Using Bayesian Networks for Paternity Calculations: Adding an Evolutionary Perspective

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Using Bayesian Networks for Paternity
CalculationsAdding an Evolutionary Perspective

• Amanda B. Hepler
• E-MAIL abhepler_at_stat.ncsu.edu
• WEBSITE http//www4.ncsu.edu/abhepler

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What is a Bayesian Network (BN)?

• A graphical model for expressing probabilistic
  relationships among a set of variables.
• Each node in the graph represents a variable (or
  event), and has a table of probabilities
  associated with that variable.
• Arrows between nodes describe associations
  between the variables in the graph.
• In this study, the software program HUGIN is used
  to create all of our BNs. HUGIN provides a free
  evaluation version available at
  http//www.hugin.dk.

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Why Use Bayesian Networks?

• BNs provide
• simple graphical representations of very complex
  problems.
• a computational alternative to complex algebraic
  manipulations
• A vehicle for communication between practitioners
  when discussing very complex cases.

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What is a Paternity Index?

• Denote M as mother of child C, and the putative
  (alleged) father as PF.
• The plaintiffs (Hp), and the defenses (Hd)
  hypotheses are
• Hp PF is the father of C.
• Hd Some other man is the father of C.
• The likelihood ratio, or paternity index (PI), is

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What are Evolutionary Effects?

• Evolutionary theory states that subpopulations
  exist within most human populations.
• People could tend to mate within their
  subpopulation, resulting in inbreeding.
• Leads to different allelic frequencies within a
  subpopulation than those estimated from the
  overall population.
• Introduce the coancestry coefficient, ?, which is
  a measure of relatedness among the individuals
  within the subpopulation.

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How Can We Account for Evolution?

• The probability of observing an allele Ai will
  now depend on how many Ai alleles we have already
  seen.
• Balding and Nichols1 proposed the following
  method to calculate this conditional probability.

pi frequency of the ith allele in the popn 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.

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Typical Paternity Case

• Consider a single locus with two alleles, A1 and
  A2.
• Assume the genotypes of the M, C, and PF are
  known.

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Probability Tables
Childs Maternal Gene
Childs Genotype
Childs Paternal Gene
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Probability Tables, Cont.
Putative Fathers Maternal Gene
True Father Putative Father?
True Fathers Maternal Gene
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Incorporating Evolution into the BN

• To implement BNs method, several new nodes are
  introduced

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Bayesian Network Using Theta
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Probability Calculations

• HUGIN allows the user to enter in an expression
  that will generate conditional probabilities
  without having to enter in each value.
• The user types the followingDistribution(Formula
  for A1, Formula for A2).
• As an example, consider mpg_3. The formula for
  A1 is taken directly from BNs formula, with n2
  and i1

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Paternity Calculations By Hand

• Suppose ? 0.03 and p1 0.1. Also suppose we
  observe the following as evidence mgt A1A1,
  cgt A1A1, pfgt A1A2.
• Evett and Weir2 obtain the following PI for this
  case

• This same result can be obtained using HUGIN.

• I.W. Evett and B.S. Weir. Interpreting DNA
  Evidence. Sinauer, Sunderland,MA., 1998.

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Paternity Calculations Using HUGIN

• This is HUGINs display after entering in the
  evidence

• The PI is obtained by taking the value shown in
  the tfpf? table next to Yes and dividing it by
  the value displayed next to No, as shown below

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Effect of Introducing ?

• If we were to assume there was no evolutionary
  relatedness (? 0)

• Thus, by incorporating evolutionary relatedness
  we are arriving at a more conservative value for
  PI (2.91 vs. 5.00).
• Conservative in the sense of favoring the
  defense, as opposed to the plaintiff.

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Other Examples We Considered

• Multiple loci case
• Assume loci are independent.
• Obtain a PI for each individual locus using the
  previous BN.
• Multiply those PI s together to obtain the
  overall PI.
• Multiple Allele Case M and PF could have at most
  four distinct alleles at a particular locus.
• Missing Father Case When PFs genotype is not
  available, but we have his brothers genotype.

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Areas for Future Research

• Apply BN technology to various forensic cases
  while taking into account evolutionary
  relatedness
• Mixture Analysis
• Cross-Transfer Evidence
• Remains Identification
• Software needs to be developed with the forensic
  scientist in mind.