It is important to be very clear about our definitions of probabilities.

Transcription

1 Use Bookmarks for electronic content links 7.6 Bayesian odds Introduction The basic ideas of robability have been introduced in Unit 7.3 in the book, leading to the concet of conditional robability. We now introduce the Bayesian aroach to robability that uses a 'likelihood ratio' to quantify the way in which new information can change the exectation, or 'odds', that a articular condition is true. In forensic terms, a court would be interested in whether the evidence, E, is sufficiently strong to ersuade a jury to accet the statement, S, of events roosed by the rosecution, or to accet the roosition from the defence that the statement is false, S. We introduce the concets of 'rior odds' and 'osterior odds' as the 'odds' before and after introducing a new iece of evidence. Bayesian statistics have articular advantages due to their simlicity in combining the cumulative 'odds' of a range of different contributing factors Conditional and Joint Probabilities It is imortant to be very clear about our definitions of robabilities. (A B) is the conditional robability (7.4.7) that outcome A will occur, given that condition B already exists. We now define (A B) as the joint robability that both outcomes A and B are true simultaneously. The joint robability of A occurring with B must be the same as the joint robability of B occurring with A, and we can deduce that: (A B) (B A) [7.32] Examle 7.30 Using the data from examle Examle 7.20, calculate the joint robability that a man, selected at random, will both have the articular disease AND test ositive, (P D), i.e. the robability of recording a true ositive. The robability that a man, selected at random, will have the disease, (D) Given that he has the disease, the man must then test ositive to satisfy the joint robability. The conditional robability that he will test ositive, given that he has the disease, is (P D) 0.9. Hence the joint robability of having the disease AND testing ositive is given by: (P D) (P D) (D)

2 We can write the general relationshi: (A B) (A B) (B) [7.33] Examle 7.31 In a grou of 10 eole, one erson, G, was resonsible for breaking a window. The robability that G will have glass on his/her clothing is estimated to be (glass G) The robability that a different erson, G, may have glass on his/her clothing is estimated to be (glass G ) 0.2. Calculate the robabilities that a erson selected at random from the grou of 10: i) has glass on his/her clothing and is the erson (G) who broke the window (glass G) ii) has glass on his/her clothing and is not the erson (G ) who broke the window (glass G ) Answer i) Probability that the erson G will be selected at random, (G) 1/ Given that (glass G) 0.99 we can use [7.33] to write: (glass G) (glass G) (G) ii) Probability that a erson selected at random will not be G, (G ) 1 - (G) Given that (glass G ) 0.2 we can use [7.33] to write: (glass G ) (glass G ) (G ) Q7.33 A oulation of 200,000 ossible susects is screened using a DNA test to match a secimen collected from a crime scene. It is estimated that the DNA results would roduce a random, false ositive, match for 1 erson in 1.0 million. Assume that the erson, K, who left the secimen, would definitely rovide a DNA match. Calculate the robabilities that a erson selected at random from the 200,000 ossible susects: i) has a DNA match and is the erson who left the DNA secimen, ii) has a DNA match but is not the erson who left the DNA secimen Bayes' Rule From [7.33] we have: (A B) (A B) (B) Similarly using [7.32] we could write: (B A) (B A) (A) 7.6.2

3 and, using [7.31], we can combine the two equations above, to give: (A B) (B) (B A) (A) which then gives Bayes' Rule: ( A) ( B) ( A B) ( B A) [7.34] The above equation is very imortant because it 'transoses the conditional'. It changes from the robability of B, given that A is true, to the robability of A, given that B is true. This is a subtle, but very significant, change. For examle, in a diagnostic test (e.g. in Examle 7.20) the scientist is rimarily concerned with the accuracy of the test, i.e. "what is the robability that the test will give a ositive result, given that the atient does have the disease?". However, from the atient's oint of view, the more imortant question (with the transosed conditional) is "what is the robability that I have the disease, given that I have just tested ositive?". The robability, (B), of getting outcome B will be the sum of the robability of getting B with A true lus the robability of getting B with A false ( A or not A): ( B) ( B A) + ( B A ) ( B A) ( A) + ( B A ) ( A ) [7.35] We can substitute (B) from [7.35] into [7.34] and obtain: ( A B) ( B A) ( A) [7.36] { ( B A) ( A) + ( B A ) ( A )} This is a very useful rearrangement of Bayes' Rule, as we will see in the following examle

