  We critically review three major theories of machine learning and provide a
new theory according to which machines learn a function when the machines
successfully compute it. We show that this theory challenges common assumptions
in the statistical and the computational learning theories, for it implies that
learning true probabilities is equivalent neither to obtaining a correct
calculation of the true probabilities nor to obtaining an almost-sure
convergence to them. We also briefly discuss some case studies from natural
language processing and macroeconomics from the perspective of the new theory.
