Bayesian Methods

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Bayesian Methods
Christopher Smith
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Introduction

• Thomas Bayes
• Conditional Probability
• Bayes Propositions
• Frequentists
• Bayesian Methods at School Level
• Higher level examples

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Thomas Bayes

• 1702 Born in London
• 1719 Studies logic and theology at the
• University of Edinburgh
• 1742 Elected a Fellow of the Royal Society
• 1761 Death
• 1764 Richard Price submits a paper written by
  Bayes to the
• Royal Society named Essay towards
  solving a problem
• in the doctrine of chances

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Conditional Probability

• The probability of some event A, given the
  occurrence of
• some other event B, written P(AB)
• Made up of the quotient of the joint probability
  of the two
• events occurring and the marginal probability
• We have P(AB) P(A?B)
• P(B)
• Partition Theorem
• P(B) P(BA)?P(A) P(BA)?P(A)
  A NOT A

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Derivation of Bayes Theorem

• From the last slide P(AB) P(A?B)
• P(B)
• Thus we also have P(BA) P(B?A)
• P(A)
• Since P(A?B) P(B?A) we are able to deduce
• P(B) is calculated using the partition theorem,
  and P(B)?0

P(AB) P(BA)?P(A) P(B)
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Bayes Propositions

• (v) If P(B) b and P(A?B) P/N
• ? P(AB) P(A?B) P/N P
• P(B)
  b/N b
• (iii) - (v) Bayes Theorem
• (ix) Assuming a uniform distribution for the
  prior distribution of the
• binomial parameter p, the probability that p
  is between a and b
• Billiard balls thrown on a table

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Frequentists

• Common among mathematicians
• Rely heavily well defined problems
• Degree-of-Belief is irrelevant
• Famous frequentists- Jerzy Neyman
• Egon Pearson
• John Venn
• R.A. Fisher

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Bayesian Methods at School Level

• Problem 1 Nine fair coins and one double-headed
  coin. Select one
• at random. What is P(coin is
  double-headed)?
• Problem 2 Tossed 5 times. HEADS 5 times.
• P(double-headed)? (10 times? A
  million times?)
• Problem 3 Tossed once. Comes out TAILS
• P(double-headed)?
• Basic principles of Bayesian methods are
  accessible to KS4/AS level
• The importance of prior belief is vast.
  (Weather, Sport, Blind date?)

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Biostatistics- A university level application
Person has disease Person does not have
disease Test is positive for disease Test is
negative for disease Sensitivity Specificity Fal
se positive False negative Predictive
value Background prevalence
D D- T T- P(TD) P(T-D-) P(TD-) P(T-D)
P(DT) P(TD)?P(D) P(T) P(D)
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Biostatistics- An example
Let D be event foetus has downs syndrome Let
T be event test is positive for downs
syndrome Background prevalence P(D) 1/270
0.37 Sensitivity P(TD) 0.89 ? False
negative P(TD-) 0.11 Specificity P(T-D-)
0.75 ? False positive P(T-D) 0.25
Problem A mother tests positive. What is the
chance that the foetus has downs? Solution
P(T) P(T?D) P(T?D-)
P(TD)?P(D) P(TD-)?P(D-) 0.89?1/270
0.25?269/270 0.2524 Now P(DT)
P(TD)?P(D) 0.89 ? 1/270 0.01306 1.31
P(T) 0.2524
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Prosecutors Fallacy

• P(AB) is incorrectly deemed to approximately
  equal P(BA),
• e.g. The probability that a random person matches
  DNA evidence
• is one in a million. This is P(EG), and
  shouldnt be confused
• with the probability of guilt given a DNA
  match P(GE)

• Sally Clark
• Two independent events A, B P(A?B) P(A)?P(B)
• (1/8500)2 1 in 72,225,000
• Research shows dependence, and P(BA) 1/100

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Summary

• Bayes didnt publish the work, it was published
  by a friend
• Conditional Probability
• Bayes theorem and underlying concepts should be
  introduced
• at school level
• Important alternative to frequentism
• Allows statistical analysis to accommodate
  expertise

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Thank-you