Bayesian confirmation theory, also called Bayesianism, is named in honor of the Reverend Thomas Bayes (1701 - 1761), an English mathematician and Presbyterian minister who proved an important theorem of probability on which the theory relies. Bayesian confirmation theory has gained widespread popularity among contemporary philosophers and scientists. Though not universally accepted, it is arguably the most successful theory of confirmation to date.
Bayesianism presupposes that rational (reasonable) degrees of belief should conform to the five mathematical rules of probability listed on the previous page. Sets of beliefs that follow those rules are said to be probabilistically coherent. In other words, in order to be probabilistically coherent:
Moreover, according to Bayesianism, there are also probabilistic rules for how one’s beliefs can rationally change over time. To represent changing beliefs, we’ll use subscripts to indicate different times: pr1(H) represents your credence in hypothesis H at time 1, just prior to discovering some new evidence E; and pr2(H) represents your credence in H at time 2, just after learning E. These two probabilities are called your prior credence and your posterior credence in H, respectively, relative to evidence E.
Bayesianism endorses the following rule, called the conditionalization rule (or the simple principle of conditionalization),Some authors refer to the conditionalization rule as Bayes’ rule, although the latter term has also been used in reference to Bayes’ theorem, which will be introduced on the next page. To avoid ambiguity, I prefer not to use the term Bayes’ rule in either context. which specifies how your credence in H should change when you learn E. The conditionalization rule says that upon learning E, your unconditional credence in H should be updated to match your prior conditional credence in H given E:
pr2(H) = pr1(H|E)
The right side of that equation represents your conditional credence in H at time 1, assuming the truth of some possible evidence E which you had not yet discovered at that time. The left side represents your credence in H at time 2, after learning that E is in fact true.
Then, at time 2, you gain some new evidence: you learn that the card is, in fact, a face card. In light of this new evidence, you should change your degree of belief in hypothesis Q. Now that you have learned F, your actual (unconditional) credence that the card is the Queen of Hearts should increase from 1/52 to 1/12. So, your posterior credence is pr2(Q) = 1/12, which is the same as your prior conditional credence pr1(Q|F).
The process of updating one’s beliefs in this way is called conditionalization. So, Bayesianism says that you should change your beliefs over time by conditionalizing on any new evidence you acquire. When you conditionalize on new evidence, your credence in any hypothesis may increase, decrease, or stay the same, depending on your prior conditional credences. If conditionalization increases your credence, the evidence is said to confirm the hypothesis. (The term ‘confirm’ is used even if the resulting credence is still quite low. ‘Confirm’ is not a synonym of ‘prove.’) Conversely, if conditionalization decreases your credence in H, the evidence is said to disconfirm H.
Since confirmation is defined in terms of an individual person’s credences, according to Bayesianism, confirmation is subjective: the very same evidence might confirm a hypothesis for one person but disconfirm it for someone else. Such subjectivity may appear to be a serious drawback of Bayesian confirmation theory. However, some Bayesians see it as an advantage, because it explains why reasonable people sometimes disagree even when presented with the same evidence (as often happens with jurors in a courtroom, for example). Moreover, many advocates of Bayesianism argue that the theory can be supplemented with additional rules or principles to make confirmation more objective. We’ll return to this idea at the end of the chapter. (See the page on the Problem of the Priors.)