4 Examle 7.32 Using the data again from examle Examle 7.20, we know that the robability that a man with the disease, will test ositive in a diagnostic test is given by, (P D) 0.9. i.e. there is a 90% chance that a man with the disease will test ositive. But what is the robability, (D P), that a man selected at random, who then tests ositive, will actually have the disease? (Note the change of the conditional in this roblem) To answer this we use [7.36]: ( D P) ( P D) ( D) { ( P D) ( D) + ( P D ) ( D )} We have the necessary data from Examle 7.15: (P D) 0.9, (P D ) 0.2, (D) 0.05 and (D ) 1 - (D) 0.95 Combining these gives: ( D P) { } This is the same result as was achieved in [7.32]. Examle 7.33 Using the data from examle Examle 7.31, calculate the robability, (G glass), that a erson, selected at random, who is found to have glass on his/her clothing, is the erson, G, who broke the window Using [7.35] and data from Examle 7.31: ( G glass) ( glass G) ( G) { ( glass G) ( G) + ( glass G ) ( G )} Q7.34 Using the data from examle Q7.33, calculate the robability that a erson, selected at random from the 200,000, and who is found to have a DNA match, will be the erson, K, who was resonsible for leaving the secimen Bayesian Odds If the ossible outcomes of a statistical trial are either A or A, then: (A) 1 - ( A ) The Odds of A, O(A), are an alternative way of exressing the likelihood of a articular event, A, being true, and are defined: O A ( A) ( A ) 1 ( A) ( A) [7.37] 7.6.4

5 This equation can be rearranged to give: ( A) O A [7.38] 1 + O A The relationshis between robabilities, '', and odds, 'O', are illustrated in Table 7.3: (A) ( A ) Odds, O(A) Odds For - - Evens 9:1 99:1 Odds Against 99:1 9:1 Evens - - Table 7.3 Odds and Probabilities Examle 7.34 Using the data again from examle Examle 7.20, i) Calculate the odds, before being tested, that a man, selected at random, has the articular disease, given that (D) 0.05 ii) Use the result from Examle 7.32 to calculate the conditional odds, O(D P), that a man who has tested ositive does indeed have the articular disease. iii) How significant is a ositive test result to a man selected at random? Answer: i) Using [7.36], we can calculate the odds: O(D) (D) / {1 - (D)} 0.05 / ii) From Examle 7.18, (D P) O(D P) (D P) / {1- (D P)} / iii) As far as the man is concerned, the ositive result from the test has made it 0.236/ times more likely that he does have the disease. From examle Examle 7.20: Odds before the test are called Prior Odds O(D) (D) / {1- (D)} Odds after testing ositive are called Posterior Odds O(D P) (D P) / {1- (D P)} [7.39] [7.40] Using Bayes Rule from [7.33], we can write: ( D P) ( P D) ( D) ( P) ( D ) D P P D ( P) 7.6.5

6 ( D) ( P) ( P D) ( D P) ( P D) O D P ( D ( D ) ( P D ) D P D ( P) which simlifies to ( P D) ( D ) ( D) O ( D P) O( D) [7.41] P We can now define the Likelihood Ratio by using the equations: Posterior Odds Likelihood Ratio Prior Odds [7.42] Likelihood Ratio, LR ( P D) P ( D ) [7.43] The effect of the test has been to change the Odds of the man having the disease by multilying the Prior Odds by the Likelihood Ratio Calculating Likelihood Ratios The Likelihood Ratio (LR) can be calculated directly by using a quantitative assessment of the evidence, i.e. calculating the alternative robabilities of a given outcome, P, given two exclusive conditions, D and D. In a courtroom situation, a jury may be faced with the roblem of deciding whether a statement describing a situation or an event, is true. The rosecution may claim that the statement is true, S, while the defence claims that the statement is not true, S. Evidence, E, may be resented which changes the odds, O(S E), of the statement being true, S, given the evidence. A 'new' iece of evidence, E, will change the odds that the statement is true. The Likelihood Ratio (LR) for this new iece of evidence is given by the ratio of the robabilities of the 'evidence' occurring given that the statement is either true, (E S), E S : or not true, Then using [7.40] we get: Likelihood Ratio, LR ( E S) ( S ) E [7.44] ( S E) O ( E S) E ( S ) O ( S) Posterior Odds Likelihood Ratio Prior Odds [7.45] 7.6.6

