Bayesian reasoning is a particular style of reasoning which involves starting with some initial prior probability of an event occurring, and then updating this probability on the basis of new evidence to produce a posterior probability. In essence, Bayesian methods dictate exactly how much one’s views should change in response to the new evidence. Though the details can be rather technical, having some understanding of Bayesian reasoning can be very useful when evaluating evidence and making decisions.
One prominent application of Bayesian reasoning occurs in interpreting the results of medical test. Suppose there is a test for some disease with only a 1% false positive and 1% false negative rate. This means that only 1% of people who actualyl have the disease will be falsely declared to be healthy, and likewise only 1% of people who are actually healthy will be falsely declared to have the disease. Suppose furthermore that the disease is also rare, affecting only one in two hundred people in the population. Given such low false positive and false negative rates, most people intuitively think that the test is very reliable, so that if you test positive, you almost certainly have the disease. In fact this is not the case, since this reasoning exhibits the fallacy of failing to properly consider the prior probability. Since this particular disease is so rare, false positives will actually outnumber true positives, even though the test is very accurate. To use some concrete numbers as an example, if one thousand people are tested, about 10 false positives are expected (1% of 1000), while about 5 true positives are expected (since one in two hundred people have the disease). Thus of all the people who test positive, only about one third of them actually have the disease (the 5 true positives among 15 people who tested positive). Even those who test positively for this disease, therefore still probably do not really have the disease, though their probability of having it is now much higher (around 33%) than it was before (around 0.5%). This is exactly how Bayesian reasoning works – it allows us to update our estimates of the probability of some event happening or some state of affairs pertaining, given some new evidence.
What is Bayesian analysis?: A simple introduction to the key idea of Bayesianism
How Bayes’ rule can make you a better thinker: Blog post discussing Bayes’ theorem and outlining why it is useful
An intuitive (and short) explanation of Bayes’ theorem: Very helpful introductory article with clear examples