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Improving Forensic Identification in Bayesian Networks : Accounting for Population Substructure. Amanda B. Hepler. Outline. Population Substructure (PS) Bayesian Networks Introduction Incorporating PS into Paternity Networks Example. What Is Population Substructure?.

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improving forensic identification in bayesian networks accounting for population substructure

Improving Forensic Identification in Bayesian Networks :Accounting for Population Substructure

Amanda B. Hepler

outline
Outline
  • Population Substructure (PS)
  • Bayesian Networks
    • Introduction
    • Incorporating PS into Paternity Networks
    • Example
what is population substructure
What Is Population Substructure?
  • Any deviation from random mating
  • Commonly due to geographical subdivision
  • Mating pairs often have remote relatives in common
  • Inbreeding coefficient ():

- measures the extent of common ancestry

why should we account for ps
Why Should We Account for PS?
  • Ignoring PS “would unfairly overstate the strength of the evidence against the defendant.” (Balding & Nichols, 1995)
  • “If the allele frequencies for the subgroup are not available…[forensic] calculations should use the population-structure equations.” (1996 NRC Report)
assumptions
Assumptions
  • Population allele frequencies are known
  • Inbreeding coefficient is known
  • Loci are independent
  • Within a subpopulation:
        • Mating is random
        • Migration and mutation events independent and constant
what is a bayesian network bn

Graphical Portion

Hair Color

Eye Color

Red

0.5

Hair Color:

Red

Brown

Numerical Portion

Brown

0.5

Blue

0.2

0.9

Green

0.8

0.1

What is a Bayesian Network (BN)?
  • A graphical model that expresses probabilistic relationships among variables or events1
  • HUGIN used to create BNs, free version available at http://www.hugin.dk
why use bayesian networks
Why Use Bayesian Networks?

BNs provide:

  • Simple representations of complex problems
  • Automation of complex algebraic manipulations
  • Communication aide
notation for paternity case
Notation for Paternity Case
  • M = mother, C = child, PF = putative father
  • Hp: PF is the father of CHd: Some other man is the father of C
  • Likelihood ratio, or paternity index (PI):
  • Interpretation: “The evidence is PI times more probable if PF is the father of C than if some other man is the father.”
genotype and allele nodes

Genotype Nodes(A1A1, A1A2, A2A2)

Mother

Putative Father

Allele/Gene Nodes(A1, A2)

Child

Genotype and Allele Nodes
  • One locus, two alleles: A1and A2
  • Observe genotypes of M, C, and PF
original paternity network 2

Hypothesis Node (Yes, No)

Original Paternity Network2
  • A.P. Dawid, J. Mortera, V.L. Pascali, and D. Van Boxel. Probabilistic expert systems for forensic inference from genetic markers. Scandinavian Journal of Statistics, 29:577-595. 2002.
accounting for population substructure
Accounting for Population Substructure
  • Probability of allele Aidepends on how many Aialready observed
  • Modified allele frequencies3:
  • pi = frequency of the ith allele in the pop’n
  • ni = number of observed alleles of type Ai
  • n = total number of alleles observed
  • D.J. Balding and R.A. Nichols. DNA profile match probability calculation. Forensic Science International, 64(2-3):125-140, 1994.
new network nodes
New Network Nodes
  •  : p1:
  • Keep track of founder genes:
  • Counting nodes:

is the value of n1 after founder 2

is the value of n1 after founder 3, etc.

new probability table
New Probability Table

now depends on

paternity calculations by hand
Paternity Calculations By Hand
  • θ = 0.03, p1 = 0.10
  • M = A1A1, C = A1A1, PF = A1A2
  • PIfor this case4:
  • I.W. Evett and B.S. Weir. Interpreting DNA Evidence. Sinauer, Sunderland,MA., 1998.
paternity calculations using hugin
Paternity Calculations Using HUGIN

This same result can be obtained using HUGIN:

effect of introducing
Effect of Introducing θ
  • Assume no population substructure (θ = 0):
  • 2.91 more “conservative” than 5.00
other examples considered
Other Examples Considered
  • Multiple loci case:
    • Assume loci independent
    • Multiply PI
  • Multiple Allele Case:
    • M and PF have at most four distinct alleles
  • Missing Father Case:
    • Brother’s genotype available
areas for future research
Areas for Future Research
  • Apply same methodology to other BNs:
    • Mutation
    • Cross-Transfer Evidence
    • Mixtures
    • Remains Identification
  • Software improvements
    • Need software for the forensic scientist
    • Improvements needed for run time
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