Bayesian Statistics: A Biologist

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Bayesian Statistics A Biologists Interpretation
Marguerite Pelletier URI Natural Resources
Science / U.S. EPA
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How have Bayesian Methods been used?

• Federal allocation of money Bayesian analysis
  of population characteristics such as poverty
  in small geographic areas
• Microsoft Windows Office Assistant Bayesian
  artificial intelligence algorithm
• It has been suggested that Bayesian statistics
  be used in environmental science because it
  addresses questions about the probability of
  events occurring, which allows better
  decision-making

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Bayesian Statistics vs. Frequentist Statistics

• Frequentist (Traditional) Statistics
• Assumes a fixed, true value for parameter of
  interest (e.g., mean, std dev)
• Expected value average value obtained by
  random sampling repeated ad infinitum
• Can only reject the null hypothesis (Ho), not
  support the alternative hypothesis (Ha)
  p-values indicate statistical rareness
• Large sample sizes make rejection of Ho more
  likely
• Confidence intervals generated shows
  confidence about value of parameter, not how
  likely that parameter is in real life

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Bayesian Statistics vs. Frequentist Statistics,
cont.

• Bayesian Statistics
• Assumes parameter of interest (e.g., mean, std
  dev) variable and based on the data
• Can test the probability of the alternate
  hypothesis (Ha) or hypotheses given the data
  (which is what most scientists really care
  about)
• Generates probability for any hypothesis being
  true
• Sample sizes taken into account large sample
  size alone wont cause acceptance of the
  hypothesis
• Creates credible intervals rather than
  confidence intervals tells how likely the
  answer is in the real world

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How do Bayesian Statistics Work?
Posterior probability Fishers Likelihood
function Prior probability
Expected
likelihood function Likelihood function Given
data, with a known (or predicted) distribution
(i.e., Normal, Poisson), a likelihood
function (probability distribution) can be
calculated Prior probability based on
existing data or a subjective indication of
what the investigator believes to be
true Expected likelihood function marginal
distribution of data given hyperparameter
takes sample size into account
Bayes Rule Posterior ? Likelihood Priors
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Problems with Bayesian Statistics

• Computationally intense (integration of complex
  functions) Howeverbetter computers and
  development of Markov Chain Monte Carlo methods
  made techniques more accessible
• Not directly applicable for many complex
  statistical analyses Can be used for certain
  regression techniques and to generate posterior
  distn given a prior. Attempts to utilize it in
  clustering unsuccessful
• Not readily available in most common
  statistical software (SPSS, SAS)
• Not applicable to very rare events priors
  dominate the function so the posterior
  doesnt change implies that further study is
  not needed/useful

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So When are Bayesian Statistics Useful?

• When limited data available formalizes the
  use of Best Professional Judgment (Case
  Study 1)
• When Bayesian algorithms have been developed
  for a statistic e.g., regression (Case
  Study 2)
• After using more traditional statistical
  methods develop a probability
  distribution (Case Study 3)
• When the answer is a single number rather than
  a complex function (e.g., simple calculation
  not complex multivariate analysis)

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Case Study 1 Development of a Bayesian
Probability Network in the Neuse River Estuary,
N.C.
(Borsuk ME, Stow CA, Reckhow KH 2003. An
integrated approach to TMDL development for the
Neuse River estuary using a Bayesian probability
network. Journal of Water Resources Planning and
Management, accepted)
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Summary of Project

• Neuse River estuary impaired due to nitrogen
  (eutrophication problems), requiring a Total
  Maximum Daily Load (TMDL) to be developed
• For development of a TMDL, links must be
  developed between pollutant load ( N ), and
  water quality impairment
• Because of the range of endpoints and the need
  to determine probability of impact, a
  Bayesian Network was developed
• Data for the model came from routine water
  quality monitoring and from elicited judgment
  of scientific experts

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River N
River Flow
Algal Density
Pfisteria abundance
CarbonProduction
WaterTemperature
Bayesian Network
Sediment OxygenDemand
System variable
Node or Submodel
Oxygen Concentration
Duration of Stratification
Association
ShellfishAbundance
Days ofHypoxia
Frequency of Cross-Channel Winds
Frequency of Fish Kills
Fish Population Health
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Use of Bayesian Network (focus on Fish Kills)

