Combining chains of Bayesian models with Markov melding

A challenge for practitioners of Bayesian inference is specifying a model that incorporates multiple relevant, heterogeneous data sets. It may be easier to instead specify distinct submodels for each source of data, then join the submodels together. We consider chains of submodels, where submodels directly relate to their neighbours via common quantities which may be parameters or deterministic functions thereof. We propose chained Markov melding, an extension of Markov melding, a generic method to combine chains of submodels into a joint model. One challenge we address is appropriately capturing the prior dependence between common quantities within a submodel, whilst also reconciling differences in priors for the same common quantity between two adjacent submodels. Estimating the posterior of the resulting overall joint model is also challenging, so we describe a sampler that uses the chain structure to incorporate information contained in the submodels in multiple stages, possibly in parallel. We demonstrate our methodology using two examples. The first example considers an ecological integrated population model, where multiple data sets are required to accurately estimate population immigration and reproduction rates. We also consider a joint longitudinal and time-to-event model with uncertain, submodel-derived event times. Chained Markov melding is a conceptually appealing approach to integrating submodels in these settings.

Using multiple data types and integrated population models to improve our knowledge of apex predator population dynamics

Current management of large carnivores is informed using a variety of parameters, methods, and metrics; however, these data are typically considered independently. Sharing information among data types based on the underlying ecological, and recognizing observation biases, can improve estimation of individual and global parameters. We present a general integrated population model (IPM), specifically designed for brown bears (Ursus arctos), using three common data types for bear (U. spp.) populations: repeated counts, capture–mark–recapture, and litter size. We considered factors affecting ecological and observation processes for these data. We assessed the practicality of this approach on a simulated population and compared estimates from our model to values used for simulation and results from count data only. We then present a practical application of this general approach adapted to the constraints of a case study using historical data available for brown bears on Kodiak Island, Alaska, USA. The IPM provided more accurate and precise estimates than models accounting for repeated count data only, with credible intervals including the true population 94% and 5% of the time, respectively. For the Kodiak population, we estimated annual average litter size (within one year after birth) to vary between 0.45 [95% credible interval: 0.43; 0.55] and 1.59 [1.55; 1.82]. We detected a positive relationship between salmon availability and adult survival, with survival probabilities greater for females than males. Survival probabilities increased from cubs to yearlings to dependent young ≥2 years old and decreased with litter size. Linking multiple information sources based on ecological and observation mechanisms can provide more accurate and precise estimates, to better inform management. IPMs can also reduce data collection efforts by sharing information among agencies and management units. Our approach responds to an increasing need in bear populations’ management and can be readily adapted to other large carnivores.

Decline in territory size and fecundity as a response to carrying capacity in an endangered songbird

Density-dependent processes are fundamental mechanisms for the regulation of populations. Ecological theories differ in their predictions on whether increasing population density leads to individual adjustments of survival and reproductive output or to dominance and monopolization of resources. Here, we use a natural experiment to examine which factors limit population growth in the only remaining population of the endangered pale-headed brush finch (Atlapetes pallidiceps). For three distinct phases (a phase of population suppression, 2001-2002; expansion due to conservation management, 2003-2008; and equilibrium phase, 2009-2014), we estimated demographic parameters with an integrated population model using population size, the proportion of successfully breeding pairs and their productivity, territory size, and mark-recapture data of adult birds. A low proportion of successful breeders due to brood parasitism (0.42, 95% credible interval 0.26-0.59) limited population growth before 2003; subsequent culling of the brood parasite resulted in a two-fold increase of the proportion of successful breeders during the 'expansion phase'. When the population approached the carrying capacity of its habitat, territory size declined by more than 50% and fecundity declined from 1.9 (1.54-2.27) to 1.3 (1.12-1.53) chicks per breeding pair, but the proportion of successful breeders remained constant (expansion phase: 0.85; 0.76-0.93; equilibrium phase: 0.86; 0.79-0.92). This study demonstrates that limiting resources can lead to individual adjustments instead of despotic behavior, and the individual reduction of reproductive output at high population densities is consistent with the slow life-history of many tropical species.