Abstract

In some modern applications the requirement to define the true sampling distribution, or likelihood, becomes a challenge. Sometimes it is impossible or very expensive to evaluate the likelihood, and is in those cases where the use of generalized Bayesian inference has proven to be useful.
Generalized Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and when the true likelihood is known, the procedure coincides with the original Bayesian updating.

Stochastic population dynamics models constitute an important component of applied and theoretical ecology. Statistical inference for these models can be difficult when, in addition to the process error, there is also sampling error. Ignoring the presence of sampling variability can lead to biases in the estimations, resulting in erroneous conclusions of the system behavior. The Gompertz model is well known for its use to describe the growth of animals, plants or cells, and it is possible to adapt it in order to consider sampling variability. However, when the sampling error is explicitly considered, the exact likelihood formulation becomes further complicated as it involves a T-dimensional integral.

In this work, we discuss the use of the composite likelihood as a loss function under the generalized Bayesian inference approach for estimation of the parameters of the Gompertz model in the presence of sampling variability. To obtain a pseudo- posterior distribution we propose a Metropolis-Hasting algorithm that computes an approximation of the composite likelihood in each Markov chain step.
It is possible to compute the composite likelihood as product of the 1,2,..., or n dimensional- marginal distributions. Although the calculation of the one dimension formulation is easier and faster, as it does not include all the parameters, it implies a 2-step inference procedure, leading to error propagation.
We consider both, one and two dimensions formulations, and using simulations we test and compare the ability of this approach to recover the real parameters of the model.