• Fish kills low bottom D.O. cross-channel
  winds (force bottom water fish to shores)
  fish health (influences susceptibility)
• Two expert fisheries biologists asked about the
  likelihood of fish kill given certain
  conditions (various wind/hypoxia/fish health
  scenarios)
• All probabilistic relationships (including fish
  kill info) incorporated into Bayesian
  network.
• Four nitrogen reduction scenarios assessed 0,
  15, 30, 45 and 60 (relative to 1991-1995
  baseline) using Latin Hypercube sampling
• As N inputs decreased, mean chl and exceedance
  frequency also reduced.
• Fish kills dont change substantially with N
  reduction fish kills relatively rare,
  effect of reduced C production is damped out
  further along the causal chain

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Case Study 2 Assessing Spatial Population
Viability Models using Bayesian Statistics
(Mac Nally R, Fleishman E, Fay JP, Murphy DD
2003. Modeling butterfly species richness using
mesoscale environmental variables model
construction and validation for the mountain
ranges in the Great Basin of western North
America. Biological Conservation 11021-31.
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Summary of Project

• Species richness ? local environmental
  variables
• Over large scales these variables hard to
  collect
• This study (14) environmental variables from
  GIS and remote sensing used to predict
  butterfly species richness
• Poisson regression used to develop appropriate
  models from the 28 variables (IV IV2)
  Schwartz Information Criteria used for selection
• Appropriate variables then used in Bayesian
  Poisson model
• Model output validated against additional field
  data

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Bayesian Poisson Regression
log ?i ? ? ?kXik ? Yi Poisson (
?i )
where ?i mean (unobservable, true) spp
richness at site i ?, ?k regression
coefficients non-informative priors ? model
error Yi observed spp richness

• Markov Chain-Monte Carlo algorithm 1000
  iteration burn-in, 3000 iterations to
  generate parameter estimates and mean spp
  richness estimates
• New model run using validation data and
  regression-coefficient distn from the 1st
  model
• Model worked well for same mountain range, but
  not for new range

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Case Study 3 Assessing Spatial Population
Viability Models using Bayesian Statistics
(McCarthy MA, Lindenmayer DB, Possingham HP 2001.
Assessing spatial PVA models of arboreal
marsupials using significance tests and Bayesian
statistics. Biological Conservation 98191-200.
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Summary of Project

• Population Viability Analysis used in
  Conservation Biology to assess potential for
  species extinction
• Many models based on limited data assessed
  via significance tests or Bayesian methods
• Metapopulation models (for 4 arboreal
  marsupials) were developed
• 2 competing null models also developed
• No effect of fragmentation
• No dispersal between patches
• Models were compared using likelihood and
  Bayesian methods

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Model Comparison

• Predicted presence in patches was compared to
  observed presence using logistic
  regression ln (o/(1 o)) ? ?ln(p/(1 -
  p)) where o observed presence p predicted
  presence ?, ? regression coefficients
• Significant differences between predicted and
  observed if ? significantly different from 0
  or ? significantly different from 1
• Models compared using log-likelihood models
  with higher log-likelihood values (closer to
  0) more closely match data
• Bayesian posterior probabilities used to
  compare models higher probabilities more
  closely match data prior all 3 models equally
  plausible Probability of Model likelihood of
  model / sum of all likelihoods

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Conclusions

• Comparison with actual data
• Full model best for greater glider,
  yellow-bellied glider
• No fragmentation model best for mountain
  brushtail possum, ringtail possum (but
  predicted values ½ observed values)
• Log-likelihood values
• Confirm no fragmentation model best for 2
  possum spp
• Confimed full model best for the greater
  glider
• Yellow bellied glider equally represented by
  full model and no dispersal model
• Bayesian statistics confirmed log-likelihood
  results
• Authors indicated that significance tests
  useful to assess model accuracy Bayesian
  methods useful for comparing models but
  computationally intense

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