Introspective Forecasting

Science ultimately seeks to reliably predict aspects of the future; but, how is this even possible in light of the logical paradox that making a prediction may cause the world to evolve in a manner that defeats it? We show how learning can naturally resolve this conundrum. The problem is studied within a causal or temporal version of the Probably Approximately Correct semantics, extended so that a learner's predictions are first recorded in the states upon which the learned hypothesis is later applied. On the negative side, we make concrete the intuitive impossibility of predicting reliably, even under very weak assumptions. On the positive side, we identify conditions under which a generic learning schema, akin to randomized trials, supports agnostic learnability.