Decision-Theoretic Interpretations

Classical decision theory is concerned with single-step games. Among its key results are the complete class and minimax theorems. The asymptotic formulation of the (infinite-horizon) prediction problem considered in this book can be also viewed as single-shot game, where one player (which one may call the Nature or adversary) selects a measure ν that generates an infinite sequence of data, and the other player (the statistician) selects a predictor ρ. The infinite game is then played out step-by-step, with the resulting payout being the asymptotic loss, as measured by either expected average KL divergence \(\bar L(\nu ,\rho )\) or the asymptotic total variation loss ltv(ν, ρ). Here we disregard the finite-time losses and only consider this final payout. In this chapter we consider the realizable case. Thus, the strategies of the statistician are unrestricted, \(\rho \in \mathcal P\), and the strategies ν of the adversary are limited to a given set of measures \(\mathcal C\subset \mathcal P\). The non-realizable case would require the introduction of a third player and this falls out of scope of the decision-theoretic framework that we consider here.