Can the Best-Alternative-Justification solve Hume's Problem?
On the Limits of a Promising Approach

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3 Schurz' central results

Within the just sketched framework, is it possible to find a provably optimal meta-inductivist strategy? Schurz believes it is and he presents two important theorems regarding the optimality of certain types of meta-inductivists. But he also relates an argument (Schurz 2008, p. 298) by Cesa-Bianchi and Lugosi (CesaBianchi/Lugosi 2006, p. 67) that points out certain limits. The argument is the following:

Impossibility Theorem 1: In the binary prediction game, a single meta-inductivist cannot be optimal in all possible worlds.

The proof of this theorem can informally be stated as follows: Assume a world with one meta-inductivist and two alternative predictors, one of which always predicts $1$ and the other always $0$. Whatever the sequence of world events is, the succes rates of the two assumed alternative predictors will always add up to 1. It follows that in any round the success rate of either the one or the other of the two assumed alternative predictors is at least 50%. Now, if the sequence of world events is a demonic sequence that always delivers the event that was not predicted by the meta-inductivist, the meta-inductivist's success rate will always remain 0%. Thus, in any round the meta-inductivist's success rate is significantly lower than that of the best player, which means that the meta-inductivist's strategy is not optimal.

Because of this, Schurz understands the different types of meta-inductivists which he develops in section four through to section six of his article (Schurz 2008, p. 285-296) as only partial solutions to the problem of induction. They are optimal in (large) classes of prediction games but not in all prediction games.

The “impossibility theorem” stated before does not cover the real valued prediction game. And indeed it can be demonstrated for the real valued prediction game that a meta-inductivist that averages over the alternative predictor's predictions weighted by the alternative predictor's relative success-advantage over the meta-inductivist will quickly approximate the maximal success rate of the alternative predictors. This is Schurz' theorem 4 (Schurz 2008, p. 297), which shall be termed “optimality theorem 1” here:

Optimality Theorem 1 for a weighted average metainductivist $wMI$ in the real valued prediction game:

1. Long run: The success rate $suc_n$ of $wMI$ approximates the alternative predictors maximal success rate $maxsuc_n(P)$, i.e. $lim_{n\rightarrow\infty} (maxsuc_n(P) - suc_n(wMI)) = 0$.
    
2. Short run: In any round $n$ for the success rate of $wMI$ holds: $suc_n(wMI) \geq maxsuc_n(P) - \sqrt{m/n}$ with $m$ being the number of alternative predictors.

Here $P$ denotes the set of all alternative predictors and $maxsuc_n(P)$ denotes the maximal success rate of the alternative predictors in round $n$. For the precise definition of the strategy of the weighted average meta-inductivist and the proof of the theorem, see Schurz' paper (Schurz 2008, p. 296ff.).

As can be seen, the long run success of the meta-inductivists does not depend on how many alternative predictors are present. Only in the short run, the number or alternative predictors $m$ is important in so far as the more alternative predictors are present, the more “distracted” the weighted average metainductivist can become in the short run.

Schurz is able to prove a similar theorem for the binary valued prediction game. The impossibility theorem stated earlier precludes that this will work for a single meta-inductivist in the binary prediction game. But, as Schurz is able to demonstrate, the mean success rate of a collective of meta-inductivists $cwMI$ can approximate the maximal success of the alternative predictors - be they as demonic as they may - almost equally well as the weighted average meta-inductivist can in the real valued prediction game. The more $cwMI$ predictors are present, the better the approximation. Or, more precisely:

Optimality Theorem 2 for the mean success rate of a collective of $k$ collective weighted-average meta-inductivists $cwMI$ in the binary prediction game:

1. Long run: The mean success rate $meansuc_n$ of the collective meta-inductivists $\frac{1}{2k}$-approximates the alternative predictor's maximal success rate $maxsuc_n$, i.e. $lim_{n\rightarrow\infty} (maxsuc_n(P) - meansuc_n(cwMI)) \leq \frac{1}{2k}$.
    
2. Short run: In any round $n$ for the mean success rate of $cwMI$ holds: $meansuc_n(wMI) \geq maxsuc_n(P) - \sqrt{m/n} - \frac{1}{2k}$ with $m$ being the number of alternative predictors.

For the proof of this theorem and for the precise algorithm of the collective weighted-average meta-inductivists ($cwMI$), see Schurz' article (Schurz 2008, p. 297-299). Again, the long run success of the collective meta-inductivists (with regard to their mean success rates!) does not depend on the number alternative predictors. This renders the finding non-trivial. For no matter how large the number or alternative predictors is, their maximal success can be approximated by a comparatively smaller collective of meta-inductivists, the precise number of which is only determined by what level of approximation is regarded as satisfactory. It is only in the short run that a large number of possibly demonic alternative predictors can effectively deceive the collective meta-inductivists.

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