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  1. Bayesian Methods in Ecology and Resource Management
  2. 1st Edition
  3. Bayesian Models in Ecology
  4. Bayesian Analysis for Population Ecology
  5. Bayesian data analysis in population ecology: motivations, methods, and benefits

The minimum purchase order quantity for the product is 1. Add to cart. The interest in using Bayesian methods in ecology is increasing, however many ecologists have difficulty with conducting the required analyses. McCarthy bridges that gap, using a clear and accessible style.

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The text also incorporates case studies to demonstrate mark-recapture analysis, development of population models and the use of subjective judgement. The advantages of Bayesian methods, are also described here, for example, the incorporation of any relevant prior information and the ability to assess the evidence in favour of competing hypotheses.

Free software is available as well as an accompanying web-site containing the data files and WinBUGS codes. Bayesian Methods for Ecology will appeal to academic researchers, upper undergraduate and graduate students of Ecology. All new products. All specials. Reduced price! Dark grey bars in background are for models with vague priors. Transparent white bars in foreground are for models with empirical data-derived priors.

Adding prior information did not systematically sacrifice the accuracy of models with respect to the validation data set. Neither group of models was more or less likely to over-or underpredict mortality in the validation sets see Appendix S1. The predictions of models with informative priors were more often 56 of 90 closer to the validation mortality rate than models with vague priors. There was no relationship between the accuracy lost or gained in having an informative prior and the increase in precision see Appendix S1.

In general, models with large increases in precision due to their informative priors were no more or less likely to be more accurate than models that had only modest increases in precision. One exception was the understory species Murraya paniculata Rutaceae , which we discuss in greater detail below.

One of the 45 single-species model species, Murraya paniculata Rutaceae , a short tree, common in the understory and lower midstory of the forest, had large disagreement between prior and likelihood Figs. Across single-species models, the posterior estimates of average mortality for the models with and without informative priors were more similar to each other than the prior was to the posterior of the model without an informative prior.

Therefore, for most models, the data had a greater influence on the posterior estimates, than did the prior. Posterior predictive distributions and observed rate for overall mortality in the validation datasets of the 90 single-species models. Straight black lines show the observed proportion of dead individuals for each validation data set. Thick gray unbroken curves in the background are the posterior predictive distribution produced by models with vague priors. Thin black broken curves are posterior predictive distributions for equivalent models that included empirical data-derived priors.


Bayesian Methods in Ecology and Resource Management

A gray background to the panel indicates the empirical data-derived prior improved model accuracy with respect to the validation data, while a white background indicates that the model with vague priors was more accurate. The horizontal axes are plotted on the complementary log—log scale to aid visualization of the probability distributions. Panels are ordered by increasing observed mortality in the validation set from top to bottom then from left to right. Our analyses illustrate that the type of data-derived priors we used increase model precision and effective sample size without forgoing accuracy.

In 56 of 90 cases, the prior drew the posterior of the predicted mortality rate toward the mortality rate observed in the validation data. In these instances, the prior and training data estimates agreed and the prior simply increased the confidence in the estimated mortality rate. But in the remaining cases, the prior estimate made the posterior estimate less accurate than the estimate without an data-derived prior.

Overall, though, there was no evidence that a prior, constructed in the manner we have here, would lead to systematic bias and inaccurate models. The influence on parameter location is proportional to the precision of the prior and data. To ensure unbiased parameter estimates, one should take as much care in specifying the prior distribution as when collecting the data McCarthy and Masters The use of Bayesian statistics for ecology has sometimes been criticized e.

Much of this sentiment stems from the idea that Bayesian priors are overly subjective. Subjective priors can be used and have been demonstrated to work effectively when data are scarce and experts are available to provide information e. But, a prior need not be subjective. There are a number of examples of ecological studies where the same level of objectivity used to collect the model training data has been used to formulate the prior e.

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The method of forming a prior we have used here is a clear example of an non-subjective empirical data-derived prior. Collecting information on growth rate could be of great benefit when mortality data are scarce or costly to collect, which is typically the case for long-lived organisms that occur at low densities such as tree species. However, this approach could be extended to link functional traits e. In our study, an empirical data-derived prior only introduced bias in an extreme situation in which the informative prior made the mortality rate prediction less accurate.

We observed this bias for the species M. The reason for the disparity highlights the varied impacts of different agents of mortality and shows that care must be taken to ensure that prior information is relevant to the data that are being modeled. In this case, the prior ignored the extreme sensitivity of M. However, for M. The species' thin bark meant that the fires directly killed many individuals, but a widespread fungal infection associated with fire-induced basal wounding led to further widespread mortality across the population.

