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Bayesian inference for statistical models with a hierarchical structure is often characterized by specification of priors for parameters at different levels of the hierarchy. When higher level parameters are functions of the lower level parameters, specifying a prior on the lower level parameters leads to induced priors on the higher level parameters. However, what are deemed uninformative priors for lower level parameters can induce strikingly non-vague priors for higher level parameters. Depending on the sample size and specific model parameterization, these priors can then have unintended effects on the posterior distribution of the higher level parameters. Here we focus on Bayesian inference for the Bernoulli distribution parameter $\theta$ which is modeled as a function of covariates via a logistic regression, where the coefficients are the lower level parameters for which priors are specified. A specific area of interest and application is the modeling of survival probabilities in capture-recapture data and occupancy and detection probabilities in presence-absence data. In particular we propose alternative priors for the coefficients that yield specific induced priors for $\theta$. We address three induced prior cases. The simplest is when the induced prior for $\theta$ is Uniform(0,1). The second case is when the induced prior for $\theta$ is an arbitrary Beta($\alpha$, $\beta$) distribution. The third case is one where the intercept in the logistic model is to be treated distinct from the partial slope coefficients; e.g., $E[\theta]$ equals a specified value on (0,1) when all covariates equal 0. Simulation studies were carried out to evaluate performance of these priors and the methods were applied to a real presence/absence data set and occupancy modelling.

State-space models (SSMs) are an important modeling framework for analyzing ecological time series. These hierarchical models are commonly used to model population dynamics, animal movement, and capture-recapture data, and are now increasingly being used to model other ecological processes. SSMs are popular because they are flexible and they model the natural variation in ecological processes separately from observation error. Their flexibility allows ecologists to model continuous, count, binary, and categorical data with linear or nonlinear processes that evolve in discrete or continuous time. Modeling the two sources of stochasticity separately allows researchers to differentiate between biological variation (e.g., in birth processes) and imprecision in the sampling methodology, and generally provides better estimates of the ecological quantities of interest than if only one source of stochasticity is directly modeled. Since the introduction of SSMs, a broad range of fitting procedures have been proposed. However, the variety and complexity of these procedures can limit the ability of ecologists to formulate and fit their own SSMs. We provide the knowledge for ecologists to create SSMs that are robust to common, and often hidden, estimation problems, and the model selection and validation tools that can help them assess how well their models fit their data. In this paper, we present a review of SSMs that will provide a strong foundation to ecologists interested in learning about SSMs, introduce new tools to veteran SSM users, and highlight promising research directions for statisticians interested in ecological applications. The review is accompanied by an in-depth tutorial that demonstrates how SSMs models can be fitted and validated in R. Together, the review and tutorial present an introduction to SSMs that will help ecologists to formulate, fit, and validate their models., Comment: 71 pages, 4 figures

State-space models (SSMs) are a popular tool for modeling animal abundances. Inference difficulties for simple linear SSMs are well known, particularly in relation to simultaneous estimation of process and observation variances. Several remedies to overcome estimation problems have been studied for relatively simple SSMs, but whether these challenges and proposed remedies apply for nonlinear stage-structured SSMs, an important class of ecological models, is less well understood. Here we identify improvements for inference about nonlinear stage-structured SSMs fit with biased sequential life stage data. Theoretical analyses indicate parameter identifiability requires covariates in the state processes. Simulation studies show that plugging in externally estimated observation variances, as opposed to jointly estimating them with other parameters, reduces bias and standard error of estimates. In contrast to previous results for simple linear SSMs, strong confounding between jointly estimated process and observation variance parameters was not found in the models explored here. However, when observation variance was also estimated in the motivating case study, the resulting process variance estimates were implausibly low (near-zero). As SSMs are used in increasingly complex ways, understanding when inference can be expected to be successful, and what aids it, becomes more important. Our study illustrates (i) the need for relevant process covariates and (ii) the benefits of using externally estimated observation variances for inference for nonlinear stage-structured SSMs.

Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ``building block'' approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population., Comment: Published at http://dx.doi.org/10.1214/088342306000000673 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)