Editing No free lunch theorem (section)

==Motivation==
The NFL theorems were explicitly ''not'' motivated by the question of what can be inferred (in the case of NFL for machine learning) or found (in the case of NFL for search) when the "environment is uniform random". Rather uniform randomness was used as a tool, to compare the number of environments for which algorithm A outperforms algorithm B to the number of environments for which B outperforms A. NFL tells us that (appropriately weighted){{clarify|date=May 2018}} there are just as many environments in both of those sets.

This is true for many definitions of what precisely an "environment" is. In particular, there are just as many prior distributions (appropriately weighted) in which learning algorithm A beats B (on average) as vice versa.{{citation needed|reason=dubious; the set of possible prior distributions is infinite|date=May 2018}} This statement about ''sets of priors'' is what is most important about NFL, not the fact that any two algorithms perform equally for the single, specific prior distribution that assigns equal probability to all environments.

While the NFL is important to understand the fundamental limitation for a set of problems, it does not state anything about each particular instance of a problem that can arise in practice. That is, the NFL states what is contained in its mathematical statements and it is nothing more than that. For example, it applies to the situations where the algorithm is fixed a priori and a worst-case problem for the fixed algorithm is chosen a posteriori. Therefore, if we have a "good" problem in practice or if we can choose a "good" learning algorithm for a given particular problem instance, then the NFL does not mention any limitation about this particular problem instance. Though the NFL might seem contradictory to results from other papers suggesting generalization of learning algorithms or search heuristics, it is important to understand the difference between the exact mathematical logic of the NFL and its intuitive interpretation.<ref name=KA-deep>Kawaguchi, K., Kaelbling, L.P, and Bengio, Y.(2017) "Generalization in deep learning", https://arxiv.org/abs/1710.05468</ref>