## Introduction

Aerial surveys are frequently used to monitor wildlife populations over large spatial scales, and several methods are available for adjusting counts of observed animals when detection rates are <1 (see Lancia *et al*. 2005, for a review). A popular approach, pioneered by Steinhorst & Samuel (1989), uses auxiliary ‘test trial’ or ‘sightability trial’ data collected from radiocollared animals to model the detection process as a function of individual-level covariates (e.g. group size, measures of visual obstruction); see Fieberg (2012) for a list of example applications. Abundance is estimated from ‘operational’ surveys using a modified Horvitz–Thompson (mHT) estimator, in which animal counts are divided by their estimated detection probabilities from fitted logistic regression models applied to the auxiliary data, along with plot-level sampling probabilities. The variance of the estimator can be expressed as the sum of three variance components that are due to: (i) random sampling (of aerial plots); (ii) random detection (and missed detection) of independent groups of individuals; and (iii) estimation of parameters related to the detection process (Steinhorst & Samuel 1989).

Inference using the mHT estimator is ‘design-based’ – that is, confidence intervals are formed by considering variation across repeated samples (which never occur), under the assumption that the estimator follows an approximate Gaussian distribution when the sample size is large. Yet, operational surveys typically involve small numbers of sample units, and as a result, the sampling distribution of the mHT estimator is typically right-skewed. In addition, the variance estimator developed by Steinhorst & Samuel (1989) includes several terms that are derived under an asymptotic normality assumption for the logistic regression parameter estimators. Fieberg & Giudice (2008) showed that for small numbers of sightability trials (e.g. ≤100), these approximate terms tend to underestimate the variance. In simulation studies, normal-based confidence intervals have often resulted in less than nominal coverage (Cogan & Diefenbach 1998), whereas intervals based on a log-normal assumption often perform better (Wong 1996). Lastly, as is often the case with Horvitz–Thompson estimators, the mHT is inefficient when inclusion probabilities are small (either due to small probabilities of sampling the aerial unit or low detection probabilities; Little 2009).

Model-based estimators for survey data, often formulated within a Bayesian framework, have been suggested as an alternative to traditional design-based estimators (Ericson 1969; Valliant, Dorfman & Royall 2000; Little 2009). Popularity of these methods has increased with advances in computing power and the availability of Markov Chain Monte Carlo (MCMC) samplers capable of approximating posterior distributions. Inferences based on Bayesian models that use vague, or noninformative, priors are often quite similar to inferences made using design-based methods (Ericson 1969). Models should, however, incorporate important survey design features to ensure estimators are consistent (e.g. covariates and random effects may be used to account for stratification and clustering, respectively). Model-based estimators provide a natural approach to modelling survey non-response and can also lead to efficiency gains compared with traditional design-based estimators (Little 2009). Little (2009) lists several other advantages of Bayesian (model-based) inference for survey data, including the potential for better inference in small samples and the ability to incorporate prior information. These advantages, in part, motivated us to develop a Bayesian alternative to the mHT estimator.

Non-detection of individuals in wildlife surveys is similar to non-response in traditional sample surveys (Steinhorst & Samuel 1989); however, an important difference is that a sampling frame (i.e. a list of all individuals in the population) is lacking in the former case. As a result, the dimension of the problem (i.e. size of the population) is itself an unknown parameter that must be estimated. Similar problems arise with other popular abundance estimators, including capture–recapture methods that include individual covariates (Royle 2009) and models of species richness that account for detection rates that are <1 (Dorazio *et al*. 2006). Maximizing the complete data likelihood, formulated to capture both the sampling and detection processes, requires numerical integration techniques that are particularly challenging when the number of covariates in the detection model is large (Royle 2009). Alternatively, Dorazio *et al*. (2006) and Royle, Dorazio & Link (2007) developed a Bayesian data augmentation scheme for abundance estimation. We show how a similar approach can be applied to sightability data and discuss the advantages afforded by the Bayesian approach.