7 suorting statement, given the new evidence due to new evidence suorting statement An imortant asect of the above equation, [7.44], for the forensic scientist, is the fact that the data required to calculate the Likelihood Ratio is in a form that is roduced by scientific investigation. It is necessary only to calculate the robabilities of observing the evidence, E, given the two conditions that the statement is either true or not true. The equation then rovides the osterior odds, O(S E), with the transosed conditional, i.e. the robability of the statement being true, given the new evidence. The osterior odds are resented in a form aroriate for consideration in court. Examle 7.35 Use the data from Examle 7.20 to calculate the Likelihood Ratio for having the disease, given a ositive result from the diagnostic test. From Examle 7.20, (P D) 0.9 and (P D ) 0.2 Using [7.42] ( P D) 0.9 Likelihood Ratio, LR 4.5 ( P D ) 0.2 This agrees with the increased odds calculated in Examle Examle 7.36 Assuming that the roortion of left-handed eole is 10%, and that it is known that the erson, G, who committed a crime was definitely left-handed, calculate the likelihood ratio for the evidence of left-handedness. The robability of being left-handed for the culrit, (L G) 1.0 For a randomly selected erson, the robability of being left-handed, (L G ) 1/ Hence, using [7.43] the likelihood ratio for the evidence of left-handedness becomes: LR ( L G) L ( G ) Q7.35 For the roblem outlined in Q7.34 and Q7.33, calculate i) Prior Odds that a randomly selected erson is the erson, K, who left the samle, O(K), ii) Likelihood Ratio due to the evidence, E, of a DNA match, iii) Posterior Odds that a randomly selected erson, who gives a DNA match, is the erson, K, who left the samle, O(KIE), iv) Does the result in (iii) agree with the result in Q7.34? 7.6.7

8 Q7.36 A articular blood grou is resent in only 5% of the oulation. If it is known that the erson who committed a articular crime had this blood grou, calculate the likelihood ratio for this evidence when used to assess whether a susect committed the crime. Assume that the exerimental measurement of the blood grou is 100% accurate. The advantage of using the Bayesian, 'Odds', aroach is that it is easy to combine different factors that lead to an overall robability, by simly multilying the relevant Likelihood Ratios. However, for combination by simle multilication, it is necessary that the robabilities of the two tests are indeendent of each other. Examle 7.37 In examle Examle 7.31, a witness states that the erson, G, who broke the window was left-handed. Assuming that the roortion of left-handed eole is 10%, calculate the osterior odds that G is the erson who has been randomly selected from the grou of 10, and who is both left-handed and has glass on his clothing. Probability of selecting the erson, G, at random from a grou of 10, (G) 0.1. Hence the rior odds for randomly selecting, G, before considering the evidence: O ( G) ( G) G Using data from Examle 7.31, the Likelihood Ratio for discovery of glass on clothing, ( glass G) 0.99 LR ( glass) ( glass G ) 0.2 From Examle 7.22, the Likelihood Ratio for 'left-handedness', ( left G) 1.0 LR ( left) 10 ( left G ) 0.1 The osterior odds, taking both tests into account, (we assume that the robability of glass on the clothing is indeendent of whether the culrit is left-handed or not) O(G glass,left) LR(glass) LR(left) O(G) It is not ossible to use simle multilication of likelihood ratios if the two ieces of evidence are not indeendent. For examle, if the ieces of evidence were that a erson was believed to be male and that he (or she) was over 6 feet tall, then the likelihood ratio for the height will be different deending on whether the erson is actually male or female. In these situations it is necessary to investigate the robabilities of the different otions using a form of 'robability tree'. Such calculations are very deendent on the articular situation involved and are not covered in this book