1. INTERPRETING COMPLEX DNA PROFILE EVIDENCE:  BAYESIAN NETWORKS TO THE RESCUE
Philip Dawid
University of Cambridge
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2. Difficulties of Formalizing Reasoning
Classical logic does not readily handle “non-monotonic” reasoning
Reasoning with uncertainty is especially delicate
but specification and manipulation of probabilities appears problematic

3. Example: “Explaining Away”
Burglar alarm is ringing
Break-in?
Earthquake?
Radio reports earthquake in vicinity
report earthquake
earthquake alarm
alarm break-in
So report break-in
???

4. PROBABILISTIC REASONING IN INTELLIGENT SYSTEMS Networks of Plausible Inference Pearl 1988

5. Go with the (causal) flow ?

6. BAYESIAN NETWORKS
Handle complex problems involving probabilistic uncertainty
Modular structure
Intuitive graphical representation
Precise semantics
relevance (conditional independence)
Correct accounting for evidence
Computational algorithms
elegant and efficient

7. AN APPLICATION Forensic Identification DNA Profiling
Disputed Paternity

8. FORENSIC USES FOR DNA PROFILES
Murder/Rape/…: Is A the culprit?
Paternity: Is A the father of B?
Immigration: Is A the mother of B? How are A and B related?
Disasters: 9/11, tsunami, Romanovs,…

9. DNA Profile From blood, saliva, semen, hair root,…
Can be amplified from a single cell
Record genotypes for 12–20 DNA markers
unlinked (different chromosomes)

10. A typical DNA profile

11. Short Tandem Repeat markers:
hypervariable “junk” (nuclear) DNA
**|GTAC|GTAC|GTAC|GTAC|**  4 repeats
(allele)
genotype, e.g. 7/13 or 14

12. D7S820
D7S880 is one of the 13 core CODIS STR genetic loci. This DNA is found on human chromosome 7. The DNA sequence of a representative allele of this locus is shown below. The tetrameric repeat sequence of D7S280 is GATA. Different alleles of this locus have from 6 to 15 tandem repeats of the GATA sequence.
Repeat number = 12 – or possibly 14??
001 AATTTTTGTATTTTTTTTAGAGACGGGGTTTCACCATGTTGGTCAGGCTGACTATGGAGT
061 TATTTTAAGGTTAATATATATAAAGGGTATGATAGAACACTTGTCATAGTTTAGAACGAA
121 CTAACGATAGATAGATAGATAGATAGATAGATAGATAGATAGATAGATAGATAGATAGAT
181 TGATAGTTTTTTTTTATCTCACTAAATAGTCTATAGTAAACATTTAATTACCAATATTTG
241 GTGCAATTCTGTCAATGAGGATAAATGTGGAATCGTTATAATTCTTAAGAATATATATTC
301 CCTCTGAGTTTTTGATACCTCAGATTTTAAGGCC

13. Disputed Paternity (Essen-Möller 1938)
We have DNA data D from a disputed child c, its mother m and the putative father pf
If the true father tf is not pf, he is a “random” alternative father af
Straightforward to compute the evidence (LIKELIHOOD RATIO) in favor of paternity
(Essen-Möller 1938)

14. Disputed Paternity
LIKELIHOOD RATIO
(Essen-Möller 1938)

15. MISSING DNA DATA
What if we can not obtain DNA from the suspect ? (or other relevant individual?)
Sometimes we can obtain indirect information by DNA profiling of relatives
But analysis is complex and subtle…

16. Network Representation
We have DNA data D from a disputed child c, its mother m and the putative father pf
child
founder
query
founder
If pf is not the true father tf, this is a “random” alternative father af
, query
hypothesis
Building blocks: founder, child

17. Disputed Paternity Case
founder
founder
query
hypothesis
child
Building blocks: founder, child, query

18. Complex Paternity Case
We have DNA from a disputed child c1 and its mother m1 but not from the putative father pf. We do have DNA from c2 an undisputed child of pf, and from her mother m2 as well as from two undisputed full brothers b1 and b2 of pf.
founder
child
query
hypothesis
Building blocks: founder, child, query

19. Criminal Identification Case
A body has been found, burnt beyond recognition, but there is reason to believe it might be that of a missing criminal CR. DNA is available from the body, from the wife of CR, and from two children c1 and c2 of CR and wife
founder
founder
query
founder
hypothesis
child
child
Building blocks: founder, child, query

20. Object-Oriented Bayesian Network
HUGIN 6
Each building block (founder / child / query) in a pedigree can be an INSTANCE of a generic CLASS network — which can itself have further structure
The pedigree is built up using simple mouse clicks to insert new nodes/instances and connect them up
Genotype data are entered and propagated using simple mouse clicks

21. Under the microscope…
Each CLASS is itself a Bayesian Network, with internal structure
Recursive: can contain instances of further class networks
Communication via input and output nodes

22. Single-marker analysis
(multiply LR’s across markers) 12
.0003 13
.0018 14
.1009 15
.1004 16
.1949 17
.2834 18
.2162 19
.0866 20
.0137 21
.0015 22
Marker vWA
(Austro-German population allele frequencies)

23. Lowest Level Building Blocks
DNA MARKER
having associated repertory of alleles together with their frequencies
gene
GENOTYPE
consisting of maximum and minimum of paternal and maternal genes
genotype
gene:
mendel
MENDELIAN SEGREGATION
Child’s gene copies paternal or maternal gene, according to outcome of fair coin flip

24. founder
FOUNDER INDIVIDUAL represented by a pair of genes pgin and mgin (instances of gene) sampled independently from population distribution, and combined in instance gt of genotype
gene
genotype

25. child CHILD INDIVIDUAL
paternal [maternal] gene selected by instances fmeiosis [mmeiosis] of mendel from father’s [mother’s] two genes, and combined in instance cgt of genotype
mendel
genotype

26. query
query
QUERY INDIVIDUAL
Choice of true father’s paternal gene tfpg [maternal gene mfpg] as either that of f1 or that of f2, according as tf=f1? is true or false.

27. Complex Paternity Case
Measurements for 12 DNA markers on all 6 individuals
Enter data, “propagate” through system
founder
child
Overall Likelihood Ratio in favour of paternity:
1300
query
hypothesis

28. MORE COMPLEX DNA CASES Mutation Silent/missed alleles,…
Mixed crime stains
rape
scuffle
Multiple perpetrators and stains
Database search
Contamination, laboratory errors …
Mixed stains in O. J. Simpson case. Interpretation contested between Defence and Prosecution – but both got it wrong!
OJS case also featured alternative explanation – framed by police.

29. MUTATION mendel + appropriate network mut to describe mutation process

30. e.g. proportional mutation:
founder
Prob(otherg)
~ mutation rate
mut
– or build other, more realistic, models

31. SILENT ALLELES Code by additional allele (99) gene genotype
unobserved + inherited
e.g. 5 = 5/5 or 5/s
Code by additional allele (99)
gene
genotype
gene
probsil – 9 values from 0 to 0.01:
silent := Binomial (1, probsil)
gene := max (99 * silent, gene0_gene)
genotype
gtmax := if (gtmax0 == 99, gtmin, gtmax0)

32. unobserved + non-inherited
MISSED ALLELES
unobserved + non-inherited
geneobs
genotype
geneobs

33. COMBINATION
Can combine any or all of above features (and others), by using all appropriate subnetworks
Can use any desired pedigree network
no visible difference at top level
Simply enter data (and desired parameter-values) and propagate…

34. Effect of accounting for silent allele
Simple paternity testing
Paternity testing with additional measured individuals

35. (Austro-German population allele frequencies)12
.0003 13
.0018 14
.1009 15
.1004 16
.1949 17
.2834 18
.2162 19
.0866 20
.0137 21
.0015 22
Marker vWA
(Austro-German population allele frequencies)

36. Simple paternity testing – allowing for silent alleles

37. Paternal incompatibility
mgt = 12/20 pfgt = 13 cgt = 12
with mutation ~ 0.005
pr(silent) LR
3.8 26 30
0.0001
125
127
0.001
203
The mother's and child's genotypes are the same as in Example 9.1, while the putative father's
observed allele is now the relatively rare allele 13, with p13 = 0:2%. The combined effects of silence
and missingness are displayed in Table 5.
The impact of introducing the possibility of silence is overwhelming: for example, when
pr(silent) = 0:01% the paternity ratio is 125. Compared with Example 9.1, the greater rarity
of the putative father's observed allele now makes the presence of a silent allele still more
plausible. However the sheer magnitude of this eect is perhaps unexpected.
p12 = – rare allele

