Bayesian Estimation Sample Clauses

Bayesian Estimation. We also consider a Bayesian framework for estimating parameters in Models (2.8) and (2.9) based on the likelihoods in Equation (2.11) and in Appendix A [26]. Inference is obtained by sampling from the joint posterior distribution of the parameters using Markov Chain Monte Carlo implemented in WinBUGS 1.4. We define parameter estimates as the posterior median and credible sets from associated 2.5 and 97.5 per- centiles. Estimates and credible sets for φ1 from Model (2.9) can be obtained directly from MCMC samples by transformation of the sampled λ’s, with φ1 = 1 . 1+exp(−λ) Care must be taken in choosing appropriate prior distributions for the parameters. In general, the priors should conform to the the plausible range of values which the parameter may take. Similarly, prior distributions of parameters may be specifically chosen to impose constraints on parameter estimates and credible sets. For Model (2.8) it is clear that 0 < φ1 < 1, and an example of an appropriate uninformative prior is φ1 ∼ Unif(0, 1). For Model (2.9), λ can reasonably take on values in (−∞, +∞); however, an uninformative prior for λ does not necessarily correlate to an uninforma- tive distribution for the parameter of interest φ1. For example, a uniform distribution for λ implies a heavy-tailed U-shaped distribution for φ1, whereas a standard normal logistic distribution for λ corresponds to a uniform distribution for φ1. Lastly, for either Model (2.8) or (2.9) the plausible values for φ2 and φ3 will vary depending on the application. Nevertheless, the prior distributions should still reflect that these are strictly positive quantities in our application of vaccination studies.