Conditional Probability and Bayes Theorem

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Conditional Probability and Bayes Theorem

• Chapter 9 of David Lucy, Forensic Statistics.

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9.1 Conditional Probability

• Independence implies that to calculate the
  probabilities for joint events (the coincidence
  of 2 or more events) the probabilities for single
  events could be multiplied.
• In this chapter, we will examine the more common
  case, where events are not thought to be
  independent.
• This will lead to the same idea for evidence
  evaluation as seen in section 8.3, although a
  more mathematical approach will be taken.
• Rogers et al (2000) give an analysis which uses
  the rhomboid fossa as an indicator of the sex of
  unknown skeletalised human remains.
• The rhomboid fossa is a groove which sometimes
  occurs at one end of the clavicle as a result of
  the attachment of the rhomboid ligament.

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9.1 Cross-tabulation of sex and presence/absence
of a rhomboid fossa
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9.2 Joint probabilities for sex and
presence/absence of a rhomboid fossa
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Conditional and Unconditional probabilities

• The values of table 9.2 were obtained by dividing
  those in table 9.1 by the grand total of 344.
• P(Ri) and P(Sj) are unconditional probabilities,
  not conditioned on any other factors in the
  dataset.
• All four P(Sj and Ri) values in the centre are
  conditional probabilities, e.g. if we know
  someone is male, what is the probability that he
  has a rhomboid fossa?
• We use the notation Pr( Rp Sm ) 155 / 231
  0.67 see table 9.3
• Conversely we can use Pr (Sm Rp), if we know
  the rhomboid fossa is present, what is the
  probability that the skeleton is male? 155 /
  167 93 see table 9.4

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9.3 Probability of the presence or absence of
rhomboid fossa given the sex of the individual
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9.4 Probability of the sex of the individual
given the presence or absence of a rhomboid fossa
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The error of the transposed conditional

• If a skeleton from the sample is observed to
  possess a rhomboid fossa then it is perfectly
  true to say that there is a 93 probability that
  it is male, and a 7 probability that it is
  female.
• If a rhomboid fossa is not observed then there is
  a 43 probability it is male, and a 57
  probability it is female.
• Thus possession of a rhomboid fossa is a good
  indicator that skeletal remains are male, but a
  poor indicator of female skeletal remains.
• The conditional probabilities Pr ( Sj Ri ) in
  table 9.3 are not the same as the conditional
  probabilities Pr ( Ri Sj ) in table 9.4.
• For example, the 67 probability that a male has
  a rhomboid fossa is not at all the same as the
  93 probability that an individual possessing a
  rhomboid fossa is male.
• Confusing the two is the error of the transposed
  conditional, easily done when describing these
  types of probabilities in words.

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Bayes Theorem (1)

• In section 3.1.4 we found that the probability of
  throwing a fair 6-sided dice and it giving a
  score which was odd (event A) and greater than 3
  (event B) could be calculated by the
  multiplication of Pr(A) and Pr(BA).
• Pr(A) ½ Pr(BA) 1/3 Pr(A,B) 1/6.
• This is an illustration of the third law of
  probability for dependent events, and is simply
  Pr(A,B) Pr(A) Pr(BA).

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Bayes Theorem (2)

• What if we now reverse the order in which we do
  this calculation?
• The probability of rolling a score gt 3 is ½, so
  now Pr(B) ½.
• If we have already rolled a score gt3, we are left
  with x 4, 5, 6 from which to have rolled an
  odd number, so Pr(AB) 1/3.
• Now Pr(A,B) Pr(B) Pr(AB)/

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Bayes Theorem (3)

• As Pr(A,B) by the first calculation must equal
  Pr(A,B) by the second calculation,
• Pr(A) Pr(BA) Pr(B) Pr(AB)
• Dividing both sides by Pr(A) gives
• Pr(BA) Pr(B) Pr(AB) / Pr(A)
• This is called Bayes therorem, which related
  conditional probabilities to unconditional
  probabilities.

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Bayes Theorem (4)

• Using a change in notation, where E indicated
  evidence and H indicates hypothesis, Bayes
  theorem can be rewritten
• Pr(HE) Pr(EH) Pr(H) / Pr(E)
• Written specifically for the rhomboid fossa
  example
• Pr(SjRi) Pr(RiSj) Pr(Sj) / Pr(Ri)

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Bayes Theorem (5)

• Thus if one were to examine a skeleton and find
  no rhomboid fossa, one could calculate the
  probability of the sex being female using Bayes
  theorem.
• Given the observation of no rhomboid fossa we
  want the probability of sex female Pr(SfRa)
  Pr(RaSf) Pr(Sf) / Pr(Ra)
• 0.89 0.33 / 0.52 0.57 approx.

