Bayes’ Theorem

On its own, the conditionalization rule isn’t very illuminating. However, it becomes much more useful when we combine it with an important theorem of the probability calculus, a theorem first proved by Thomas Bayes himself:

Bayes’ theorem:  pr(X|Y) = pr(Y|X) × pr(X)/pr(Y),  provided  pr(Y) ≠ 0.

By applying Bayes theorem, we can rewrite the conditionalization rule as follows:

pr2(H)  =  pr1(H|E)

            =  pr1(E|H) × pr1(H)/pr1(E)

            =  pr1(H) × pr1(E|H)/pr1(E)

This way of writing the equation provides unambiguous answers to two of the crucial questions that hypothetico-deductivism couldn’t answer. First, it tells us exactly how much the hypothesis is confirmed or disconfirmed by the evidence. Whenever you obtain new evidence relevant to hypothesis H, you should change your credence in H by a factor of pr1(E|H)/pr1(E), which has been called the Bayesian multiplier:
Bayesian multiplier  =   pr1(E|H)

The Bayesian multiplier is the factor by which your credence in hypothesis H should increase or decrease when you learn E. For example, if the Bayesian multiplier is equal to 1, then your credence in H shouldn’t change when you learn E. If pr1(E|H) is twice as high as pr1(E), then the Bayesian multiplier is 2, which means that your credence in H should double when you learn E. And if pr1(E|H) is only half as high as pr1(E), then the Bayesian multiplier is ½, so your credence in H should decrease: you should be only half as confident in H as you were before.

To understand intuitively how the Bayesian multiplier works, consider the following illustration. Suppose you’re planning a safari trip and you believe you’re only 10% likely to encounter elephants in Ethiopia: your prior credence in E is pr1(E) = 0.1. However, you think you’re three times more likely to encounter elephants given the hypothesis that the habitat has a huge herd: your prior conditional credence in E given H is pr1(E|H) = 0.3. Then, you go on the safari and you do encounter elephants in Ethiopia! In that case, the Bayesian multiplier is:

pr1(E|H)/pr1(E)   =   0.3/0.1   =   3

Since the Bayesian multiplier is 3, your credence that the habitat has a huge herd should triple when you encounter elephants. For instance, if you previously believed with 20% confidence that the habitat has a huge herd—that is, if your prior credence pr1(H) was 0.2—then, upon encountering elephants, your credence should increase to 60% confidence in the “huge herd” hypothesis: your posterior credence pr2(H) should be 0.6.

The rewritten equation for the conditionalization rule also answers a second important question that hypothetico-deductivism couldn’t answer: namely, why successful predictions that are surprising or unexpected provide stronger evidence for a hypothesis than unsurprising ones do. Here’s why. To say that hypothesis H “predicts” some phenomenon E means that pr1(E|H) is high, i.e., close to 1. Moreover, to say that E is “unexpected” just means that pr1(E) is low. Therefore, if hypothesis H predicts some unexpected phenomenon E, the Bayesian multiplier pr1(E|H)/pr1(E) is large, so the probability of H increases dramatically when E is discovered. This explains why a hypothesis that predicts surprising or unexpected phenomena is strongly confirmed when those predictions are tested and found to be correct. In contrast, testing an unsurprising prediction won’t give the hypothesis much support. The prior probability of an unsurprising phenomenon will be relatively high, and because this probability is in the denominator of the Bayesian multiplier, the resulting increase in your degree of belief will be small.

There are other ways of reformulating both Bayes’ theorem and the conditionalization rule to reveal further insights about confirmation. For example, the conditionalization rule can be expressed in a way that clarifies the role of background information and auxiliary hypotheses in the hypothetico-deductive method. Also, we can use another version of Bayes’ theorem to see how the probabilities of two or more rival hypotheses are affected by the same evidence, whereas hypothetico-deductivism only deals with one hypothesis at a time.For further discussion of the advantages of Bayesianism, see: Howson & Urbach (2005), Scientific Reasoning: The Bayesian Approach (3rd ed.). Open Court. These are important advantages of Bayesianism over other theories of confirmation. However, Bayesian confirmation theory also has some limitations, as we’ll see in the next section.