**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?

(2)(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.