Transcript Document

Bayesian Statistics:
A Biologist’s Interpretation
Marguerite Pelletier
URI Natural Resources Science / U.S. EPA
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
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’
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 won’t cause
acceptance of the hypothesis
• Creates ‘credible intervals’ rather than confidence intervals – tells how
likely the answer is in the ‘real world’
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
Problems with Bayesian Statistics
• Computationally intense (integration of complex functions)
However…better 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 dist’n 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 doesn’t change – implies that further study is not needed/useful
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)
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)
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
River [ N ]
River Flow
Algal Density
Pfisteria abundance
Carbon
Production
Water
Temperature
Duration of
Stratification
Shellfish
Abundance
Frequency of
Cross-Channel
Winds
Sediment Oxygen
Demand
Oxygen
Concentration
Bayesian Network
System variable
Node or Submodel
Association
Days of
Hypoxia
Frequency of
Fish Kills
Fish Population
Health
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 don’t change substantially with N reduction – fish kills
relatively rare, & effect of reduced C production is ‘damped out’ further
along the causal chain
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 110:21-31.
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
Bayesian Poisson Regression:
log i =  +  k*X’ik + 
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 dist’n
from the 1st model
• Model worked well for same mountain range, but not for new range
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 98:191-200.
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
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
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