Uncertainty Toolbox

A python toolbox for predictive uncertainty quantification, calibration, metrics, and visualizations. Also: a glossary of useful terms and a collection of relevant papers and references.

Many machine learning methods return predictions along with uncertainties of some form, such as distributions or confidence intervals. This begs the questions: How do we determine which predictive uncertanties are best? What does it mean to produce a best or ideal uncertainty? Are our uncertainties accurate and well calibrated?

Uncertainty Toolbox provides standard metrics to quantify and compare predictive uncertainty estimates, gives intuition for these metrics, produces visualizations of these metrics/uncertainties, and implements simple “re-calibration” procedures to improve these uncertainties. This toolbox currently focuses on regression tasks.

Toolbox Contents

Uncertainty Toolbox contains:

  • Glossary of terms related to predictive uncertainty quantification.
  • Metrics for assessing quality of predictive uncertainty estimates.
  • Visualizations for predictive uncertainty estimates and metrics.
  • Recalibration methods for improving the calibration of a predictor.
  • Paper list: publications and references on relevant methods and metrics.


Uncertainty Toolbox requires Python 3.6+. For a lightweight installation of the package only, run:

pip install git+https://github.com/uncertainty-toolbox/uncertainty-toolbox

For a full installation with examples and tests, run:

git clone https://github.com/uncertainty-toolbox/uncertainty-toolbox.git
cd uncertainty-toolbox
pip install -e .

To verify correct installation, you can run the test suite via:

source shell/run_all_tests.sh

Quick Start

import uncertainty_toolbox as uct

# Load an example dataset of 100 predictions, uncertainties, and ground truth values
predictions, predictions_std, y, x = uct.data.synthetic_sine_heteroscedastic(100)

# Compute all uncertainty metrics
metrics = uct.metrics.get_all_metrics(predictions, predictions_std, y)

This example computes metrics for a vector of predicted values (predictions) and associated uncertainties (predictions_std, a vector of standard deviations), taken with respect to a corresponding set of ground truth values y.

Colab notebook: You can also take a look at this Colab notebook, which walks through a use case of Uncertainty Toolbox.


Uncertainty Toolbox provides a number of metrics to quantify and compare predictive uncertainty estimates. For example, the get_all_metrics function will return:

  1. average calibration: mean absolute calibration error, root mean squared calibration error, miscalibration area.
  2. adversarial group calibration: mean absolute adversarial group calibration error, root mean squared adversarial group calibration error.
  3. sharpness: expected standard deviation.
  4. proper scoring rules: negative log-likelihood, continuous ranked probability score, check score, interval score.
  5. accuracy: mean absolute error, root mean squared error, median absolute error, coefficient of determination, correlation.


The following plots are a few of the visualizations provided by Uncertainty Toolbox. See this example for code to reproduce these plots.

Overconfident (too little uncertainty)

Underconfident (too much uncertainty)

Well calibrated

And here are a few of the calibration metrics for the above three cases:

  Mean absolute calibration error (MACE) Root mean squared calibration error (RMSCE) Miscalibration area (MA)
Overconfident 0.19429 0.21753 0.19625
Underconfident 0.20692 0.23003 0.20901
Well calibrated 0.00862 0.01040 0.00865


The following plots show the results of a recalibration procedure provided by Uncertainty Toolbox, which transforms a set of predictive uncertainties to improve average calibration. The algorithm is based on isotonic regression, as proposed by Kuleshov et al.

See this example for code to reproduce these plots.

Recalibrating overconfident predictions

  Mean absolute calibration error (MACE) Root mean squared calibration error (RMSCE) Miscalibration area (MA)
Before Recalibration 0.19429 0.21753 0.19625
After Recalibration 0.01124 0.02591 0.01117

Recalibrating underconfident predictions

  Mean absolute calibration error (MACE) Root mean squared calibration error (RMSCE) Miscalibration area (MA)
Before Recalibration 0.20692 0.23003 0.20901
After Recalibration 0.00157 0.00205 0.00132


We welcome and greatly appreciate contributions from the community! Please see our contributing guidelines for details on how to help out.


If you found this toolbox helpful, please cite the following paper:

  title={Uncertainty Toolbox: an Open-Source Library for Assessing, Visualizing, and Improving Uncertainty Quantification},
  author={Chung, Youngseog and Char, Ian and Guo, Han and Schneider, Jeff and Neiswanger, Willie},
  journal={arXiv preprint arXiv:2109.10254},

Additionally, here are papers that led to the development of the toolbox:

  title={Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification},
  author={Chung, Youngseog and Neiswanger, Willie and Char, Ian and Schneider, Jeff},
  journal={arXiv preprint arXiv:2011.09588},

  title={Methods for comparing uncertainty quantifications for material property predictions},
  author={Tran, Kevin and Neiswanger, Willie and Yoon, Junwoong and Zhang, Qingyang and Xing, Eric and Ulissi, Zachary W},
  journal={Machine Learning: Science and Technology},
  publisher={IOP Publishing}


Development of Uncertainty Toolbox is supported by the following organizations.