What you have labelled Cox’s theorem is not Cox’s theorem; it’s some version of the Dutch book argument and/or Savage’s theorem. Cox’s theorem is more like this. (I am the author of those blog posts.)
Also, you’ve misstated the complete class theorem—it’s actually weaker than you claim. It says that every estimator is either risk-equivalent to some Bayesian minimum posterior expected loss estimator (BMPELE) or has worse risk in at least one possible world. Conversely, no non-BMPELE risk-dominates any BMPELE. (Here “risk” is statistical jargon for expected loss, where the expectation is taken with respect to the sampling distribution.)
Thanks. What do you think I should call it instead of Cox’s theorem? Should I just call it “Dutch book argument”?
For the complete class theorem, is the beef with my use of “strictly worse” when I really mean “weakly worse” / “at least as bad”? That was me being sloppy and I’ll fix it now, but let me know if there’s a further issue.
For the complete class theorem, is the beef with my use of “strictly worse” when I really mean “weakly worse” / “at least as bad”? That was me being sloppy and I’ll fix it now, but let me know if there’s a further issue.
Yup, “strictly worse” overstates things. Basically, the complete class theorem says the class of Bayesian minimum posterior expected loss estimators is the risk-Pareto frontier of the set of all estimators.
What you have labelled Cox’s theorem is not Cox’s theorem; it’s some version of the Dutch book argument and/or Savage’s theorem. Cox’s theorem is more like this. (I am the author of those blog posts.)
Also, you’ve misstated the complete class theorem—it’s actually weaker than you claim. It says that every estimator is either risk-equivalent to some Bayesian minimum posterior expected loss estimator (BMPELE) or has worse risk in at least one possible world. Conversely, no non-BMPELE risk-dominates any BMPELE. (Here “risk” is statistical jargon for expected loss, where the expectation is taken with respect to the sampling distribution.)
Thanks. What do you think I should call it instead of Cox’s theorem? Should I just call it “Dutch book argument”?
For the complete class theorem, is the beef with my use of “strictly worse” when I really mean “weakly worse” / “at least as bad”? That was me being sloppy and I’ll fix it now, but let me know if there’s a further issue.
Yeah, stick with “Dutch book”.
Yup, “strictly worse” overstates things. Basically, the complete class theorem says the class of Bayesian minimum posterior expected loss estimators is the risk-Pareto frontier of the set of all estimators.
Thanks!