One of the more convenient forms of uses . Bayes' rule says that, when you observe a piece of evidence your odds for your hypothesis given is just your odds on times the
For example, suppose we're trying to solve a mysterious murder, and we start out thinking the odds of Professor Plum vs. Miss Scarlet committing the murder are 1 : 2, that is, Scarlet is twice as likely as Plum to have committed the murder . We then observe that the victim was bludgeoned with a lead pipe. If we think that Plum, if he commits a murder, is around 60% likely to use a lead pipe, and that Scarlet, if she commits a murder, would be around 6% likely to us a lead pipe, this implies of 10 : 1 for Plum vs. Scarlet using the pipe. The odds for Plum vs. Scarlet, after observing the victim to have been murdered by a pipe, are . We now think Plum is around five times as likely as Scarlet to have committed the murder.
Let denote a of hypotheses. An odds function is a function that maps to a set of . For example, if then might be which says that is 3x as likely as and 6x as likely as An odds function captures our relative probabilities between the hypotheses in for example, (6 : 2 : 1) odds are the same as (18 : 6 : 3) odds. We don't need to know the absolute probabilities of the in order to know the relative odds. All we require is that the relative odds are proportional to the absolute probabilities:
In the example with the death of Mr. Boddy, suppose denotes the proposition "Reverend Green murdered Mr. Boddy", denotes "Mrs. White did it", and denotes "Colonel Mustard did it". Let be the vector If these propositions respectively have probabilities of 80%, 8%, and 4% (the remaining 8% being reserved for other hypotheses), then represents our relative credences about the murder suspects — that Reverend Green is 10 times as likely to be the murderer as Miss White, who is twice as likely to be the murderer as Colonel Mustard.
Suppose we discover that the victim was murdered by wrench. Suppose we think that Reverend Green, Mrs. White, and Colonel Mustard, if they murdered someone, would respectively be 60%, 90%, and 30% likely to use a wrench. Letting denote the observation "The victim was murdered by wrench," we would have This gives us a defined as
Let denote the odds of the hypotheses after observing evidence then states:
This says that we can multiply the relative prior credence by the likelihood to arrive at the relative posterior credence Because odds are invariant under multiplication by a positive constant, it wouldn't make any difference if the likelihood function was scaled up or down by a constant, because that would only have the effect of multiplying the final odds by a constant, which does not affect them. Thus, only the are necessary to perform the calculation; the absolute likelihoods are unnecessary. Therefore, when performing the calculation, we can simplify to the relative likelihoods
In our example, this makes the calculation quite easy. The prior odds for Green vs White vs Mustard were The relative likelihoods were = Thus, the relative posterior odds after observing = Mr. Boddy was killed by wrench are Given the evidence, Reverend Green is 40 times as likely as Colonel Mustard to be the killer, and 20/3 times as likely as Mrs. White.
Bayes' rule states that this relative proportioning of odds among these three suspects will be correct, regardless of how our remaining 8% probability mass is assigned to all other suspects and possibilities, or indeed, how much probability mass we assigned to other suspects to begin with. For a proof, see .
, , and may be helpful for explaining or visualizing the odds form of Bayes' rule.