# How is the "conformal prediction" conformal?

The question is clarified by Prof.V.Vovk. See his answer below for discussion.

Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a promising candidate towards distribution-free inference coming along with confidence level guarantee.

Given the current literature, especially [2,3], the main improvement in the conformal inference(implemented in support vector machine) is providing a confidence region with certain credibility. For example, in the classical regression model we can obtain the pointwise prediction interval at each of observation sample points. The CP

... Unlike traditional regression methods which produce point predictions, Conformal Predictors output predictive regions that satisfy a given confidence level. The regions produced by any Conformal Predictor are automatically valid, however their tightness and therefore usefulness depends on the nonconformity measure used by each CP ... what we call in this paper “Conformal Prediction” (CP).[2]

... We also obtain a measure of “credibility” which serves as an indicator of the reliability of the data upon which we make our prediction.[3]

And in their introductory book, they explained their usage of the name "Conformal Prediction" by saying

Most of this book is devoted to a particular method that we call "conformal prediction". When we use this method, we predict that a new object will have a label that makes it similar to the old examples in some specified way, and we use the degree to which the specified type of similarity holds within the old examples to estimate our confidence in the prediction. Our conformal predictors are, in other words, "confidence predictors".[5]pp.7-8

To be more precise and in response to @RHahn comment below, I do not think "conformal" is an arbitrary choice of word, since Vovk mentioned that

"In 1963–1970 Andrei Kolmogorov suggested a different approach to modelling uncertainty based on information theory; its purpose was to provide a more direct link between the theory and applications of probability. On-line compression models are a natural adaptation of Kolmogorov's programme to the technique of conformal prediction. Working Paper 8 introduces three on-line compression models: exchangeability (equivalent to the iid model), Gaussian and Markov."[6]

Therefore I do believe there is a deeper motivation of the "conformal prediction" from complex analysis that I was not aware of. Thanks!

Therefore my question is,

1. Given the name "Conformal prediction", is this method more or less associated with the concept of conformal mapping in (multivariate) complex analysis?

Does it mean that the old sample can be mapped locally conformally to the new samples? Since most of CP are implemented using SVM, is this related to the shape of classifying hyperplane determined by the SVM?

1. (More like a opinion-based question) How is conformal predictor different from the existing robust predictors while they both come with a guarantee that the true value will fall into the confidence region with high probability?

Reference

[1]Lei, Jing, et al. "Distribution-free predictive inference for regression." Journal of the American Statistical Association just-accepted (2017).

[2]Papadopoulos, Harris, Vladimir Vovk, and Alexander Gammerman. "Regression conformal prediction with nearest neighbours." Journal of Artificial Intelligence Research 40 (2011): 815-840.

[3]Saunders, Craig, Alexander Gammerman, and Volodya Vovk. "Transduction with confidence and credibility." Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI'99). Vol. 2. 1999.

[4]Shafer, Glenn, and Vladimir Vovk. "A tutorial on conformal prediction." Journal of Machine Learning Research 9.Mar (2008): 371-421.

[5]Vovk, Vladimir, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world. Springer Science & Business Media, 2005.

• I hope someone knowledgable will answer. fwiw, I was under the impression that "conformal" was being used informally, basically as a synonym for "similarity", just to mean prediction based on which observations "match, fit, suit, answer, agree with, be like, correspond to, be consistent with, measure up to, tally with, square with" (to quote my thesaurus). But I haven't read any other sources besides what you have. Apr 12, 2017 at 14:38
• @RHahn I thought so at the first sight, but later I just feel it very unnatural to use such a word "conformal", if we are in statistical literature, "similar" would definitely be a better choice; and if the authors are to emphasize the simultaneous protection nature of CP, then "simultaneous" would be a better choice. Consider Vovk's background, I think there must be some deeper motivation from complex analysis that I was not aware of(which is a branch I am not good at). See my updates too. Apr 12, 2017 at 18:54
• @RHahn I am wrong, but we have a really nice answer now! Apr 13, 2017 at 10:44
• The term "conformal" in differential geometry refers to the property of angle preservation, and it is in this sense that the term is used in complex analysis, conformal field theory, conformal differential geometry, and all other uses of the term that I am familiar with in mathematics. I concede that the term is not well chosen, because its ordinary English use does not concern angles. Mar 10, 2023 at 20:46

## 1 Answer

Thanks for your interest. The term “conformal prediction” was suggested by Glenn Shafer, and at first I did not like it exactly for the reason that you mention: it has nothing (or very little) to do with conformal mappings in complex analysis. But then I discovered other meanings, even in maths; e.g., Wikipedia has five on its disambiguation page for “conformal”:

• Conformal film on a surface (same thickness)
• Conformal fuel tanks on military aircraft
• Conformal coating in electronics
• Conformal hypergraph, in mathematics
• Conformal software, in ASIC Software

So the word did not look taken to me anymore. The expression that we had used before Glenn proposed “conformal prediction” was even worse (“transductive confidence machine”).

Thanks to Hengrui Luo for drawing my attention to this question.

As for question (2), the answer depends on which robust predictors you have in mind. The predictors with most similar properties are the ones in classical statistics (such as the standard prediction intervals in linear regression based on Student's t distribution); the main difference is that they are parametric. There is a predictive version of tolerance intervals in nonparametric statistics, but their treatment of objects (x parts of observations (x,y), where y are labels) is limited. Upper bounds on the probability of error given by standard PAC predictors are often too high to be useful.

• Thanks so much for the answer and welcome to MO! Will award with a bounty for the nice clarification! The kind of robust estimators I have in mind is Huber's estimators about the regression coefficients; since CP proposed a simultaneous confidence region when Huber's robust interval seems doing the same thing. I feel that your CP is closely related to upper bounds of a given stochastic process, but definitely will dig deeper into it! Thanks! Apr 13, 2017 at 11:06
• PS: I kind of like transductive confidence machine Dec 13, 2017 at 0:13