Statistics as Inductive Inference

Abstract

An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on all possible data. Statistics is a mathematical discipline that describes procedures for deriving results about a population from sample data. These results include predictions on future samples, decisions on rejecting or accepting a hypothesis about the population, the determination of probability assignments over such hypotheses, and the selection of a statistical model for studying the population. Both inductive logic and statistics are calculi for getting from the given data to propositions or results that transcend the data. This suggests that there is a strong parallel between statistics and inductive logic. The first statistical procedure is Neyman-Pearson hypotheses testing. This procedure is introduced as explicitly non-inferential. Power and significance are often interpreted inferentially. The second statistical procedure is parameter estimation. It briefly discusses Fisher's theory of maximum likelihood estimators, and shows that there is a certain relation with the inductive logic developed by Carnap. This leads to a discussion of Bayesian statistics in relation to Bayesian inductive logic.