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I'm reading through these notes I found online, and I'm having a bit of trouble distinguishing Bayes's Theorem and Bayes's Rule. The central idea of Bayesian Confirmation Theory is that the our credences conform to the axioms of probability calculus, so that our credence that hypotheses H is true is represented by a probability function C(H). Bayes's Theorem, that for two outcomes A,B, P(A|B)=P(B|A)P(A)/P(B), can be derived from the definition of conditional probability alone. Bayes's Rule says that, in light of new evidence E, our updated credence should be C*(H)=C(H|E). The author of the notes linked above claims that Bayes's Rule does not follow from the axioms of the probability calculus, so we need to justify it in some other way.

Imagine for a second that Bayes's Rule did not exist/had never been postulated. Surely the symbol P(A|B) would still have meaning beyond its mathematical definition P(A|B):=P(A AND B)/P(B); all definitions have some motivation behind them. It is the probability of A given we know B has obtained. Back to the world of credences then, if we accept that credences are modelled by probabilities, i.e. that my credence for a hypothesis can be respresented by a function C(H) which behaves like a probability function, then surely along with that comes the idea of conditional credences i.e. C(H|E) where E is some evidence. Isn't Bayes's rule just stating the obvious?

  • Yes, it is essentially just the multiplication rule for probabilities, but the idea expands into a breath-taking conception that takes on many related forms in different contexts, although it has a lot of baggage. E.T. Jaynes' book (a preview) is a good read; he kind of describes the deep and shallow sides of things. (Be warned, though, he takes many pop-shots at Fisher.) – dwn Jan 16 '15 at 22:23
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On page 22 (emphasis mine):

Suppose that your probability for rain tomorrow, conditional on a sudden drop in temperature tonight, is 0.8, whereas your probability for rain given no temperature drop is 0.3. The temperature drops. What should be your new subjective probability for rain? It seems intuitively obvious that it ought to be 0.8.
The Bayesian conditionalization rule simply formalizes this intuition. It dictates that, if your subjective probability for some outcome d conditional on another outcome e is p, and if you learn that e has in fact occurred (and you do not learn anything else), you should set your unconditional subjective probability for d, that is, C(d), equal to p.

So indeed, this is stating the obvious, or, 'formalizing an intuition'.

So why is this important? I'm guessing here, but I think it has to do with chronology. Before, the notation P(A|B) was only used when A would be an effect of cause B (and thus, B would occur before A). Now Bayes' rule, talking about hypothesis and evidence rather than cause and effect, says that it also works the other way around: we can also use P(A|B) when A is an hypothesis which implies B (and thus, in some sense, A would occur before B). Bayes' rule teaches us that conditional probability is independent of time.

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