Bayesian ideas revisited

In the last chapter, we talked very briefly about Bayesian ways of thinking. Recall that the Bayesian way of thinking is to let our data shape and update our beliefs. We start with a prior probability, or what we naïvely think about a hypothesis, and then we have a posterior probability, which is what we think about a hypothesis, given some data.

Bayes’ theorem

Bayes’ theorem is arguably the most well-known part of Bayesian inference. Recall that we previously defined the following:

• P(A) = the probability that event A occurs
• P(A|B) = the probability that A occurs, given that B occurred
• P(A, B) = the probability that A and B occur
• P(A, B) = P(A) * P(B|A)

That last bullet can be read as “the probability that both A and B occur is equal to the probability that A occurs x times the probability that B occurred, given that A has already occurred.”

Starting from the last bullet points, we know the...