Rules for building statistical models

“Statistics uses the empirical hypothesis that apparatus (‘lotteries’) exist, admitting random choices of one among any given number of elements. Such apparatus do not exist in absolute perfection and their degree of perfection can only be defined after development of their theory. Their role is analogous to that of rigid bodies in euclidean geometry and of perfect clocks in dynamics. Empirical interpretation of probability statements is only possible with reference to such random apparatus or to natural phenomena empirically found to behave statistically sufficiently tike these”. D. van Dantzig [1] Summary One of the fundamental questions in statistical model building is when to use the same model for different situations or experiments. Axioms, however useful mathematically, say nothing about this. The author therefore proposes to introduce rules for the choice of a statistical model which have the character of instructions for use of the statistical toolkit. One of the basic rules proposed is the “principle of equivalence”. Two repeatable experiments are called statistically equivalent if they cannot be distinghuished from one another by means of sequences of outcomes of arbitrary length. This principle is elaborated and illustrated by means of an example. If experiments are (deemed to be) statistically equivalent the use of the same statistical model for all of them is justified. This principle is then used for the introduction of conditional probabilities and composite models, with symmetric probability spaces as models for randomizers as a startingpoint.