Martin Larsson and Johannes Ruf’s contribution to the Discussion of ‘Estimating means of bounded random variables by betting’ by Waudby-Smith and Ramdas
本文讨论了一种通过赌博策略构建置信序列的方法，指出任何非负鞅资本过程都可表示为单一资产的交易策略，并举例说明多样化凯利策略与Cover通用投资组合的联系。
We congratulate Ian Waudby-Smith and Aaditya Ramdas on their comprehensive and insightful paper. The authors construct time-uniform confidence sequences for the mean of a sequence of [0,1]-valued random variables X1,X2,… that all have the same conditional mean, assumed to be deterministic. Specifically, fix m∈(0,1) and define Pm as the set of probability measures on the canonical sequence space under which, for each t∈N, the conditional expectation of Xt given the history X1,…,Xt−1 is equal to the deterministic number m. The authors construct nonnegative processes (Kt) that satisfy K0=1 and are Pm-martingales, i.e. P-martingales for each P∈Pm. These martingales are then used to construct anytime-valid statistical tests that in turn can be transformed into confidence sequences (see also Ramdas et al., 2022). In keeping with the game-theoretic probability literature, the authors refer to the processes (Kt) as capital processes. Let us ponder this terminology, starting with the following simple but interesting observation made by the authors: every nonnegative Pm-martingale (Kt) with K0=1 is of the form for some predictable process (λt) with values in [−(1−m)−1,m−1]. Predictable means that each λt only depends on X1,…,Xt−1. One may interpret λt as the proportion of one’s capital Kt−1 that is invested in an asset with return Xt−m, keeping whatever is left over ‘in the pocket’. The fact that λt can be greater than one, or negative, poses no issue as this simply means that one may borrow cash to purchase more of the asset than one could otherwise afford, or sell the asset short to generate additional cash income. Crucially, one’s capital must always remain nonnegative. The upshot is this: not only is (Kt) the capital process produced by repeated betting; thanks to the representation (1) there is always an explicit trading strategy, operating on one single asset, that generates the capital process. Indeed, one has λt=(Kt/Kt−1−1)/(Xt−m). Given any particular (Kt) of interest, we believe insight can be gained by computing the associated trading strategy. For example, the ‘diversified Kelly’ capital process considered by the authors is built from D separate strategies (λtd), d=1,…,D. This is equivalent to the single strategy where (Ktd) is the capital process generated by the dth strategy. In other words, diversified Kelly arises from executing the capital-weighted average of the given strategies. This links it to Cover’s universal portfolios (Cover, 1991). Many of the capital processes proposed in the paper are specified in terms of a strategy (λt). What we find worth emphasising is that such a (λt) can always be found, and is likely to yield insights. Finally, let us point out that the representation (1) is of course specific to the particular structure of Pm. An interesting question is to what extent analogous representations exist for other, more complex, statistical hypotheses.