The important recent book by G. Schurz appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior -- they prove that there are ``as many priors'' (loosely speaking) for which any induction algorithm $A$ out-generalizes some induction algorithm $B$ as vice-versa. Importantly though, in addition to the NFL theorems, there are many \textit{free lunch} theorems. In particular, the NFL theorems can only be used to compare the \textit{marginal} expected performance of an induction algorithm $A$ with the marginal expected performance of an induction algorithm $B$. There is a rich set of free lunches which instead concern the statistical correlations among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocate as a ``solution to Hume's problem'' are just an example of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference.
翻译:G. Schurz最近的重要著作《G. Schurz》认识到,无自由中链理论(NFL)对(甲型)上岗问题有重大影响。我在这里回顾NFL理论,强调这些理论不仅涉及有统一的前科的案例,而且强调它们不仅涉及“许多前科(粗略地说)”的预期性能。对于这些案例,任何上岗算法都把一些上岗算法($A$)作为反向推算法($B$)。重要的是,除了NFL理论外,还有许多Textit{free lunch}理论。特别是NFL理论,它们不仅能够用来比较上岗算法(textit{marginal)的预期性能和前科算法的预期性能。对于上岗算法的通缩算法中,Schurz的元数算法(Schurz)的元数比平级化法性化的正反常数,而后期的进化法学算法(I)则是前期的直译法(I)的正反正变法(I)中,其前期的反正反正变法(I)的一个例子。