Bayesian inference

Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. The name comes from the frequent use of the Bayes' theorem in this discipline. Bayes' theorem is named after the Reverend Thomas Bayes.

Evidence and the scientific method

Bayesian statisticians claim that methods of Bayesian inference are a formalisation of the scientific method involving collecting evidence which points towards or away from a given hypothesis; some go further and claim that it is the only method which applies logic to this method. There can never be certainty, but as more evidence accumulates, the degree of belief in a hypothesis will usually become very high (almost 1) or very low (near 0). Bayes theorem provides a method for adjusting degrees of belief in the light of new information.

In many cases, the impact of evidence can be summarised in a likelihood ratio, as expressed in the the law of likelihood. This can be combined with the prior probability to reflect the original degree of belief and any earlier evidence already taken into account. Before a decision is made, the loss function also needs to be considered to reflect the consequences of making an erroneous decision.

Simple examples of Bayesian inference

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 a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. Let H₁ corresponds to bowl #1, and H₂ to bowl #2. It is given that the bowls are identical from Fred's point of view, thus P(H₁) = P(H₂), and the two must add up to 1, so both are equal to 0.5. The "data" D consists in the observation of a plain cookie. From the contents of the bowls, we know that P(D | H₁) = 30/40 = 0.75 and P(D | H₂) = 20/40 = 0.5. Bayes' formula then yields

{\begin{matrix}P(H_{1}|D)&=&{\frac {P(H_{1})\cdot P(D|H_{1})}{P(H_{1})\cdot P(D|H_{1})+P(H_{2})\cdot P(D|H_{2})}}\\\\\ &=&{\frac {0.5\times 0.75}{0.5\times 0.75+0.5\times 0.5}}\\\\\ &=&0.6\end{matrix}}

Before observing the cookie, the probability that Fred chose bowl #1 is the prior probability, P(H₁), which is 0.5. After observing the cookie, we revise the probability to P(H₁|D), which is 0.6.

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 positive 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).

Suppose also, however, that only 0.1% of the population have that disease (i.e. with probability 0.001). We now have all the information required to use Bayes' theorem to calculate the probability that, given the test was positive, that it is a false positive.

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 alternative form of Bayes' theorem (above), the probability of a true positive is

{\begin{matrix}P(A|B)&=&{\frac {0.99\times 0.001}{0.99\times 0.001+0.05\times 0.999}}\,,\\~\\&\approx &0.019\,.\end{matrix}}

and hence the probability of a false positive is about (1 − 0.019) = 0.981.

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 particular, a test must be very reliable in reporting a negative result when the patient does not have the disease, if it is to avoid the problem of false positives. In mathematical terms, this would ensure that the second term in the denominator of the above calculation is small, relative to the first term. For example, if the test reported a negative result in patients without the disease with probability 0.999, then using this value in the calculation yields a probability of a false positive of roughly 0.5.

In this example, 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 possibility of false positives. (The probability of a 'false negative' could also be calculated using Bayes' theorem, to completely characterise the possible errors in the test results.)

In the courtroom

Bayesian inference can be used to coherently assess additional evidence of guilt in a court setting.

Let G be the event that the defendent is guilty.

Let E be the event that the defendent's DNA matches DNA found at the crime scene.

Let p(E | G) be the probability of seeing event E assuming that the defendent is guilty. (Usually this would be taken to be unity.)

Let p(G | E) be the probability that the defendent is guilty assuming the DNA match event E

Let p(G) be the probability that the defendent is guilty, based on the evidence other than the DNA match.

Bayesian inference tells us that if we can assign a probability p(G) to the defendent's guilt before we take the DNA evidence into account, then we can revise this probability to the conditional probability p(G | E), since

p(G | E) = p(G) p(E | G) / p(E)

Suppose, on the basis of other evidence, a juror decides that there is a 30% chance that the defendent is guilty. Suppose also that the forensic evidence is that the probability that a person chosen at random would have DNA that matched that at the crime scene was 1 in a million, or 10^-6.

The event E can occur in two ways. Either the defendent is guilty (with prior probability 0.3) and thus his DNA is present with probability 1, or he is innocent (with prior probability 0.7) and he is unlucky enough to be one of the 1 in a million matching people.

Thus the juror could coherently revise his opinion to take into account the DNA evidence as follows:

p(G | E) = 0.3 × 1.0 /(0.3 × 1.0 + 0.7 × 10^-6) = 0.99999766667.

In the United Kingdom, Bayes' theorem was explained by an expert witness to the jury in the case of Regina versus Denis Adams. The case went to Appeal and the Court of Appeal gave their opinion that the use of Bayes' theorem was inappropriate for jurors.

More mathematical examples

Naive Bayes classifier

See: naive Bayesian classification.

Posterior distribution of the binomial parameter

In this example we consider the computation of the posterior distribution for the binomial parameter. This is the same problem considered by Bayes in Proposition 9 of his essay.

We are given m observed successes and n observed failures in a binomial experiment. The experiment may be tossing a coin, drawing a ball from an urn, or asking someone their opinion, among many other possibilities. What we know about the parameter (let's call it a) is stated as the prior distribution, p(a).

For a given value of a, the probability of m successes in m+n trials is

p(m,n|a)={\begin{pmatrix}n+m\\m\end{pmatrix}}a^{m}(1-a)^{n}

Since m and n are fixed, and a is unknown, this is a likelihood function for a. From the continuous form of the law of total probability we have

p(a|m,n)={\frac {p(m,n|a)\,p(a)}{\int _{0}^{1}p(m,n|a)\,p(a)\,da}}={\frac {{\begin{pmatrix}n+m\\m\end{pmatrix}}a^{m}(1-a)^{n}\,p(a)}{\int _{0}^{1}{\begin{pmatrix}n+m\\m\end{pmatrix}}a^{m}(1-a)^{n}\,p(a)\,da}}

For some special choices of the prior distribution p(a), the integral can be solved and the posterior takes a convenient form. In particular, if p(a) is a beta distribution with parameters m₀ and n₀, then the posterior is also a beta distribution with parameters m+m₀ and n+n₀.

A conjugate prior is a prior distribution, such as the beta distribution in the above example, which has the property that the posterior is the same type of distribution.

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. Interestingly, Bayes actually states his question in a way that might make the idea of assigning a probability distribution to a parameter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial parameter p depend on a random event, he cleverly escapes a philosophical quagmire that he most likely was not even aware was an issue.

Computer applications

Bayesian inference has applications in artificial intelligence and expert systems. Bayesian inference techniques have been a fundamental part of computerized pattern recognition techniques since the late 1950s.

There is growing interest in using Bayesian inference to filter spam. For example: Bogofilter, SpamAssassin and Mozilla.

In some applications fuzzy logic is an alternative to Bayesian inference. Fuzzy logic and Bayesian inference, however, are mathematically and semantically not compatible: You cannot, in general, understand the degree of truth in fuzzy logic as probability and vice versa.

External links

On-line textbook: Information Theory, Inference, and Learning Algorithms, by David MacKay, has many chapters on Bayesian methods, including introductory examples; compelling arguments in favour of Bayesian methods; state-of-the-art Monte Carlo methods, message-passing methods, and variational methods; and examples illustrating the intimate connections between Bayesian inference and data compression.
Naive Bayesian learning paper
A Tutorial on Learning With Bayesian Networks