One Form of Bayes’ Theorem
Bayes’ theorem is often used to mathematically show the probability of some hypothesis changes in light of new evidence. Bayes’ theorem is named after Reverend Thomas Bayes, an ordained Christian minister and mathematician, who presented the theorem in 1764 in his Essay towards solving a problem in the doctrine of chances. Before showing what the theorem is, I’ll recap some basic probability symbolism.
|Pr(A) =||The probability of A being true; e.g. Pr(A) = 0.5 means “The probability of A being true is 50%.”|
|Pr(A|B) =||The probability of A being true given that B is true. For example:
Pr(I am wet|It is raining) = 0.8This means “The probability that I am wet given that it is raining is 80%.”
|Pr(¬A) =||The probability of A being being false (¬A is read as “not-A”); e.g. Pr(¬A) = 0.5 means “The probability of A being false is 50%.”|
|Pr(B ∪ C) =||The probability that B or C (or both) are true.|
|Pr(B ∩ C) =||The probability that B and C are both true.|
|Pr(A|B ∩ C) =||The probability of A given that both B and C are true.|
Some alternate forms:
|One Version||Alternate Forms|
|Pr(¬A)||Pr(~A), Pr(−A), Pr(AC)|
|Pr(B ∪ C)||Pr(A ∨ B)|
|Pr(B ∩ C)||Pr(B ∧ C), Pr(B&C)|
The alternate forms can be combined, e.g. an alternate form of Pr(H|E) is P(H/E).
Bayes’ theorem comes in a number of varieties, but here’s one of the simpler ones where H is the hypothesis and E is the evidence:
In the situation where hypothesis H explains evidence E, Pr(E|H) basically becomes a measure of the hypothesis’s explanatory power. Pr(H|E) is called the posterior probability of H. Pr(H) is the prior probability of H, and Pr(E) is the prior probability of the evidence (very roughly, a measure of how surprising it is that we’d find the evidence). Prior probabilities are probabilities relative to background knowledge, e.g. Pr(E) is the likelihood that we’d find evidence E relative to our background knowledge. Background knowledge is actually used throughout Bayes’ theorem however, so we could view the theorem this way where B is our background knowledge:
To simplify it though I’ll leave the background knowledge in Bayes’ theorem implicit.
Here’s an example of Bayes’ theorem in action. Suppose we have a lottery and the odds are 1 in 5,461,512 that the following lottery numbers are chosen:
(4) (19) (26) (42) (51)Let H be the hypothesis that the above lottery numbers were chosen. Let E be a newspaper called The Likely Truth reporting those numbers. The Likely Truth reports the lottery numbers with 99% accuracy (though it never fails to report some series of five lottery numbers of the sort that the lottery can result in, accurate or not), thereby making, making Pr(E|H) = 0.99. The odds that any particular series of five lottery numbers will be reported is likewise 1 in 5,461,512, making Pr(E) = 1 in 5,461,512. With that, we have the following probabilities:
Another Form of Bayes’ Theorem
Keeping in mind I’ll leave the background knowledge in Bayes’ theorem implicit, another form of is Bayes’ theorem is this:
In my next blog entry I’ll show how Bayes’ theorem can be used for theism.