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The secret technical codeword that cognitive scientists use to mean "rational". Bayesian probability theory is the math of epistemic rationality, Bayesian decision theory is the math of instrumental rationality. Right up there with cognitive bias as an absolutely fundamental concept on Less Wrong.

Philosophy

Classical statistics is a bucket of assorted methods; different "methods" may give different answers for whether, e.g., an experimental result is "statistically significant". In contrast, as the famous Bayesian E. T. Jaynes emphasized, probability theory is math and its results are theorems, every theorem consistent with every other theorem; you cannot get two different results by doing the derivation two different ways.

So is the project of rationality solved? Indeed not. First, probability theory and decision theory are often too computationally expensive to run in practice - it wouldn't take a galaxy-sized computer, so much as an unphysical computer (much larger than the known universe). And second, it's not always clear how the math applies - even in theory, let alone the practice.

But we do know that violations of Bayesianism - even "unavoidable" violations due to lack of computing power - carry a price; a family of theorems demonstrates that anyone who does not choose according to consistent probabilities can be made to accept combinations of bets that are sure losses, or reject bets that are sure wins (the Dutch Book arguments); similarly, Cox's Theorem and its extensions show that anyone who obeys various "common-sensical" constraints on their betting probabilities must be representable in standard probability theory.

In other words, Bayesianism isn't just a good idea - it's the law, and if you violate it, you'll pay some kind of price.

When cognitive psychologists identify a cognitive bias, they know it's an error by comparison to the Bayesian gold standard.

Math

(Needs to be fleshed out.) For introductions see probability theory, decision theory, and this introduction to Bayes' theorem. A widely lauded technical book on this subject is E. T. Jaynes's "Probability Theory: The Logic of Science".

Other usages

"Bayesian" in philosophical usage often describes someone who adheres to the Bayesian interpretation of probability, viewing probability as a level of certainty in a potential outcome or idea. This is in contrast to a frequentist who views probability as a representation of how frequently a particular outcome will occur over any number of trials.

The term "Bayesian" may also refer to an ideal rational agent implementing precise, perfect Bayesian probability theory and decision theory (see, for example, Aumann's agreement theorem).

Blog posts

See also