Sometimes I hear people say their thought process is Bayesian. They seem to mean that they often rate their beliefs on a scale of [0,1] and they "update" by which they seem to mean that they change their mind when they encounter new information.

I don't think this is Bayesian and I don't think rating beliefs is special. The [0,1] scale makes it seem technical but otherwise, rating your feelings is a normal part of modern life. It's just that usually we use scales like 1-3, 1-5 or 1-10.

Changing one's mind based on new information seems even less special. It seems to me literally every human being alive does this. So this all just looks like normal thinking to me and not any more reliable or technical.

I think for humans to update Bayesianly, it would mean using a formula like below and filling in unknown probabilities with intuitions. It should require some arithmetic and a commitment to constrain new statements of belief to whatever numbers come out of this process.

I don't really know how I would make myself not feel like a 0.6 after I realized based on my calculations that I should be feeling like a 0.7 but that's probably why I don't label my mental processes as "Bayesian".

I have heard that there are risk estimation companies and betting houses that do try to update human intuitions in a formal Bayesian way with some real world success. So I think it is possible to have a truly Bayesian process involving human intuition.

However, I think it would require a lot of discipline and focus to pull off!