I just don't see any practical examples of people successfully betting by doing calculations with probability numbers derived from their intuitive feelings of confidence that would go beyond what a mere verbal expression of these feelings would convey. Can you think of any?

I'd speculate that bookies and professional sports bettors are doing something like this. By bookies here I mean primarily the kind of individuals who stand with a chalkboard at race tracks rather than the large companies. They probably use some semi-rigorous / scientific techniques to analyze past form and then mix it with a lot of intuition / expertise together with lots of detailed domain specific knowledge and 'insider' info (a particular horse or jockey has recently recovered from an illness or injury and so may perform worse than expected, etc.). They'll then integrate all of this information together using some non mathematically rigorous opaque mental process and derive a probability estimate which will determine what odds they are willing to offer or accept.

I've read a fair bit of material by professional investors and macro hedge fund managers describing their thinking and how they make investment decisions. I think they are often doing something similar. Integrating information derived from rigorous analysis with more fuzzy / intuitive reasoning based on expertise, knowledge and experience and using it to derive probabilities for particular outcomes. They then seek out investments that currently appear to be mis-priced relative to the probabilities they've estimated, ideally with a fairly large margin of safety to allow for the imprecise and uncertain nature of their estimates.

It's entirely possible that this is not what's going on at all but it appears to me that something like this is a factor in the success of anyone who consistently profits from dealing with risk and uncertainty.

The problem with discussing investment strategies is that any non-trivial public information about this topic necessarily has to be bullshit, or at least drowned in bullshit to the point of being irrecoverable, since exclusive possession of correct information is a sure path to getting rich, but its effectiveness critically depends on exclusivity.

My experience leads me to believe that this is not entirely accurate. Investors are understandably reluctant to share very specific time critical investment ideas for free but they frequently share their thought processes for free and talk in general terms about their approaches and my impression is that they are no more obfuscatory or deliberately misleading than anyone else who talks about their success in any field.

In addition, hedge fund investor letters often share quite specific details of reasoning after the fact once profitable trades have been closed and these kinds of details are commonly elaborated in books and interviews once time-sensitive information has lost most of its value.

Either your "rationality" manifests itself only in irrelevant matters, or you have to ask yourself what is so special and exclusive about you that you're reaping practical success that eludes so many other people, and in such a way that they can't just copy your approach.

This seems to be taking the ethos of the EMH a little far. I comfortably attribute a significant portion of my academic and career success to being more intelligent and a clearer thinker than most people. Anyone here who through a sense of false modesty believes otherwise is probably deluding themselves.

Where your own individual judgment falls within this picture, you cannot know, unless you're one of these people with esoteric expertise.

This seems to be the main point of ongoing calibration exercises. If you have a track record of well calibrated predictions then you can gain some confidence that your own individual judgement is sound.

Overall I don't think we have a massive disagreement here. I agree with most of your reservations and I'm by no means certain that improving one's own calibration is feasible but I suspect that it might be and it seems sufficiently instrumentally useful that I'm interested in trying to improve my own.