Table of contents |
2 Bayes' theorem in probability theory 3 Bayesianism 4 Examples 5 References 6 See also |
Historical remarks
Bayes' theorem is a result in probability theory, named after the Reverend Thomas Bayes (1702--61). Bayes worked on the problem of computing a distribution for the parameter of a binomial distribution (to use modern terminology); his work was edited and presented posthumously (1763) by his friend Richard Price, in An Essay towards solving a Problem in the Doctrine of Chances. Bayes' results were replicated and extended by Laplace in an essay of 1774, who apparently was not aware of Bayes' work.
The main result (Proposition 9 in the essay) derived by Bayes is the following: assuming a uniform distribution for the prior distribution of the binomial parameter p, the probability that p is between two values a and b is
What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the parameter p. That is, not only can one compute probabilities for experimental outcomes, but also for the parameter which governs them, and the same algebra is used to make inferences of either kind. In contrast, according to the frequentist interpretation, there is no such thing as a probability distribution for p and therefore some non-probabilistic inference mechanism must be used to reason about p.
Bayes' theorem in probability theory
Bayes' theorem is used in statistical inference to update estimates of the probability that different hypotheses are true, based on observations and a knowledge of how likely those observations are, given each hypothesis. There are discrete and continuous versions of the theorem. Bayesianism, as a school of thought, is based on the notion that Bayes' theorem can be applied to any propositions, whether they are statements about experimental variables or any other kind of statements. In contrast, according to the frequentist school, Bayes' theorem can only be applied to problems in which all statements are about experimental variables alone.
In probability theory, Bayes' theorem is a statement about conditional probabilities that allows the exchange of the order of the eventss. If A and B are two events, Bayes' theorem allows us to calculate the probability of A given B if we know the probability of B given A and the probabilities of each event alone. It is a simple theorem with broad applicability.
To derive Bayes' theorem, note first from the definition of conditional probability that
Each term in Bayes' theorem has a conventional name. The term P(A) is called the prior probability of A. It is "prior" in the sense that it precedes any information about B. Equivalently, P(A) is also called the marginal probability of A. The term P(A|B) is called the posterior probability of A, given B. It is "posterior" in the sense that it is derived from or entailed by the specified value of B. The term P(B|A), for a specific value of B, is called the likelihood function for A. The term P(B) is called the prior or marginal probability of B.
Alternative forms of Bayes' theorem
Bayes' theorem is often embellished by noting that
so the theorem can be restated as
See also the law of total probability.
Bayes' theorem for probability densities
Multiplying that by the prior probability density funtion of p and then normalizing gives the posterior probability distribution of p, and thus updates probabilities in light of the new data given by the opinion poll. Thus if the prior probability distribution of p is uniform on the interval [0,1], then the posterior probability distribution would have a density of the form
- f(p|x) = [constant] p^{x} (1 − p)^{n−x}
Bayesianism
Bayesianism is the philosophical school which holds that the rules of mathematical probability apply not only when probabilities are relative frequencies assigned to random events, but also when they are degrees of belief assigned to uncertain propositions. Updating these degrees of belief in light of new evidence almost invariably involves application of Bayes' theorem.
Examples
From which bowl is the cookie?
To illustrate, suppose there are two bowls full of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?
Intuitively, it seems clear that the answer should be more than 50%, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. Let H_{1} corresponds to bowl #1, and H_{2} to bowl #2. It is given that the bowls are identical from Fred's point of view, thus P(H_{1}) = P(H_{2}), and the two must add up to 1, so both are equal to 50%. The "data" D consists in the observation of a plain cookie. From the contents of the bowls, we know that P(D | H_{1}) = 30/40 = 75% and P(D | H_{2}) = 20/40 = 50%. Bayes' formula then yields
False positives in a medical test
False positives are a problem in any kind of test: no test is perfect, and sometimes the test will incorrectly report a positive result. For example, if a test for a particular disease is performed on a patient, then there is a chance (usually small) that the test will return a postive result even if the patient does not have the disease. The problem lies, however, not just in the chance of a false positive prior to testing, but determining the chance that a positive result is in fact a false positive. As we will demonstrate, using Bayes' theorem, if a condition is rare, then the majority of positive results may be false positives, even if the test for that condition is (otherwise) reasonably accurate.
Suppose that a test for a particular disease has a very high success rate:
- if a tested patient has the disease, the test accurately reports this, a 'positive', 99% of the time (or, with probability 0.99), and
- if a tested patient does not have the disease, the test accurately reports that, a 'negative', 95% of the time (i.e. with probability 0.95).
Let A be the event that the patient has the disease, and B be the event that the test returns a positive result. Then, using the second form of Bayes' theorem (above), the probability of a true positive is
Despite the apparent high accuracy of the test, the incidence of the disease is so low (one in a thousand) that the vast majority of patients who test positive (98 in a hundred) do not have the disease. (Nonetheless, this is 20 times the proportion before we knew the outcome of the test! The test is not useless, and re-testing may improve the reliability of the result.) In this case, Bayes' theorem helps show that the accuracy of tests for rare conditions must be very high in order to produce reliable results from a single test, due to the posibility of false positives.
References
Versions of the essay
- T. Bayes (1763), "An Essay towards solving a Problem in the Doctrine of Chances", Philosophical Transactions of the Royal Society of London, 53.
- T. Bayes (1763/1958) "Studies in the History of Probability and Statistics: IX. Thomas Bayes's Essay Towards Solving a Problem in the Doctrine of Chances", Biometrika 45:296-315 (Bayes's essay in modernized notation)
- T. Bayes "An essay towards solving a Problem in the Doctrine of Chances" (Bayes's essay in the original notation)
Commentaries
- G.A. Barnard. (1958) "Studies in the History of Probability and Statistics: IX. Thomas Bayes's Essay Towards Solving a Problem in the Doctrine of Chances", Biometrika 45:293-295 (biographical remarks)
- D. Covarrubias "An Essay Towards Solving a Problem in the Doctrine of Chances" (an outline and exposition of Bayes's essay)
- S.M. Stigler (1982) "Thomas Bayes' Bayesian Inference," Journal of the Royal Statistical Society, Series A, 145:250-258 (Stigler argues for a revised interpretation of the essay -- recommended)
Additional material
- P.S. Laplace (1774) "Mémoire sur la Probabilité des Causes par les Événements," Savants Étranges 6:621-656, also Oeuvres 8:27-65.
- P.S. Laplace (1774/1986), "Memoir on the Probability of the Causes of Events", Statistical Science, 1(3):364--378.
- S.M. Stigler (1986), "Laplace's 1774 memoir on inverse probability," Statistical Science, 1(3):359--378.
- Jeff Miller Earliest Known Uses of Some of the Words of Mathematics (B) (very informative -- recommended)
- A. Papoulis (1984), Probability, Random Variables, and Stochastic Processes, second edition. New York: McGraw-Hill.
See also