The high mortality rate and associated fungal infection also coincided with a population-wide slowing of growth rate. Thus, the relative growth rate for this species was far lower than it would have been under normal circumstances. This low relative growth rate, according to the generalization on which the prior was based, indicated that future mortality would be low, so the prior shifted the mortality lower and away from the mortality rates observed in both the training and validation data sets.

This circumstance only arose as the decrease in relative growth rate was so great that the relative growth rate for this species was misleading with respect to the mortality data. The reluctance of ecologists to use informative priors despite an increase in the use of Bayesian methods is surprising given the increasing use of hierarchical models, which have a similar logic to Bayesian informative priors. Hierarchical models are a type of formal inductive reasoning, as they enable us to make transparent and general inference from many specific cases coherently.

In hierarchical models, whether Bayesian or non-Bayesian, the group-varying parameters operate like a prior and likelihood in a single-level model Gelman and Hill If researchers are comfortable with the use of hierarchical models, they should be comfortable with using informative priors. Using priors as we have here is an extension of the logic of hierarchical modeling. For any given individual-group parameter a parameter associated with a particular group in a hierarchical model , the data of the group form the likelihood, and the prior is the global across group mean and associated variance.

Each group in a hierarchical data set contributes to the global mean estimate proportionally to the group sample size. For groups with relatively large sample sizes, the data dominates the parameter estimate for that group and will have a value with much the same location and precision as if the estimate was made without the influence of other groups. But for groups with very small sample sizes, the global-average-derived prior contributes far more to the posterior estimate.

Small-sample-size group estimates are often very different from the estimates made if their data were modeled with a simpler single-level model. A group with few data would be dominated by the global-average-derived prior and informed by the greater precision of the global-average-derived prior relative to the lower precision of the small-sample-size group.

A hierarchical model estimate would be closer to the global, across-group average, and more precise than a single small-sample-size group model estimate. Groups not present in a data set at all operate at the extreme of low sample size. Our work empirically tests the effect of empirical data-derived priors on model accuracy for the first time. Adding prior information increased the precision of our estimates without systematically biasing model estimates. When priors are appropriately formulated, they should not introduce bias and will increase precision.

Here, we have shown the gain in precision possible by recognizing the general link between growth and mortality.

Alhough, on average, accuracy was neither greater nor less for the models with empirical data-derived priors, in some cases, we identified a bias introduced by the prior because the information used to form the prior was atypical. Our findings contribute to a motivation for the use of Bayesian methods for ecology. This work provides powerful incentive to use empirical data-derived priors in models for ecology by overcoming a perception that priors could lead to systematic biases. National Center for Biotechnology Information , U. Journal List Ecol Evol v.

Ecol Evol.

Bayesian Models in Ecology

Published online Dec 5. Author information Article notes Copyright and License information Disclaimer. Correspondence William K. Funding Information William K. Morris was funded by an Australian Postgraduate Award. This article has been cited by other articles in PMC. Extended methods and results.

Bayesian Analysis for Population Ecology

Appendix S2. JAGS code for multi-species and single-species models. Abstract Despite benefits for precision, ecologists rarely use informative priors. Keywords: Ecological data, model precision, model validation, tree mortality. Introduction Ecological data are hard to acquire.

18. Bayesian Methods

Materials and Methods To test the effect of empirical data-derived priors on model accuracy requires fitting a large number of equivalent models with and without empirical data-derived priors and validating them against independent data. Open in a separate window. Figure 1. Prior specification Step A Correlations among biological rates can be used as a source of prior information. Model fitting Step B The priors derived from the hierarchical model were then applied to single-species models based on a sample of 98 individual stems of each species.

Model validation Step C Last, we validated both versions of the single-species models with and without empirical data-derived priors. Results We developed single-species models for 45 tree species from the HKK forest dynamics plot using growth and mortality data collected between and Figure 2.

Bayesian data analysis in population ecology: motivations, methods, and benefits

Figure 3. Discussion Our analyses illustrate that the type of data-derived priors we used increase model precision and effective sample size without forgoing accuracy. Conflict of Interest None declared. Supporting Information Appendix S1. Click here to view. Ashton PS. Disturbance history and historical stand dynamics of a seasonal tropical forest in western Thailand. The slow-growth-high-mortality hypothesis: a test using the cabbage butterfly. Generalized linear mixed models: a practical guide for ecology and evolution. Trends Ecol. Ashton P. Mengersen K. Elicitation by design in ecology: using expert opinion to inform priors for Bayesian statistical models.

Why environmental scientists are becoming Bayesians. Foster RB. Mortality rates of neotropical tree and shrub species and the impact of a severe drought. Discussion: should ecologists become Bayesians? Joachim J. Bayesian estimation of species richness from quadrat sampling data in the presence of prior information.

Prior distributions for variance parameters in hierarchical models. Bayesian Anal. Hill J. New York: Cambridge University Press; Rubin DB. Bayesian data analysis, Texts in stastical science.