Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.3.4: An example of legal evidence 553.4 An example of legal evidenceThe following example illustrates that there is more to Bayesian inference thanthe priors.Two people have left traces of their own blood at the scene of acrime. A suspect, Oliver, is tested and found to have type ‘O’blood. The blood groups of the two traces are found to be of type‘O’ (a common type in the local population, having frequency 60%)and of type ‘AB’ (a rare type, with frequency 1%). Do these data(type ‘O’ and ‘AB’ blood were found at scene) give evidence infavour of the proposition that Oliver was one of the two peoplepresent at the crime?A careless lawyer might claim that the fact that the suspect’s blood type wasfound at the scene is positive evidence for the theory that he was present. Butthis is not so.Denote the proposition ‘the suspect and one unknown person were present’by S. The alternative, ¯S, states ‘two unknown people from the population werepresent’. The prior in this problem is the prior probability ratio between thepropositions S and ¯S. This quantity is important to the final verdict andwould be based on all other available information in the case. Our task here isjust to evaluate the contribution made by the data D, that is, the likelihoodratio, P (D | S, H)/P (D | ¯S, H). In my view, a jury’s task should generally be tomultiply together carefully evaluated likelihood ratios from each independentpiece of admissible evidence with an equally carefully reasoned prior probability.[This view is shared by many statisticians but learned British appealjudges recently disagreed and actually overturned the verdict of a trial becausethe jurors had been taught to use Bayes’ theorem to handle complicated DNAevidence.]The probability of the data given S is the probability that one unknownperson drawn from the population has blood type AB:P (D | S, H) = p AB (3.24)(since given S, we already know that one trace will be of type O). The probabilityof the data given ¯S is the probability that two unknown people drawnfrom the population have types O and AB:P (D | ¯S, H) = 2 p O p AB . (3.25)In these equations H denotes the assumptions that two people were presentand left blood there, and that the probability distribution of the blood groupsof unknown people in an explanation is the same as the population frequencies.Dividing, we obtain the likelihood ratio:P (D | S, H)P (D | ¯S, H) = 1 1= = 0.83. (3.26)2p O 2 × 0.6Thus the data in fact provide weak evidence against the supposition thatOliver was present.This result may be found surprising, so let us examine it from variouspoints of view. First consider the case of another suspect, Alberto, who hastype AB. Intuitively, the data do provide evidence in favour of the theory S ′