1. Bayesian Statistics: A Biologist’s Interpretation Marguerite Pelletier URI Natural Resources Science / U.S. EPA

2. How have Bayesian Methods been used? Federal allocation of money: Bayesian analysis of population characteristics such as poverty in small geographic areas Federal allocation of money: Bayesian analysis of population characteristics such as poverty in small geographic areas Microsoft Windows Office Assistant: Bayesian artificial intelligence algorithm 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 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

3. Bayesian Statistics vs. Frequentist Statistics Frequentist (Traditional) Statistics Assumes a fixed, true value for parameter of interest (e.g., mean, std dev) 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 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 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 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’ Confidence intervals generated – shows confidence about value of parameter, not how likely that parameter is in ‘real life’

4. Bayesian Statistics vs. Frequentist Statistics, cont. Bayesian Statistics Assumes parameter of interest (e.g., mean, std dev) variable and based on the data 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) 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’ Generates probability for any hypothesis being ‘true’ Sample sizes taken into account; large sample size alone won’t cause acceptance of the hypothesis 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’ Creates ‘credible intervals’ rather than confidence intervals – tells how likely the answer is in the ‘real world’

5. How do Bayesian Statistics ‘Work’? Posterior probability = Fishers Likelihood function * Prior probability Expected likelihood function 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 “Bayes Rule”: Posterior  Likelihood * Priors

6. Computationally intense (integration of complex functions) However…better computers and development of Markov Chain Monte Carlo methods made techniques more accessible 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 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 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 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

7. So When are Bayesian Statistics Useful? When limited data available – formalizes the use of ‘Best Professional Judgment’(Case Study 1) 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) 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) 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) When the answer is a single number rather than a complex function (e.g., simple calculation not complex multivariate analysis)

8. 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)

9. Neuse River estuary impaired due to nitrogen (eutrophication problems), requiring a Total Maximum Daily Load (TMDL) to be developed 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 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 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 Data for the model came from routine water quality monitoring and from elicited judgment of scientific experts Summary of Project

10. 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-ChannelWinds Days of Hypoxia Water Temperature System variable Node or Submodel Duration of Stratification Bayesian Network Association

11. 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) 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) 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. 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 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. 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 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

12. 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.

13. Species richness  local environmental variables Species richness  local environmental variables Over large scales these variables hard to collect Over large scales these variables hard to collect This study: (14) environmental variables from GIS and remote sensing used to predict butterfly species richness 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 + IV 2 ); Schwartz Information Criteria used for selection Poisson regression used to develop appropriate models from the 28 variables (IV + IV 2 ); Schwartz Information Criteria used for selection Appropriate variables then used in Bayesian Poisson model Appropriate variables then used in Bayesian Poisson model Model output validated against additional field data Model output validated against additional field data Summary of Project

14. Bayesian Poisson Regression: log  i =  +   k *X’ ik +  Y i ~ Poisson (  i ) where  i = mean (unobservable, true) spp richness at site i ,  k = regression coefficients; non-informative priors  = model error Y i = observed spp richness Markov Chain-Monte Carlo algorithm; 1000 iteration ‘burn-in,’ 3000 iterations to generate parameter estimates and mean spp richness estimates 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 1 st model New model run using validation data and regression-coefficient dist’n from the 1 st model Model worked well for same mountain range, but not for new range Model worked well for same mountain range, but not for new range

15. 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.

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

17. Model Comparison Predicted presence in patches was compared to observed presence using logistic regression: ln (o/(1 – o)) =  +  *ln(p/(1 - p)) whereo = observed presence p = predicted presence ,  = regression coefficients Predicted presence in patches was compared to observed presence using logistic regression: ln (o/(1 – o)) =  +  *ln(p/(1 - p)) whereo = observed presence p = predicted presence ,  = regression coefficients Significant differences between predicted and observed if  significantly different from 0 or  significantly different from 1 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 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 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

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