FOUNDATIONS OF STATISTICS
Inference from effect to cause— inverse inference—is the main concern of this chapter. Inverse inference is a canonical form of scientific thinking. Our hypothesis predicts a certain effect under certain conditions. We set up these conditions empirically; if we observe the predicted effect, we take this as evidence for our hypothesis.
Of course, this inverse inference, from effect to cause, does not prove our hypothesis. The same effect might result from some other cause.
Statistics provides one tool for inverse inference, a tool that allows for the possibility that the effect was caused by chance. According as our observed effect is unlikely or likely to occur by chance alone, we do or do not consider it supportive evidence for our hypothesis. This argument is just common sense; the virtue of the significance test is to make it quantitative.
But this commonsense argument about chance has a serious limitation. It takes no account of other evidence, which is generally available. This other evidence must also be integrated to determine the credibility of our hypothesis.
How should this other evidence be integrated? This question is especially problematic when, as is usually true, the other evidence involves subjective belief rather than objective observations.
Two answers have been pursued in statistical theory. One answer is that statistics should confine itself to observables; subjective evidence must be handled in some extrastatistical manner. The second answer is that statistical theory has an effective way to handle subjective evidence—and that excluding it would cripple the statistical field.
There is a lot to be said for both answers. And an awful lot has been said, for the question lies at the foundation of statistics. Fortunately, the main sense of the controversy is interesting and clear, without reference to the subtleties and polemics. Even more fortunately, nearly all material covered in previous chapters remains applicable regardless of the theoretical differences of opinion.