# Kei Takeuchi

Kei Takeuchi holds Ph D in Economics. Until recently he was Professor at the Faculty of International Studies, Meiji Gakuin University. He is also Professor Emeritus, University of Tokyo; Director, Institute for International Studies, Meiji Gakuin University; Chairman, Statistics Council of Japan; President, Japan Statistical Association; President, Center for Academic Publications Japan; Former President, Japan Statistical Society.

Personal History:

- 1933 Born in Tokyo, Japan
- 1961 Graduated from the University of Tokyo graduate school
- 1963 - 1994 Associate Professor then Professor, Faculty of Economics, University of Tokyo
- 1987 - 1994 Professor, Research Center for Advanced Science and Technology (Joint Appointment)
- 1994 Retired from the University of Tokyo
- 1994 - 2006 Professor, Faculty of International Studies, Meiji Gakuin University
- 1988 - 1994 Member of Science Council of Japan
- 1997 - 2000 Member of Science Council of Japan

Research Areas:

- Mathematical Statistics
- Econometrics
- Japanese Economy
- Socio-Economic Impacts of Science and Technology
- General History of Civilization
- Global Environment Problems
- Game-theoretic Probability

In 1979 Kei Takeuchi wrote a handout (dated July 17 1979, 790717-takeuchi-memo.pdf) for a seminar at Stanford university in which he anticipated some features of conformal prediction. Developing the notion of an `⚠ $1-\alpha$`

expectation tolerance region, he posed the problem of finding the smallest (with respect to the Lebesgue measure) such region that would be valid under the randomness assumption. Invoking the standard completeness result for the order statistics, he replaced the randomness assumption by the exchangeability assumption. Assuming that the observations are generated from a probability distribution with density `⚠ $f_0$`

, he showed that the optimal solution is the conformal predictor assigning the conformity score `⚠ $f_0(z_i)$`

to an observation `⚠ $z_i$`

. This is a special case of conformal prediction not involving learning, but he also developed a more general version that does require some learning (namely, parameter estimation). If the true distribution is only supposed to belong to a parametric statistical model, he suggested using the plug-in density for computing the conformity scores `⚠ $f_0(z_i\mid\hat\theta)$`

. To preserve validity, `⚠ $\hat\theta$`

was to be computed from the extended training sequence with the test object assigned a postulated label `⚠ $y$`

, making Prof. Takeuchi's procedure an instance of conformal prediction.