38. Maternal incompatibility
mgt = pfgt = 18 cgt = 18
The mother must have passed a silent allele to the child
who must have inherited allele 18 from his father
pr(silent) LR
Impossible
4.6
0.0001
0.001
The undisputed mother is apparently incompatible with the child: she must therefore have a
missed allele, or have transmitted a silent or mutated allele to her child. Given that p18 = 21% is
much larger than any value considered for pr(silent) or pr(missed), we can be pretty sure, first
that both pfgt and cgt are truly homozygous, and then that the child inherited allele 18 from its
father. This has probability close to 1 under paternity, and to p18 = 0:2162 under non-paternity.
Correspondingly the paternity ratio is close to 1=0: :6 for any combination of the above
explanations. This can be confirmed by calculations (not shown), using our networks.

39. Paternity testing

40. Paternity testing with brother too

41. Consider additional evidence (likelihood ratio) LRB carried by the brother’s data B
Overall likelihood ratio is
where D denotes data on triplet (pf, c, m)

42. Incompatible triplet mgt = 12/15 pfgt = 14 cgt = 12 B = 16/20 12/14 1422
p(silent)
LRD
LRB 1
0.55
3334
0.5
1.00
1595
0.0001
2.5
404
0.001
7.5 46
*Maximum LRoverall is 1027, at p(silent) = *
p12 = .0003
p22 = .0003

43. Compatible triplet mgt = 12/15 pfgt = 13 cgt = 12/13 B = 13 13/16
21/22 22
p(silent)
LRD
LRB
556 1
551
1.00
0.51
0.0001
528
1.02
0.52
0.001
410
1.11
0.61
There is no effect whatsoever when the brother is heterozygous with no allele in common with the child
(bgt = f21; 22g); otherwise there is some effect, which is most apparent in column 5, where b is
apparently homozygous but different from pfgt: it then becomes more plausible that pf is in fact
heterozygous with one silent allele.

44. Extensions Estimation of mutation rates from paternity data
Peak area data
mixtures
contamination
low copy number

45. Network to estimate mutation rate

46. Mixed crime trace Marker: D8 D18 D21 Alleles: 10 11 14 13 16 17 59 6567 70
Peak
Area (RFUs):
6416
383
5659
38985
1914
1991
1226
1434
8816
8894
Suspect alleles in yellow
Excerpt of data on 6 markers from Evett et al. (1998)

47. Mixed crime trace – alleles only
Bayesian network that can be used to infer which of a suspect, victim and up to 6 possible
unknown individuals might have contributed DNA to a mixed crime trace.

48. Mixed crime trace – peak areas

49. REFERENCES
Dawid, A. P., Mortera, J., Pascali, V. L. and van Boxel, D. W. (2002). Probabilistic expert systems for forensic inference from genetic markers. Scand. J. Statist . 29, 577–595.
Dawid, A. P. (2003). An object-oriented Bayesian network for estimating mutation rates. In Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, January 3–6 2003, Key West, Florida, edited by Christopher M. Bishop and Brendan J. Frey. ISBN Online at
Mortera, J., Dawid, A. P. and Lauritzen, S. L. (2003). Probabilistic expert systems for DNA mixture profiling. Theor. Pop. Biol . 63, 191–205.
Cowell, R. G., Lauritzen, S. L. and Mortera, J. (2006). Identification and separation of DNA mixtures using peak area information. Forensic Science International (to appear).
Dawid, A. P., Mortera, J. and Vicard, P. (2007). Object-oriented Bayesian networks for complex forensic DNA profiling problems. Forensic Science International: Genetics 1 (to appear).

50. Mixed crime trace 25,000 170,000,000 Marker: D8 D18 D21 Alleles: 10 1114 13 16 17 59 65 67 70
Peak
area:
6416
383
5659
38985
1914
1991
1226
1434
8816
8894
+ 3 more…
LR (alleles only):
25,000
LR (peak areas too):
170,000,000

51. Thanks to: Julia Mortera Paola Vicard Steffen Lauritzen Robert Cowell and The Leverhulme Trust

52. and especially to JUDEA PEARL
who made it all possible