Bayesian Poisson Model Research Articles

Challenges with non-informative gamma priors in the Bayesian over-dispersed Poisson reserving model

Tables 2 and 3 in England et al. (2012) raise the conjecture that the claims reserves in the Bayesian over-dispersed Poisson (BODP) model with non-informative gamma priors are equal to the claims reserves in the chain-ladder (CL) model (the small differences in the figures could be explained by simulation inaccuracy). We give a counter-example to this conjecture. Moreover, we show that, working with non-informative priors needs a lot of care because a naïve treatment may lead to improper posterior distributions.

Bayesian spatial modeling of HIV mortality via zero‐inflated Poisson models

In this paper, we investigate the effects of poverty and inequality on the number of HIV-related deaths in 62 New York counties via Bayesian zero-inflated Poisson models that exhibit spatial dependence. We quantify inequality via the Theil index and poverty via the ratios of two Census 2000 variables, the number of people under the poverty line and the number of people for whom poverty status is determined, in each Zip Code Tabulation Area. The purpose of this study was to investigate the effects of inequality and poverty in addition to spatial dependence between neighboring regions on HIV mortality rate, which can lead to improved health resource allocation decisions. In modeling county-specific HIV counts, we propose Bayesian zero-inflated Poisson models whose rates are functions of both covariate and spatial/random effects. To show how the proposed models work, we used three different publicly available data sets: TIGER Shapefiles, Census 2000, and mortality index files. In addition, we introduce parameter estimation issues of Bayesian zero-inflated Poisson models and discuss MCMC method implications.

Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method

AbstractWe consider the Bayesian over-dispersed Poisson (ODP) model for claims reserving in general insurance. We choose two different types of prior distributions for the parameters and then study the different Bayesian predictors. This study leads, on the one hand, to the classical chain ladder predictor and, on the other hand, to Bornhuetter & Ferguson predictors. We highlight (either analytically or numerically) how these predictors are obtained and how their prediction uncertainty can be determined.

Bayesian Time-Series Analysis of a Repeated-Measures Poisson Outcome With Excess Zeroes

In this article, the authors demonstrate a time-series analysis based on a hierarchical Bayesian model of a Poisson outcome with an excessive number of zeroes. The motivating example for this analysis comes from the intensive care unit (ICU) of an urban university teaching hospital (New Haven, Connecticut, 2002-2004). Studies of medication use among older patients in the ICU are complicated by statistical factors such as an excessive number of zero doses, periodicity, and within-person autocorrelation. Whereas time-series techniques adjust for autocorrelation and periodicity in outcome measurements, Bayesian analysis provides greater precision for small samples and the flexibility to conduct posterior predictive simulations. By applying elements of time-series analysis within both frequentist and Bayesian frameworks, the authors evaluate differences in shift-based dosing of medication in a medical ICU. From a small sample and with adjustment for excess zeroes, linear trend, autocorrelation, and clinical covariates, both frequentist and Bayesian models provide evidence of a significant association between a specific nursing shift and dosing level of a sedative medication. Furthermore, the posterior distributions from a Bayesian random-effects Poisson model permit posterior predictive simulations of related results that are potentially difficult to model.

Bayesian multivariate Poisson models for insurance ratemaking

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner’s insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.

Empirical Bayes models of Poisson clinical trials and sample size determination

Bayesian methods are often used to reduce the sample sizes and/or increase the power of clinical trials. The right choice of the prior distribution is a critical step in Bayesian modeling. If the prior not completely specified, historical data may be used to estimate it. In the empirical Bayesian analysis, the resulting prior can be used to produce the posterior distribution. In this paper, we describe a Bayesian Poisson model with a conjugate Gamma prior. The parameters of Gamma distribution are estimated in the empirical Bayesian framework under two estimation schemes. The straightforward numerical search for the maximum likelihood (ML) solution using the marginal negative binomial distribution is unfeasible occasionally. We propose a simplification to the maximization procedure. The Markov Chain Monte Carlo method is used to create a set of Poisson parameters from the historical count data. These Poisson parameters are used to uniquely define the Gamma likelihood function. Easily computable approximation formulae may be used to find the ML estimations for the parameters of gamma distribution. For the sample size calculations, the ML solution is replaced by its upper confidence limit to reflect an incomplete exchangeability of historical trials as opposed to current studies. The exchangeability is measured by the confidence interval for the historical rate of the events. With this prior, the formula for the sample size calculation is completely defined.

