Reliable Predictive Inference


An important factor to guarantee a responsible use of data-driven recommendation systems is that we should be able to communicate their uncertainty to decision makers. This can be accomplished by constructing prediction intervals, which provide an intuitive measure of the limits of predictive performance.

This website contains a Python implementation of the following papers:

CQR is a technique for constructing prediction intervals that attain valid coverage in finite samples, without making distributional assumptions. It combines the statistical efficiency of quantile regression with the distribution-free coverage guarantee of conformal prediction. On one hand, CQR is flexible in that it can wrap around any algorithm for quantile regression, including random forests and deep neural networks. On the other hand, a key strength of CQR is its rigorous control of the miscoverage rate, independent of the underlying regression algorithm.


For more information, please refer to the cqr: synthetic experiment and cqr: real data experiment pages.

Y. Romano, E. Patterson, and E. J. Candès, “Conformalized quantile regression.” NeurIPS, 2019.

To support equitable treatment, the equalized coverage methodology forces the construction of the prediction intervals to be unbiased in the sense that their coverage must be equal across all protected groups of interest. Similar to CQR and conformal inference, equalized coverage offers rigorous distribution-free guarantees that hold in finite samples. This methodology can also be viewed as a wrapper around any predictive algorithm.


For more information, please refer to the equalized coverage: real data experiment and equalized coverage: detect prediction bias pages.

Y. Romano, R. F. Barber, C. Sabbatti and E. J. Candès, "With malice toward none: Assessing uncertainty via equalized coverage." arXiv:1908.05428 2019.

Conformal inference, cross-validation+, and the jackknife+ are hold-out methods that can be combined with virtually any machine learning algorithm to construct prediction sets with guaranteed marginal coverage. Here, we propose specialized versions of these techniques for categorical and unordered response labels that, in addition to providing marginal coverage, are also fully adaptive to complex data distributions, in the sense that they perform favorably in terms of approximate conditional coverage compared to alternative methods.


For more information, please refer to the classification: example 1 and classification: example 2 pages.

Y. Romano, M. Sesia, and E. J. Candès, "Classification with Valid and Adaptive Coverage," 2020.

The methods described in the paper are implemented in Python, and available under the MIT license.

Some of the code is taken from the nonconformist package available at One may refer to the nonconformist repository to view other applications of conformal prediction.

Please contact Yaniv Romano for bug reports. Email: yaniv dot romano at gmail dot com

Illustration: Regression

The figure below illustrates the marginal prediction intervals constructed by CQR on synthetic example. The heteroskedasticity of the data is evident, as the dispersion of 𝑌 (the response) varies considerably with 𝑋 (the features). The data also contains outliers.

We start by fitting two quantile regressors (random forests in this example) on the training set to obtain initial estimates of the lower and upper bounds of the prediction interval (the two black curves in the figure). Then, we use a held-out validation set to conformalize and, if necessary, correct this prediction interval. Unlike the original interval, the conformalized prediction interval (the highlighted blue area) is guaranteed to satisfy the coverage requirement (90% in this example) on unseen test data, regardless of the choice or accuracy of the quantile regression estimator.

Notice how the length of constructed interval varies with 𝑋, reflecting the uncertainty in the prediction of 𝑌.

For more details, please refer to the synthetic experiment page and to our paper.