One possible connection between "postmodern" (i.e. recent Continental) philosophy and Bayesian rationality may be found in the notion of "embodied philosophy" or "embodied cognition" — e.g. the work of George Lakoff in linguistics and Hans Moravec in robotics.
This is on my mind recently because I've been reading Lakoff's Women, Fire, and Dangerous Things.
Take meaning, for instance — the assignment of referents to symbols. The classical view of meaning is that there are objectively true meanings that are discovered, out there in the world. The embodied view of meaning is that meaning is necessarily subjective; there are no "God's-eye view" meanings: meaning only takes place in minds in bodies, which arrive at meanings not by objectively observing an exterior world, but by participating in the world.
(This connects to the Bayesian view of causality, at least so far as I understand it: reasoning about causation involves reasoning about interventions and not merely about observations. Observed correlation can only tell us about statistical, rather than causal, regularities; in order to discover authentic causes, we have to consider intervention.)
That meaning is subjective does not mean that it is arbitrary, or that you get to come up with whatever meanings you like and give them equal validity to meanings assigned through processes such as language acquisition, science, or social construction. Like Bayesian probability, meaning is subjectively objective: it takes place only inside (world-involved) minds, but you can still do it wrong by not paying attention to the world.
(This connects to the Bayesian view of causality, at least so far as I understand it: reasoning about causation involves reasoning about interventions and not merely about observations. Observed correlation can only tell us about statistical, rather than causal, regularities; in order to discover authentic causes, we have to consider intervention.)
You should read the first few chapters of Causality by Judea Pearl. He details how you can get causal information from static data (if you ignore "just so" correlations with measure 0). Causality (both the book and the pattern) is cool.
My current favourite waste of time is the concept of Bayesian postmodernism. Just putting those two words together invokes a world of delightful wrangling, as approximately anyone who understands one won't understand or will have contempt for the other. (Though I found at least one person - a programmer who's studied philosophy - who got the idea straight away when I posted it to my blog, which is one more than I was expecting.) It is currently a page of incoherent notes and isn't necessarily actually useful for anything as yet and may never be.
Anyway, that's not my point today. My point today is that as part of this, I have to somehow explain Bayesian thinking in a nutshell to people who are highly intelligent, but have no mathematical knowledge and may actually be dyscalculic - but who can and do get the feel of things. I'm trying to get across that this is how learning works already and I just want to make them aware of it. I've run it past a couple of working artists who seemed to get the idea a bit. So I am posting this here for your technical correction.
If you think it's any good, please do run it past artists or critics of your acquaintance. (I'm glancing in AndrewHickey's direction right now.)
"The meaning of a thing is the way you should be influenced by it." - Vladimir Nesov
To explain what "Bayesian postmodernism" means, I first have to try to explain Bayesian epistemology.
Bayesian epistemology is the notion of using this approach to map out the network of your degrees of certainty of your ideas and how they interact, and just how much a new idea should change your existing degrees of certainty.
The application to criticism and understanding of art should be obvious to anyone with even an enthusiast's experience in the field. (And probably not to anyone without.) Postmodernism tells us we can't be certain of anything; Bayesianism tells us precisely how uncertain we should be.
Problems:
No human who claims to be a Bayesian actually has a network mapped out in their head. They're just doing their best. But that people (a) do this and (b) get useful results from it - even in number-based fields, rather than subjective feeling-based ones - is promising.
A word on competing approaches: The model that holds that probability exists in the world, which is the version found in common everyday popular usage and which your statistics textbooks probably taught you how to use, is the frequentist approach. This is a grab-bag of tools and statistical methods to apply to the problem. The easy part is you don't have to know your precise prior. The hard part is that different methods can get different answers, of which only one (if that) can be right, so you have to know which one to apply. The entire frequentist toolkit can be mathematically derived from the Bayesian approach. The Bayesian approach is currently increasingly popular in science and economics, because it gives the right answer if you have your prior right.
If the above only requires minor fixes, I may post-edit based on comments so I can just refer people to this link.
Despite the above section being what I've posted this here for discussion of, this is going to devolve into a thread about postmodernism. So I'll answer some of the obvious here.
Post-script: No-one's coughed up their own skull in horror yet, so I assume I haven't made any glaring technical errors and, modulo a few post-edits, this'll do for now. It's still too mathematical, but diagrams may help - maybe the next version will have some.
Nor has anyone started talking about postmodernism, to my surprise.
PPS: And I'm surprised no-one's disputed "No human who claims to be a Bayesian actually has a network mapped out in their head."