscedastic models, use normalized scores:

s_i = |y_i - mu(x_i)| / sigma(x_i)

Algorithmic patterns

Pattern	Output	Best use	Main caveat
Temperature scaling	calibrated class probabilities	classification post-processing	assumes validation slice matches deployment
Isotonic or Platt scaling	calibrated scores	binary detectors	can overfit small calibration sets
Deep ensembles	predictive distribution	robust UQ, OOD detection	higher train and inference cost
MC dropout	sample-based uncertainty	approximate Bayesian retrofit	dropout distribution may be weak
Evidential models	distribution over evidence parameters	compact uncertainty head	can be miscalibrated without strong validation
Quantile regression	conditional intervals	regression bounds	quantile crossing and coverage drift
Split conformal	prediction sets or intervals	coverage guarantee on exchangeable data	guarantee is marginal, not per-slice
Mondrian conformal	group-conditional sets	class/range/weather slices	needs enough calibration data per group

AV, perception, SLAM, mapping, and planning relevance

Detection

Detector scores are often ranking scores, not calibrated probabilities. Calibrate by slices that affect sensor quality:

• class
• range
• object size
• occlusion
• weather
• time of day
• sensor modality
• map region or site

A 0.8 score for a close vehicle and a 0.8 score for a far cone may not mean the same empirical correctness.

Tracking and fusion

Kalman-style tracking requires covariance consistency. Use normalized innovation squared:

NIS = innovation^T S^-1 innovation

If the model is consistent, NIS follows an approximate chi-square distribution with degrees of freedom equal to measurement dimension. Persistent excess NIS means uncertainty is underestimated or the model is wrong.

Occupancy and mapping

For occupancy:

P(cell occupied | predicted p = 0.7) should be about 0.7

But cell-wise calibration is not enough. Evaluate connected components, object surfaces, drivable-space boundaries, and high-consequence regions separately.

Forecasting and world models

Future prediction is multimodal. A single mean trajectory can be calibrated in MSE terms and still be useless. Use probabilistic metrics:

• NLL under a mixture or sample distribution
• coverage of prediction sets
• miss rate at fixed false positive rate
• closed-loop planner regret or collision rate
• slice calibration for interactive and occluded scenarios

Planning

A planner should consume uncertainty through explicit contracts:

state estimate + covariance
object distribution or prediction set
occupancy probability and age
model confidence or OOD score
calibration domain metadata

Do not pass an uncalibrated neural score as if it were a probability in a safety cost.

Implementation notes

• Keep a separate calibration split. Do not tune temperature, thresholds, or conformal quantiles on the final test set.
• Version calibration artifacts with dataset slice, model checkpoint, label version, preprocessing, and operating domain.
• Report reliability diagrams by scenario slice, not only globally.
• Use proper scoring rules such as NLL and Brier score when evaluating predictive distributions.
• For conformal prediction, document exchangeability assumptions. Route, date, weather, sensor rig, and map version splits matter.
• Avoid treating marginal conformal coverage as per-class or per-scenario coverage. Use grouped methods or separate calibration when operationally required.
• Monitor uncertainty drift online with NIS, score histograms, set sizes, OOD rates, and post-deployment label audits.

Failure modes and diagnostics

Symptom	Likely cause	Diagnostic
Confident false detections	score is uncalibrated or OOD	reliability by class/range/weather
Conformal sets too large	base model weak or calibration slice broad	inspect nonconformity quantiles by slice
Conformal coverage fails in deployment	exchangeability broken	recalibrate by route/site/weather/time split
Ensemble disagreement low but wrong	shared training bias or blind spot	diversify data, architecture, and OOD tests
MC dropout variance meaningless	dropout not aligned with epistemic uncertainty	compare to ensembles and held-out OOD
Planner overreacts to uncertainty	uncertainty not tied to consequence	calibrate risk costs against closed-loop outcomes
Planner underreacts to uncertainty	probabilities treated as scores	enforce calibrated probability contracts
Global ECE looks good	slice errors cancel	class/range/scenario reliability diagrams

Sources

• Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q. Weinberger, "On Calibration of Modern Neural Networks": https://arxiv.org/abs/1706.04599
• Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell, "Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles": https://arxiv.org/abs/1612.01474
• Yarin Gal and Zoubin Ghahramani, "Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning": https://arxiv.org/abs/1506.02142
• Anastasios N. Angelopoulos and Stephen Bates, "A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification": https://arxiv.org/abs/2107.07511
• Glenn Shafer and Vladimir Vovk, "A Tutorial on Conformal Prediction": https://jmlr.csail.mit.edu/papers/v9/shafer08a.html