Bayesian decision theory
Bayesian decision theory refers to decision theory which is informed by Bayesian probability. An agent operating under such a decision theory uses the concepts of Bayesian statistics to estimate the expected value of its actions, and update its expectations based on new information.
Consider any kind of probability distribution - such as the weather for tomorrow (encompassing several variables such as humidity, rain or temperature). From a Bayesian perspective, that represents a prior distribution. That is, it represents how we believe today the weather is going to be tomorrow. By contrast, for a frequentist under classical probability interpretation, it would be a simple distribution function with no special meaning. A Bayesian decision rule is one that consistently tries to minimize the risk of the probability distribution. Such risk can be seen as the difference between the prior beliefs and the real outcomes - the prediction and the actual weather tomorrow.
Computer algorithms such as those studied in the subject of Machine learning can use Bayesian methods also. Besides this explicit implementantions, it also has been observed that naturally evolved [systems] mirror these probabilistic methods when they adapt to an uncertain environment.
What Less Wrong refers to as Rationality is an effort to make conscious thoughts a better approximation of Bayesian decision theory, in order to better understand the world and achieve one's goals.