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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 l.jpg

Improving Forensic Identification in Bayesian Networks :Accounting for Population Substructure

Amanda B. Hepler


Outline l.jpg
Outline

  • Population Substructure (PS)

  • Bayesian Networks

    • Introduction

    • Incorporating PS into Paternity Networks

    • Example


What is population substructure l.jpg
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


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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)


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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 l.jpg

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 l.jpg
Why Use Bayesian Networks?

BNs provide:

  • Simple representations of complex problems

  • Automation of complex algebraic manipulations

  • Communication aide


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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 l.jpg

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


Probability tables for genotype and allele nodes l.jpg

PF’s Maternal Gene

PF’s Paternal Gene

PF’s Genotype

Probability Tables for Genotype and Allele Nodes


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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 l.jpg
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 l.jpg
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 l.jpg
New Probability Table

now depends on


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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 l.jpg
Paternity Calculations Using HUGIN

This same result can be obtained using HUGIN:


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Effect of Introducing θ

  • Assume no population substructure (θ = 0):

  • 2.91 more “conservative” than 5.00


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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


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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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