Friday, August 17, 2007
Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. The name "Bayesian" comes from the frequent use of Bayes' theorem in the inference process. Bayes' theorem was derived from the work of the Reverend Thomas Bayes.
Evidence and changing beliefs
From which bowl is the cookie?
False positives result when a test falsely or incorrectly reports a positive result. For example, a medical test for a disease may return a positive result indicating that patient has a disease even if the patient does not have the disease. We can use Bayes' theorem to determine the probability that a positive result is in fact a false positive. We find that if a disease is rare, then the majority of positive results may be false positives, even if the test is accurate.
Suppose that a test for a disease generates the following results:
Suppose also that only 0.1% of the population has that disease, so that a randomly selected patient has a 0.001 prior probability of having the disease.
We can use Bayes' theorem to calculate the probability that a positive test result is a false positive.
Let A represent the condition in which the patient has the disease, and B represent the evidence of a positive test result. Then, probability that the patient actually has the disease given the positive test result is
and hence the probability that a positive result is 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 that the vast majority of patients who test positive do not have the disease. Nonetheless, the fraction of patients who test positive who have the disease (.019) is 19 times the fraction of people who have not yet taken the test who have the disease (.001). Thus the test is not useless, and re-testing may improve the reliability of the result.
In order to reduce the problem of false positives, a test should be very accurate in reporting a negative result when the patient does not have the disease. If the test reported a negative result in patients without the disease with probability 0.999, then
so that 1- 0.5 = 0.5 now is the probability of a false positive.
On the other hand, false negatives result when a test falsely or incorrectly reports a negative result. For example, a medical test for a disease may return a negative result indicating that patient does not have a disease even though the patient actually has the disease. We can also use Bayes' theorem to calculate the probability of a false negative. In the first example above,
The probability that a negative result is a false negative is about 0.0000105 or 0.00105%. When a disease is rare, false negatives will not be a major problem with the test.
But if 60% of the population had the disease, then the probability of a false negative would be greater. With the above test, the probability of a false negative would be
The probability that a negative result is a false negative rises to 0.0155 or 1.55%.
If a tested patient has the disease, the test returns a positive result 99% of the time, or with probability 0.99
If a tested patient does not have the disease, the test returns a negative result 95% of the time, or with probability 0.95. False positives in a medical test
Bayesian inference can be used in a court setting by an individual juror to coherently accumulate the evidence for and against the guilt of the defendant, and to see whether, in totality, it meets their personal threshold for 'beyond a reasonable doubt'.
Bayesian inference tells us that if we can assign a probability p(G) to the defendant's guilt before we take the DNA evidence into account, then we can revise this probability to the conditional probability P(G | E), since
Suppose, on the basis of other evidence, a juror decides that there is a 30% chance that the defendant 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.
The event E can occur in two ways. Either the defendant 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:
The benefit of adopting a Bayesian approach is that it gives the juror a formal mechanism for combining the evidence presented. The approach can be applied successively to all the pieces of evidence presented in court, with the posterior from one stage becoming the prior for the next.
The juror would still have to have a prior for the guilt probability before the first piece of evidence is considered. It has been suggested that this could be the guilt probability of a random person of the appropriate sex taken from the town where the crime occurred. Thus, for a crime committed by an adult male in a town containing 50,000 adult males the appropriate initial prior probability might be 1/50,000.
For the purpose of explaining Bayes' theorem to jurors, it will usually be appropriate to give it in the form of betting odds rather than probabilities, as these are more widely understood. In this form Bayes' theorem states that
Posterior odds = prior odds x Bayes factor
In the example above, the juror who has a prior probability of 0.3 for the defendant being guilty would now express that in the form of odds of 3:7 in favour of the defendant being guilty, the Bayes factor is one million, and the resulting posterior odds are 3 million to 7 or about 429,000 to one in favour of guilt.
In the United Kingdom, Bayes' theorem was explained to the jury in the odds form by a statistician expert witness in the rape case of Regina versus Denis John Adams. A conviction was secured but the case went to Appeal, as no means of accumulating evidence had been provided for those jurors who did not want to use Bayes' theorem. The Court of Appeal upheld the conviction, but also gave their opinion that "To introduce Bayes' Theorem, or any similar method, into a criminal trial plunges the Jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task." No further appeal was allowed and the issue of Bayesian assessment of forensic DNA data remains controversial.
Gardner-Medwin argues that the criterion on which a verdict in a criminal trial should be based is not the probability of guilt, but rather the probability of the evidence, given that the defendant is innocent. He argues that if the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. This will depend on the incidence of the crime and this is an odd piece of evidence to consider in a criminal trial. Consider the following three propositions:
A: The known facts and testimony could have arisen if the defendant is guilty,
B: The known facts and testimony could have arisen if the defendant is innocent,
C: The defendant is guilty.
Gardner-Medwin argues that the jury should believe both A and not-B in order to convict. A and not-B implies the truth of C, but the reverse is not true. It is possible that B and C are both true, but in this case he argues that a jury should acquit, even though they know that they will be letting some guilty people go free.
Other court cases in which probabilistic arguments played some role were the Howland will forgery trial, the Sally Clark case, and the Lucia de Berk case.
Let G be the event that the defendant is guilty.
Let E be the event that the defendant's DNA matches DNA found at the crime scene.
Let P(E | G) be the probability of seeing event E assuming that the defendant is guilty. (Usually this would be taken to be unity.)
Let P(G | E) be the probability that the defendant is guilty assuming the DNA match event E
Let P(G) be the juror's personal estimate of the probability that the defendant is guilty, based on the evidence other than the DNA match. This could be based on his responses under questioning, or previously presented evidence. In the courtroom
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