Empirical Bayesian inference on Poisson processes with a Clayton prior distribution

Incorrect date on title page. Date of award is 2022.Dependency between rates of occurrence of events can exist for a variety of reasons. For example, management culture within organisations can have a similar impact on multiple outcomes. Modelling approaches that assume independence between event rates can be mathematically convenient, but they might also fail to account for all the information within the data since the existence of dependency means that data fromone process can provide information about the rate of occurrence on similar processes. However, estimating correlated event rates is challenging. We address this challenge by developing an inference framework to account for such dependency using copulas in order to make full use of available data. We develop an empirical Bayesian inference method based on a multivariate Poisson – Clayton with Gamma marginals probability model. The proposed model aims to capture both aleatory and epistemic uncertainties. We assume that events are generated from a homogeneous Poisson process capturing the pure inherent randomness in the observations, i.e. the aleatory uncertainty. Epistemic uncertainty is represented by the prior where the marginal distributions of event rates are Gamma, and the underlying correlation is captured by the Clayton copula. Of particular interest are situations where we might anticipate low rates of occurrence. The Clayton copula is appropriate for situations with left tail dependence, that is where low rates are considered relatively more correlated compared to high rates. However, estimating copulas dependence parameter using count data can be challenging. Hence, we provide analytical expressions for estimating dependency of the Clayton copula as a function of the count data realised from Poisson processes. We examine the relative accuracy of the model and investigate the robustness of results under different parameter settings. To support comparison between the proposed model and existing theory, we consider the classic empirical Bayes method assuming independent Gamma priors. Findings are based on simulation experiments. We also evaluate our method when applied for supplier ranking using de - sensitised real data. We explicitly discuss the ranking problem from a Bayesian perspective, and we propose multiple ranking methods. We identify cases with different final rankings which further enhance the importance of not choosing to ignore dependency.Dependency between rates of occurrence of events can exist for a variety of reasons. For example, management culture within organisations can have a similar impact on multiple outcomes. Modelling approaches that assume independence between event rates can be mathematically convenient, but they might also fail to account for all the information within the data since the existence of dependency means that data fromone process can provide information about the rate of occurrence on similar processes. However, estimating correlated event rates is challenging. We address this challenge by developing an inference framework to account for such dependency using copulas in order to make full use of available data. We develop an empirical Bayesian inference method based on a multivariate Poisson – Clayton with Gamma marginals probability model. The proposed model aims to capture both aleatory and epistemic uncertainties. We assume that events are generated from a homogeneous Poisson process capturing the pure inherent randomness in the observations, i.e. the aleatory uncertainty. Epistemic uncertainty is represented by the prior where the marginal distributions of event rates are Gamma, and the underlying correlation is captured by the Clayton copula. Of particular interest are situations where we might anticipate low rates of occurrence. The Clayton copula is appropriate for situations with left tail dependence, that is where low rates are considered relatively more correlated compared to high rates. However, estimating copulas dependence parameter using count data can be challenging. Hence, we provide analytical expressions for estimating dependency of the Clayton copula as a function of the count data realised from Poisson processes. We examine the relative accuracy of the model and investigate the robustness of results under different parameter settings. To support comparison between the proposed model and existing theory, we consider the classic empirical Bayes method assuming independent Gamma priors. Findings are based on simulation experiments. We also evaluate our method when applied for supplier ranking using de - sensitised real data. We explicitly discuss the ranking problem from a Bayesian perspective, and we propose multiple ranking methods. We identify cases with different final rankings which further enhance the importance of not choosing to ignore dependency