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Bayes Theorem (6)

• Rewriting the equation for sex male given
  rhomboid fossa present
• Pr(SmRp) Pr(RpSm) Pr(Sm) / Pr(Rp)
• From table 9.3, Pr(RpSm) 0.67
• From table 9.2, Pr(Sm) 0.67, Pr(Rp) 0.48
• So Pr(SmRp) 0.67 0.67 / 0.48 0.93.
• Check this in table 9.4.

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Bayes Theorem (7)

• A useful variant on the equation is to expand the
  denominator (Ri)
• Pr(Rp) Pr(Rp,Sm) Pr(Rp,Sf) 2nd law
• Pr(Rp,Sm) Pr(RpSm) Pr(Sm) 3rd law
• Pr(Rp,Sf) Pr(RpSf) Pr(Sf) 3rd law
• Pr(SmRp) Pr(RpSm) Pr(Sm) / (Pr(RpSm)
  Pr(Sm)) (Pr(RpSf) Pr(Sf)) Bayes
• Pr(SmRp) 0.67 0.67 / (0.67 0.67) (0.11
  0.33) 0.93.

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Bayes Theorem (8)

• The probability of the sex of a skeleton given
  the presence or absence of a rhomboid fossa is
  called the posterior probability.
• Why do we need Bayes theorem to calculate the
  posterior probability when we could just as well
  look it up in Table 9.4?
• The unconditional probabilities Pr(Sm) and Pr(Sf)
  are called prior probabilities and are the
  probability of the sex of a skeleton before any
  information about the rhomboid fossa is taken
  into account.

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Bayes Theorem (9)

• The prior probabilities used here are dependent
  on the sample of skeletons to which we have had
  access, and are not prior probabilities from the
  number of males and females in a living
  population.
• From survey data we know that the population is
  about 48 male and 52 female, not 67 male and
  33 female as in the skeletal sample.
• So when inspecting a skeleton from an unknown
  source, rather than the sample, it would be
  reasonable for Pr(Sm) 0.48 and Pr(Sf) 0.52.
• Pr(SmRp) 0.67 0.48 / (0.11 0.52) (0.67
  0.48) 0.85.

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Bayes Theorem (10)

• This is not the end of the process for a Bayesian
  type analysis.
• If there were some other sex indicator unrelated
  to the rhomboid fossa, then this could be used as
  an independent measure of sex.
• One way is to use the posterior probability of
  sex given the observation of the rhomboid fossa
  as the prior probability in another similar
  analysis.
• In this way pieces of independent evidence as to
  the sex of an unknown skeleton can be combined to
  give a probability of sex given all the
  indicators.

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The Value of Evidence (1)

• Rather than thinking about male / female, let us
  now think about the propositions of guilt or
  innocence of a suspect given some form of
  evidence.
• The selection of priors can be problematic.
• If a crime were committed in Rochdale (pop.
  205,000), should the prior probability of guilt
  be 1/205,000 or 1/49,000,000 (pop. UK).?
• Realistically such questions are impossible to
  answer, and not really a question with which the
  forensic scientist should be too concerned.
• The role of the forensic scientist is not to
  give probabilities of guilt or innocence their
  function is to comment solely on the evidence
  presented to them.

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The Value of Evidence (2)

• A way in which statisticians have overcome this
  problem is to derive a measure of the importance
  of evidence. This is the same measure of
  evidential value given in section 8.3, but here
  is a more mathematical treatment of the
  underlying rationale.
• The first thing to consider are the odds of
  guilt. Odds are defined for an event w as
• Odds probability of w occurring / probability
  of w not occurring.
• See equations on p113 of Lucy.

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Conclusion

• The prior odds and the posterior odds of guilt
  are for the consideration of the court in
  criminal cases, the scientist being in no
  position to evaluate either.
• The only feature of equation 26 which is directly
  dependent on the evidence and the propositions to
  which that evidence lends support, but not
  influenced by the prior odds, is the likelihood
  ratio
• LR Pr( E G) / Pr( E not G)
• And exactly the measure of evidential value
  discussed in section 8.3.