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 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: Michael Strevens uses this term in his article “The Bayesian Approach to the Philosophy of Science,” in Donald M. Borchert (ed.), Macmillan Encyclopedia of Philosophy, Second Edition (Detroit: Macmillan Reference USA, 2006); preprint available here. To my knowledge, this is the earliest published use of the term “Bayesian multiplier.” This concept should not be confused with the widely-used “Bayes factor,” which will be introduced in the next chapter.
| Bayesian multiplier = | pr1(E|H) |
| pr1(E) |
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.
In this video, YouTube educator Grant Sanderson explains Bayes’ theorem using helpful animations and geometrical diagrams to make the mathematics more intuitive.This video was uploaded to YouTube by the copyright holder, who enabled the option to embed the video in the YouTube “Embeddable Player.” According to the Terms of Service, section 6C, this constitutes an agreement to “grant each user of the Service a non-exclusive license to access your Content through the Service, and to use, reproduce, distribute, display and perform such Content as permitted through the functionality of the Service and under these Terms of Service.”
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, Scientific Reasoning: The Bayesian Approach, Third Edition (Chicago: Open Court, 2006). 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 final sections of this chapter.