Abstract
Computational convenience has led to widespread use of Bayesian inference with vague or flat priors to analyze statistical models in ecology. Vague priors are claimed to be objective and to `let the data speak'. However, statisticians have long disputed these claims and have criticized the use of vague priors from philosophical to computational to pragmatic reasons. One of the major criticisms is that the inferences based on non-informative priors are generally dependent on the parameterization of the models. Ecologists, unfortunately, often dismiss such criticisms as having no practical implications. One argument is that for large sample sizes, the priors do not matter. The problem with this argument is that, in practice, one does not know whether or not the observed sample size is sufficiently large for the effect of the prior to vanish. It intricately depends on the complexity of the model and the strength of the prior. We study the consequences of parameterization dependence of the non-informative Bayesian analysis in the context of population viability analysis and occupancy models and at the commonly obtained sample sizes. We show that they can have significant impact on the analysis, in particular on prediction, and can lead to strikingly different managerial decisions. In general terms, the consequences are: (1) All subjective Bayesian inferences can be masqueraded as objective (flat prior) Bayesian inferences, (2) Induced priors on functions of parameters are not flat, thus leading to cryptic biases in scientific inferences, (3) Unrealistic independent priors for multiparameter models lead to unrealistic priors on induced parameters, (4) Bayesian prediction intervals may not have correct coverage, thus leading to errors in decision making, (5) Reparameterization to facilitate MCMC convergence may influence scientific inference. Given the wide spread applicability of the hierarchical models and uncritical use of non-informative Bayesian analysis in ecology, researchers should be cautious about using vague priors as a default choice in practical situations.
Highlights
Hierarchical models, known as state-space models, mixed effects models or mixture models, have proved to be extremely useful for modeling and analyzing ecological data (e.g., Bolker, 2008; Kery and Schaub, 2011)
Due to lack of invariance, analysis of the same data under the same statistical model can lead to substantially different conclusions under a non-informative Bayesian framework
We have studied in concrete terms the consequences of the lack of parameterization invariance in important ecological problems at commonly observed sample sizes, especially in wildlife management
Summary
Hierarchical models, known as state-space models, mixed effects models or mixture models, have proved to be extremely useful for modeling and analyzing ecological data (e.g., Bolker, 2008; Kery and Schaub, 2011). Due to lack of invariance, analysis of the same data under the same statistical model can lead to substantially different conclusions under a non-informative Bayesian framework. Priors for the (a, K) parameterization: a ∼ LN(0, 10), K ∼ Gamma(100, 100), σ 2 ∼ LogNormal(0, 10) These are some of the commonly used distributions for representing non-information on the appropriate ranges of the parameters (e.g., Kery and Schaub, 2011). We use the data cloning algorithm (Lele et al, 2007, 2010) to compute the maximum likelihood estimators (MLE) based frequentist predictions to obtain PPIs under these two parameterizations. For the non-informative Bayesian analysis, instead of the Gamma distribution, we used a uniform distribution prior for the carrying capacity parameter. Contrary to what is commonly claimed, the non-informative priors do not lead to inferences that are similar to the likelihood inferences
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