Estimation of the within-herd transmission parameter of bovine leukemia virus

In this study, we estimated the transmission parameter of bovine leukemia virus (BLV) infection within herds by following up the serological status of cattle in beef and dairy herds. Two consecutive serological tests using ELISA were conducted for cattle at four beef breeding farms and nine dairy farms with an interval of 5 months. Assuming that sero-converted animals were newly infected, transmission parameters were estimated using a hierarchical Bayesian Poisson model. All tested herds had at least one positive animal at the first testing, but sero-prevalence considerably varied between herds. It is estimated that 0.62 animals were infected by one infected animal introduced into a fully susceptible population for both beef breeding and dairy farms during the 5-month interval. The 95% credible interval of the transmission parameter for the combined herds was 0.37–0.89 over this period. Asymptomatically infected animals with BLV can become the source of lifelong infection in herds. These estimated transmission parameters indicated that the early removal of infected animals with periodical tests could reduce the dissemination of BLV infection within herds.

Community-Level Correlates of Hospitalizations for Persons With Schizophrenia

Community-Level Correlates of Hospitalizations for Persons With Schizophrenia

Geographical spread of gastrointestinal tract cancer incidence in the Caspian Sea region of Iran: Spatial analysis of cancer registry data

BackgroundHigh incidence rates of gastrointestinal tract cancers have been reported in the Caspian region of Iran. This study aimed to: 1) describe the geographical spatial patterns of gastrointestinal tract cancer incidence based on cancer registry data and, 2) determine whether geographical clusters of statistical significance exist.MethodsThe Babol Cancer Registry, which covers the two major northern Iranian provinces of Mazandaran and Golestan (total population = 4,484,622) was used to identify new gastrointestinal tract cancer cases during 2001 to 2005. Age-specific cancer incidence rates were calculated for 7 gastrointestinal tract cancer sites in 26 wards of the Mazandaran and Golestan provinces. Spatial autocorrelation indices, hierarchical Bayesian Poisson models, and spatial scan statistics were used in measuring the geographic pattern and clusters.ResultsThere were non-random spatial patterns in esophageal and stomach cancers that were similar for both sexes. Clusters of high incidence were identified in esophageal, stomach, colorectal and liver cancer for both sexes, as well as a possible cluster of pancreas cancer in males.ConclusionGastrointestinal tract cancers exhibit significant spatial clustering of risk in northern Iran. Further work is needed to relate these geographical patterns to information on potential life-style and environmental factors.

On Models for Binomial Data with Random Numbers of Trials

A binomial outcome is a count s of the number of successes out of the total number of independent trials n=s+f, where f is a count of the failures. The n are random variables not fixed by design in many studies. Joint modeling of (s, f) can provide additional insight into the science and into the probability pi of success that cannot be directly incorporated by the logistic regression model. Observations where n= 0 are excluded from the binomial analysis yet may be important to understanding how pi is influenced by covariates. Correlation between s and f may exist and be of direct interest. We propose Bayesian multivariate Poisson models for the bivariate response (s, f), correlated through random effects. We extend our models to the analysis of longitudinal and multivariate longitudinal binomial outcomes. Our methodology was motivated by two disparate examples, one from teratology and one from an HIV tertiary intervention study.

Bayesian Poisson Models for the Graphical Combination of Dependent Expert Information

Summary A supra-Bayesian (SB) wants to combine the information from a group of k experts to produce her distribution of a probability θ. Each expert gives his counts of what he thinks are the numbers of successes and failures in a sequence of independent trials, each with probability θ of success. These counts, used as a surrogate for each expert's own individual probability assessment (together with his associated level of confidence in his estimate), allow the SB to build various plausible conjugate models. Such models reflect her beliefs about the reliability of different experts and take account of different possible patterns of overlap of information between them. Corresponding combination rules are then obtained and compared with other more established rules and their properties examined.