Falsifiable implies Learnable

The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable – i.e. admits a strategy that predicts optimally. An analogous result is shown for universal induction. A theory that explains everything, [predicts] nothing. – attributed to Karl Popper. To what extent are theory-based predictions justified by prior observations? The question is known as the problem of induction and is fundamental to scientific inference. We address the problem of induction from the perspective of learning theory. That is, we consider which theories, and under what assumptions, can be applied to make optimal predictions. Our main result is that the more hypotheses a theory falsifies, suitably quantified, the closer the predictive performance of the best strategy (based on the theory) will be to the theory's post hoc explanatory performance on observed data. Learning theorists have characterized the generalization performance of algorithms in a wide range of scenarios. Although none of these scenarios adequately captures the practice of scientific inference, they form a family of minimal models of prediction. An intuitive understanding of the main results of learning theory therefore belongs in every scientist's conceptual toolkit. Unfortunately, the results are phrased in opaque terminology that depends on specialized concepts such as Rademacher complexity, shattering coefficients and VC-dimensions. This paper presents basic results from learning theory in terminology that is meaningful to the broader scientific community. The results cover three scenarios. In each scenario, Forecaster uses a theory (or theories) to predict Nature's next move(s) based on Nature's previous moves. The paper develops the following account. A. The risk. — The risk of a theory is how accurately it explains a sequence of events. A theory explains a sequence of events perfectly if it contains a predictor that correctly labels every element. In general, the accuracy of an explanation is the fraction of the sequence that its best predictor explains correctly. — The risk of a strategy is how accurately it predicts a sequence of events. A strategy specifies picks a predictor based on previously observed events, which it then applies to future events. The strategy's predictive accuracy is the fraction of