Introduction to Bayesian Epistemology

1. Deriving Bayes' Theorem Bayes' theorem is a piece of mathematics. It is called a theorem because it is derivable from a simple definition in probability theory. As a piece of mathematics, it is not controversial. Bayesianism, on the other hand, is a controversial philosophical theory in epistemology. It proposes that the mathematics of probability theory can be put to work in explicating various concepts connected with issues about evidence and confirmation. Before I make an observation, I assign a probability to the hypothesis H; this probability may be high, medium, or low. After I make the observation, thereby learning that some observation statement O is true, I want to update the probability I assigned to H, to take account of what I have just learned. The probability that H has before the observation is called its prior probability; it is represented by Pr(H). The probability that H has in the light of the observation O is called its posterior probability; it is represented by the conditional probability Pr(H*O) (read this as " the probability of H, given O "). Bayes' theorem shows how the prior and the posterior probability are related. This definition is intuitive. What is the probability that a card drawn at random from a standard deck is a heart, given that it is red? Well, the probability that it is a red heart is 1/4; the probability that it is red is ½. Thus, the answer is: ½. By switching A's and B's with each other, it will also be true that Pr(B*A) = Pr(A&B)/Pr(A). These two expressions allow the probability of the conjunction (A&B) to be expressed in two different ways: