Modeling complex legal cases as a Bayesian network using idioms and sensitivity analysis with the Collins case as a complete example

Transcription

1 Modeling complex legal cases as a Bayesian network using idioms and sensitivity analysis with the as a complete example FENTON, NE; DE ZOETE, J; Lagnado, D; 10th International Conference on Forensic Inference and Statistics The Author(s) 2017 For additional information about this publication click this link. Information about this research object was correct at the time of download; we occasionally make corrections to records, please therefore check the published record when citing. For more information contact scholarlycommunications@qmul.ac.uk

2 See discussions, stats, and author profiles for this publication at: Modeling complex legal cases as a Bayesian network using idioms and sensitivity analysis with the as a complete example Presentation September 2017 DOI: /RG CITATIONS 0 READS 3 3 authors: Jacob De Zoete Queen Mary, University of London 18 PUBLICATIONS 26 CITATIONS Norman Elliott Fenton Queen Mary, University of London 248 PUBLICATIONS 11,924 CITATIONS SEE PROFILE SEE PROFILE David Lagnado University College London 106 PUBLICATIONS 1,951 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Modelling crime linkage View project PAMBAYESIAN (PAtient Managed decision-support using Bayesian networks) View project All content following this page was uploaded by Jacob De Zoete on 26 January The user has requested enhancement of the downloaded file.

3 Modeling complex legal cases as a Bayesian network using idioms and sensitivity analysis with the as a complete example Norman Fenton, David Lagnado, Queen Mary University of London, University College London September 6, 2017

4 Evidence model friendship statement witness 1 friend of suspect witness 2 reliable fingermark recovery transfer witness 1 reliable offender persistence transfer exonerating statement DNA evidence recovery persistence red hair same donor DNA evidence 2 recovery

5 Case model friendship statement witness 1 friend of suspect witness 2 reliable fingermark recovery transfer witness 1 reliable offender persistence transfer exonerating statement DNA evidence recovery persistence red hair same donor DNA evidence 2 recovery

6 Case model friendship statement witness 1 friend of suspect witness 2 reliable fingermark recovery transfer witness 1 reliable offender persistence transfer exonerating statement DNA evidence recovery persistence red hair same donor DNA evidence 2 recovery

7 Case model friendship statement witness 1 friend of suspect witness 2 reliable recovery fingermark transfer witness 1 reliable offender persistence transfer exonerating statement DNA evidence recovery persistence red hair same donor DNA evidence 2 recovery

8 Evidence model Case model overseeable uncertainty Simple more uncertainties structured (sub)-problem more relevant to fact finder situation specific Realistic case specific pre-trial

9 On June 18, 1964, a woman named Juanita Brooks was walking home from grocery shopping. As she made her way down an alley, she suddenly felt herself being pushed to the ground. When she looked up, she saw a young woman with a blond ponytail running away down the alley with her purse. Near the end of the alley, a man named John Bass saw a woman run out of the alley and jump into a yellow car. The car took off and passed close by him. Bass subsequently described the driver as black, with a beard and a mustache. He described the woman as Caucasian, slightly over five feet tall, with dark in a ponytail. Several days later, the police arrested Janet Collins and her husband Malcolm Collins.

10 Juanita Brooks witness statement

11 John Bass witness statement

12 Witness statement - idiom witness reliable

13 Witness statement - idiom witness reliable witness accurate (statement)

14 Witness statement - idiom witness reliable witness accurate (statement) witness statement

15 Witness statement - idiom witness reliable witness accurate (statement) witness statement event happened

16 Witness statement - idiom John Bass reliable John Bass accurate blond John Bass: woman blond woman blond

17 Witness statement - idiom John Bass reliable John Bass reliable John Bass accurate John Bass accurate pony blond tail John Bass: woman John blond Bass: pony tail woman woman pony blond tail

18 Witness statement - idiom John Bass reliable John Bass reliable Juanita reliable John Bass accurate John Bass accurate Juanita pony blond tail accurate blond John Bass: woman John blond Bass: pony Juanita: tail mugger blond woman woman pony blond tail mugger blond

19 Blond woman evidence /

20 Blond woman evidence crying screaming from alley mugger woman is mugger woman

21 Blond woman evidence crying screaming from alley mugger woman is mugger woman mugger woman

22 Blond woman evidence crying screaming from alley woman is mugger juanita reliable mugger woman juanita accurate mugger woman juanita states

23 Blond woman evidence crying screaming from alley woman is mugger juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

24 Blond woman evidence crying screaming from alley woman is mugger juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

25 Blond woman evidence crying screaming from alley john states crying screaming woman is mugger john accurate crying screaming juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

26 Blond woman evidence crying screaming from alley john states crying screaming woman is mugger john accurate crying screaming juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

27 Blond woman evidence crying screaming from alley john states crying screaming woman is mugger john accurate crying screaming juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

28 Blond woman evidence crying screaming from alley john states crying screaming woman is mugger john accurate crying screaming juanita reliable mugger woman john reliable juanita accurate mugger woman john accurate juanita states john states

29 Malcolm - Money on June 18 While in custody, the police searched Malcolm and found receipts for $35 in traffic fines paid on the day after the robbery. This amount matched the amount in the stolen purse. Malcolm said he had won the money gambling, but Janet said the money came from her earnings.

