Poisson Likelihood and Noninformative Prior Sample Clauses

Poisson Likelihood and Noninformative Prior. Suppose r = y/n, where r is the death rate, y is the number of deaths, and n is the size of the population at risk. A typical assumption for count data is that they take on a Poisson distribution. Thus, if we assume that y ∼ Poisson with parameter θts—1n where t is the age, s is the number of stages, and θ is another unknown parameter, then the data distribution y|s, x xxx xx xxxxxxx xx x (x|x, x) ∝ (θts—1n)y exp(−(θts—1n)). The conjugate prior for Poisson likelihood is the Gamma distribution. However, the Poisson model parameter θts—1n has two unknown parameters θ and x. It is difficult to derive specific distributions for θ and s in order to obtain the Gamma distribution for θts—1n. Therefore, we consider the noninformative prior distributions for θ and s then use Markov chain Monte Carlo (MCMC) sampler methods to obtain the posterior estimates for them. Prior 1: θ ∼ N (0, σ2) A weak normal prior with large variance can be assigned to the unknown parameter θ. Given little information we know about this parameter, the prior mean is set to zero. Gamma or Beta priors are not used for the unknown parameter θ since possible negative value is observed for θ. The variance for the normal prior can be fixed at a large value (i.e., 106), or defined as a hyperprior with Gamma distribution (i.e., σ2 ∼ Gamma(0.1, 100)).