It is not a fee-for-service relationship. The price system in medicine has been mangled beyond recognition. Patients are not told prices; doctors avoid, even disdain, any discussion of prices; and the prices make no rational sense even if and when you do discover them. This destroys all ability to make rational economic choices about healthcare.
I think pricing of medical services faces somewhat of a breakdown in the normal price-setting mechanism of markets. For some random good like a sandwich or whatever the buyer can at least have a reasonable sense of how much they want it, the seller understands their costs to produce it, and the price gets established by this balance. But how is someone who seeks medical care really supposed to know how much they value a particular medical service? They would presumably have to rely on their provider, who is on the opposite side of the transaction. Insurers could somewhat serve this role, but I think people often look down upon this, and also it seems likely to be a difficult and imperfect process.
I’m definitely not representative of lesswrong in my my views above I don’t think. In fact in some sense I think I’m shadowboxing with lesswrong in some of my comments above, so sorry about any confusion that introduced.
I don’t think I’ve ever seen maximum likelihood vs maximum a-posteriori discussed on lesswrong, and I’m kind of just griping about it! I don’t have a references off to top of my head but I recall this appearing in debates elsewhere (i.e. not on lesswrong) like in more academic/stats settings. I can see if I can find examples. But in general it addresses an estimation perspective instead of hypothesis testing.
I think I’m in agreement with you here. My “methodological” was directed at what I view as a somewhat more typical lesswrong perspective, similar to what is expressed in the Eliezer quote. Sure, if we take some simple case we can address a more philosophical question about frequentism vs bayesianism, but in practical situations there are going to so many analytical choices that you could make that there are always going to be issues. In an actual analysis you can always do stuff like look at multiple versions of an analysis and trying to use that to refine your understanding of a phenomenon. If you fix the likelihood but allow the data to vary then p-values are likely to be highly correlated with possible alternatives like bayes factors, a lot of the critiques I feel are focused on making a clean philosophical approach while ignoring the inherent messiness that would be introduced if you ever want to infer things from reasonably complicated data or observations. I don’t think swapping likelihood ratios for p-values would sudden change things all that much, a lot of the core difficulties of inferring things from data would remain.