30 Malcolm - Money on June 18

31 Malcolm - Money on June 18

32 Malcolm - Money on June 18

33 Malcolm - Money on June 18

34 Malcolm - Money on June 18

35 net accomplice (handbag) illegal source lottery winnings money on June 18 janet earnings savings 35$ paid on June 18

36 Distribution f (money on June 18 = x) 35 x handbag Figure: Conditional distribution money on June 18

37 Distribution f (money on June 18 = x) 35 x handbag Figure: Conditional distribution money on June 18

38 Distribution f (money on June 18 = x) 35 x 100% handbag 35 Figure: Conditional distribution money on June 18

39 Distribution f (money on June 18 = x) 35 x earnings Figure: Conditional distribution money on June 18

40 Distribution f (money on June 18 = x) 35 x earnings + lottery Figure: Conditional distribution money on June 18

41 Distribution f (money on June 18 = x) 84% 16% 35 x earnings + lottery < 35 earnings + lottery 35 Figure: Conditional distribution money on June 18

42 Money evidence Pr( 35$ on June 18 accomplice (handbag) ) Pr( 35$ on June 18 [lottery, janet earnings] ) = = 6.25

43 jb.r < cptable( john bass reliable, levels = c( true, false ), values = c(0.7, 0.3)) 1 2 3

44 jb.r < cptable( john bass reliable, node name levels = c( true, false ), values = c(0.7, 0.3)) 1 2 3

45 jb.r < cptable( john bass reliable, levels = c( true, false ), states values = c(0.7, 0.3)) 1 2 3

46 jb.r < cptable( john bass reliable, levels = c( true, false ), values = c(0.7, 0.3)) probability table 1 2 3

47 jb.r < cptable( john bass reliable, levels = c( true, false ), values = c(0.7, 0.3)) jb.a < cptable( john bass accurate + john bass reliable, levels = c( true, false ), values = c(0.9, 0.1, 0.0, 1.0))

48 jb.r < cptable( john bass reliable, levels = c( true, false ), values = c(0.7, 0.3)) jb.a < cptable( john bass accurate + john bass reliable, levels = c( true, false ), node parents values = c(0.9, 0.1, 0.0, 1.0))

49 - reliable f (bass reliable = x) x 0.70 Figure: Prior probability John Bass reliable

50 - reliable f (bass reliable = x) x 0.70 Figure: Prior probability John Bass reliable

51 - accurate f (bass accurate = x) x 0.90 Figure: Cond. probability John Bass accurate

52 - accurate f (bass accurate = x) x 0.90 Figure: Cond. probability John Bass accurate

53 4 jb.r < cptable( john bass reliable, 5 levels = c( true, false ), 6 values = c(0.7, 0.3)) 7 8 jb.a < cptable( john bass accurate + john bass reliable, 9 levels = c( true, false ), 10 values = c(0.9, 0.1, 0.0, 1.0)) 11 12

54 N < 1000 p.reliable < rnorm(n, 0.70, 0.05) # normal distribution p.accurate < runif(n, 0.80, 1.00) # uniform distribution 4 jb.r < cptable( john bass reliable, 5 levels = c( true, false ), 6 values = c(0.7, 0.3)) 7 8 jb.a < cptable( john bass accurate + john bass reliable, 9 levels = c( true, false ), 10 values = c(0.9, 0.1, 0.0, 1.0))

55 N < p.reliable < rnorm(n, 0.70, 0.05) # normal distribution 2 p.accurate < runif(n, 0.80, 1.00) # uniform distribution 3 for ( i in 1:N){ 4 jb.r < cptable( john bass reliable, 5 levels = c( true, false ), 6 values = c(p.reliable[i], 1 p.reliable[i])) 7 8 jb.a < cptable( john bass accurate + john bass reliable, 9 levels = c( true, false ), 10 values = c(p.accurate[i], 1 p.accurate[i], 0.0, 1.0)) 11 } 12

56 f (Malcolm accomplice = x) x 0.81 Figure: Post. probability Malcolm accomplice

57 f (Malcolm accomplice = x) x Figure: Post. probability Malcolm accomplice

58 f (Malcolm accomplice = x) x Figure: Post. probability Malcolm accomplice

59 Outliers reliable all low posterior high posterior Pr(bass reliable)

60 Outliers reliable all low posterior high posterior Pr(bass accurate )

61 Idioms for pieces of evidence simplify the construction of complex case models. The proposed method of sensitivity analysis can help - especially for complex cases - to determine whether there is sufficient evidence for a case to be brought to court (R) programming allows for a more thorough sensitivity analysis, for example by allowing for ranges or distributions for probability estimates instead of point estimates.

62 Acknowledgments This work is supported by the ERC project BAYES KNOWLEDGE (ERC-2013-AdG339182)