Where is all this “local optimum” / “global optimum” stuff coming from? While I’m not familiar with the complete class theorem, going by the rough statement given in the article… local vs. global optima is just not the issue here, and is entirely the wrong language to talk about this here?
That is to say, talking about local maximum requires that A. things are being measured wrt some total order (though I suppose could be relaxed to a partial order, but you’d have to be clear whether you meant “locally maximum” or just “locally maximal”; I don’t know that this is standard terminology) and B. some sort of topological structure on the domain so that you can say what’s near a given position. The statement of the complete class theorem, as given, doesn’t say anything about any sort of topological structure, or any total order.
Rather, it’s a statement about a partial order (or preorder? Since people seem to use preorders in decision theory). And I mean I said above, the definition of “local maximum” could certainly be relaxed to that case, but that’s not relevant here because there’s just no localness anywhere there. Rather it’s simply saying, “Every maximal element is Bayesian.”
In particular, this implies that if there is a maximum element, it must be Bayesian, as certainly a maximum element must be maximal. Of course there’s no guarantee that there is a maximum element, but I suppose you’re considering the case where the partial order is extended to a total (pre)order with some maximum element.
Hell, even in the original post, with the local/global language that makes no sense, the logic is wrong: If we assume that the notions of “local optimum” and “global optimum” make sense here, well, a global maximum certainly is a local maximum! So if every local maximum is Bayesian, every global maximum is Bayesian.
None of this takes away from your point that knowing there exists a better Bayesian method doesn’t tell you how to find it, let alone find it with bounded resources. And that just because a maximum, if it exists, must be Bayesian, doesn’t imply there is anything good about other Bayesian points, and you may well be better off with a known-good frequentist method. But as best I can tell, all the stuff about local optima is just nonsense, and really just distracts from the underlying point. (So basically, you’re wrong about Myth #2.)
Where is all this “local optimum” / “global optimum” stuff coming from? While I’m not familiar with the complete class theorem, going by the rough statement given in the article… local vs. global optima is just not the issue here, and is entirely the wrong language to talk about this here?
That is to say, talking about local maximum requires that A. things are being measured wrt some total order (though I suppose could be relaxed to a partial order, but you’d have to be clear whether you meant “locally maximum” or just “locally maximal”; I don’t know that this is standard terminology) and B. some sort of topological structure on the domain so that you can say what’s near a given position. The statement of the complete class theorem, as given, doesn’t say anything about any sort of topological structure, or any total order.
Rather, it’s a statement about a partial order (or preorder? Since people seem to use preorders in decision theory). And I mean I said above, the definition of “local maximum” could certainly be relaxed to that case, but that’s not relevant here because there’s just no localness anywhere there. Rather it’s simply saying, “Every maximal element is Bayesian.”
In particular, this implies that if there is a maximum element, it must be Bayesian, as certainly a maximum element must be maximal. Of course there’s no guarantee that there is a maximum element, but I suppose you’re considering the case where the partial order is extended to a total (pre)order with some maximum element.
Hell, even in the original post, with the local/global language that makes no sense, the logic is wrong: If we assume that the notions of “local optimum” and “global optimum” make sense here, well, a global maximum certainly is a local maximum! So if every local maximum is Bayesian, every global maximum is Bayesian.
None of this takes away from your point that knowing there exists a better Bayesian method doesn’t tell you how to find it, let alone find it with bounded resources. And that just because a maximum, if it exists, must be Bayesian, doesn’t imply there is anything good about other Bayesian points, and you may well be better off with a known-good frequentist method. But as best I can tell, all the stuff about local optima is just nonsense, and really just distracts from the underlying point. (So basically, you’re wrong about Myth #2.)