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Abstract. The existing methods of predicting with confidence give good accuracy and confidence values, but quite often are computationally inefficient. Some partial solutions have been suggested in the past. Both the original method and these solutions were based on transductive inference. In this paper we make a radical step of replacing transductive inference with inductive inference and define what we call the Inductive Confidence Machine (ICM); our main concern in this paper is the use of ICM in regression problems. The algorithm proposed in this paper is based on the Ridge Regression procedure (which is usually used for outputting bare predictions) and is much faster than the existing transductive techniques. The inductive approach described in this paper may be the only option available when dealing with large data sets.

Summary A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive on‐line algorithms, has arisen over the last decade in computer science (to a large degree, under the influence of Dawid's prequential statistics). In this approach, which we call “competitive on‐line statistics”, it is not assumed that data are generated by some stochastic mechanism; the bounds derived for the performance of competitive on‐line statistical procedures are guaranteed to hold (and not just hold with high probability or on the average). This paper reviews some results in this area; the new material in it includes the proofs for the performance of the Aggregating Algorithm in the problem of linear regression with square loss.

Conformal predictors are set predictors that are automatically valid in the sense of having coverage probability equal to or exceeding a given confidence level. Inductive conformal predictors are a computationally efficient version of conformal predictors satisfying the same property of validity. However, inductive conformal predictors have been only known to control unconditional coverage probability. This paper explores various versions of conditional validity and various ways to achieve them using inductive conformal predictors and their modifications.

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