An introduction to Bayesian analysis

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An introduction to Bayesian analysis

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The objective of risk assessment

• To evaluate the probabilities of alternative
  consequences of different management actions
• To do this we need to make probabilistic
  statements about alternative hypotheses (states
  of nature)

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Traditional statistics

• Make no claims about probabilities
• The 95 confidence interval on a regression slope
  does not claim there is a 95 probability the
  slope is in that range, rather only that if the
  experiment was repeated many times, 95 of the
  time the estimated slope would be in that range
• frequentists argue that probabilities do not
  exist, there is only 1 true slope

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Likelihood

• calculates the likelihood of the data given the
  hypothesis --- not the probability of the
  hypothesis given the data
• to calculate probability of hypothesis you need
  Bayes Law - this is mathematically proven and not
  contested by anyone!

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Bayes Law
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Restated in English

• The relative belief in a hypothesis given some
  data
• is the support the data provide for the
  hypothesis
• times the prior belief in the hypothesis
• divided by the support times the prior for all
  hypotheses.

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Things to note
When the data are given then is the likelihood
.
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So the essential elements are

• What we know about the hypotheses before the data
  are collected, the prior, or more importantly the
  sum of existing knowledge
• And the likelihood something we are all
  familiar with

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The Bayesian Rumpole

• A man in England was charged with the rape and
  murder of a woman
• The only evidence against him was the DNA match
  between tissue found on the dead womans body,
  and from the man
• Scientists testified that the probability that
  his DNA would match the tissue from the womans
  body by chance was 1 in 3 million

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• Hypothesis I - the DNA was from him
• Hypothesis II - the DNA was from someone else
• What is the probability of Hypothesis I?
• What do we need
• Alternative hypotheses (given above)
• Prior probabilities
• Pr(DH)
• Take 5 minutes to calculate probability he is
  guilty

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Thus you have to specify the prior belief in
the hypothesis!
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The missing floppy disk

• I am in my office and need a floppy disk I had at
  home last night.
• It might still be at home, it might be in my
  car, or it might be in the office - 3 hypotheses.
• Based on numerous similar experiences my prior is
  50 at home, 30 in the car, 20 in the office
• Also based on experience I know that if I search
  my office and it is there I have a 50 chance of
  finding it - the same for the home, but I have a
  90 of finding it if it is in the car
• What happens to my posterior if I search the
  office and dont find it?

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More examples

• Smiths children
• Monte Hall and the 3 doors

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Wildebeest data and regression

• Use wildebeest data to calculate a regression
  between census estimate and year
• Repeat using Bayes law assuming variance is known

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A worked example with non-linear model

• Wildebeest growth problem
• Two parameters r and k
• Assume priors are normal
• r mean .2 sd .2
• k mean 2,000 sd 2000
• Calculate numerator of Bayes law in spreadsheet
• Use EXCEL table function to obtain posterior

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Marginal posteriors

• What emerges from a Bayesian estimation is the
  posterior probability in as many dimensions as
  there are parameters
• We often want to look at what we know about a
  single parameter that is we want to know the
  marginal posterior for the parameter
• This is easily done simply by integrating (or
  summing) across all other parameters

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Alternatives to Bayesian analysis

• Bootstrapping - commonly done because it is so
  easy - but it isnt the same as a probability -
• compare bootstrap (sampling across data) to SIR
  (sampling across hypotheses)
• Likelihood and likelihood profile

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Criticisms of Bayesian analysis

• Probabilities are not real (the coin has flipped)
• need priors
• use subjective priors
• dont explore goodness of fit and residuals
• conclusions may depend more on priors than the
  data

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Distinction between a practical Bayesian and a
pure Bayesian

• Exploration of alternative model structures
• A pure Bayesian would use the most general
  model, with priors on all parameters and then let
  the data distinguish between competing model
  structures

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The myths of Bayesian analysis

• Priors
• subjectivity
• ignore model fit - residuals
• The difference between Bayesian analysis and
  likelihood is the distinction between integration
  and maximization

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Challenges in Bayesian analysis

• Defining what we know before the data are
  collected -- the priors
• Deciding on how complex our models should be
• Deciding the appropriate likelihood to use (these
  are problems in non-Bayesian analysis as well
• Integration across a high number of parameters