Peg Bayesian[1]

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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 populationcharacteristics 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



• 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

Problems with Bayesian Statistics



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)



• 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

Summary of Project



Algal Density Pfisteria abundance

Carbon

Production

Sediment Oxygen

Demand

Oxygen

Concentration

Shellfish

Abundance

Frequency of

Fish Kills

River [ N ] River Flow

Fish Population

Health

Frequency of

Cross-Channel

Winds

Days of

Hypoxia

Water

Temperature

System variable

Node or SubmodelDuration of

Stratification

Bayesian Network

Association



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.



• 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

Summary of Project



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.



• 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

Summary of Project



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



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Peg Bayesian[1]