The use of ostensibly vague inverse-gamma prior distributions for a2 (i.e, IG(e, e) for small e) is fairly conventional in Bayesian analysis of models containing random effects or other latent structure. Gelman (2006) considered the appropriateness of this prior distribution, noting that "...for datasets in which low values of a are possible, inferences become very sensitive to e in this model, and the prior distribution hardly looks non-informative " (p. 522). Royle (2008c) provided a similar analysis of prior sensitivity under the CJS model with heterogeneity, using various IG(e, e) priors for ap. The posterior of a^ appeared relatively insensitive to the prior specification.
In Figure 11.3, the three panels correspond to the estimated posterior under a uniform(0,5) prior (left panel) for ap, the estimated posterior under an IG(1,1) prior for ap (middle panel) and under an IG(.01,.01) prior for ap (right panel). In all cases, the posterior is restricted to [0, 5] where most of the posterior mass occurs (the histograms were restandardized to sum to 1 over that range). These results are
strikingly similar to those reported by Gelman (2006). In particular, note that the uniform prior does not affect the posterior so much, whereas for the conventional vague conditionally-conjugate prior, the posterior does appear to be influenced by choice of e . As summarized by Gelman (2006): "... the inverse-gamma(e,e) prior is not at all 'non-informative' for this problem since the resulting posterior distribution remains highly sensitive to the choice of e". Similarly, Figure 11.3 suggests that a proper uniform prior for the variance component is justified.
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