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Suppose the following Bayesian hierarchical model: $$ Y|\lambda\sim\text{Pois}(\lambda) $$ $$ \log(\lambda)|(\mu,\sigma)\sim N(\mu,\sigma)$$ $$ \mu\sim N(0,10) $$ $$ \sigma\sim\text{Gamma}(0.01,0.01).$$ Suppose an observation $y$ of $Y$. I am interested on the theoretical simulation of a sample from the posterior predictive distribution of $Y$, written as $p(\tilde{y}|y)$. Is the following procedure correct?

  1. Take a sample from the posterior distribution of $\mu$ and $\sigma$ : $\mu^{(1)},\ldots,\mu^{(M)}$ and $\sigma^{(1)},\ldots,\sigma^{(M)}$ (assume we have it, from WinBUGS for example).

  2. Generate an observation $\eta^{(i)}$ from $N(\mu^{(i)},\sigma^{(i)})$, $i=1,\ldots,M$.

  3. Generate an observation $\gamma^{(i)}$ from $\text{Pois}(\exp(\eta^{(i)}))$, $i=1,\ldots,M$. Then $\gamma^{(1)},\ldots,\gamma^{(M)}$ is a sample from $Y$.

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    $\begingroup$ That is correct, but I'd council strongly against your prior for $\sigma$; this is known to be problematic when there's any chance of $\sigma$ being near zero. Gelman has an article on Euclid projecteuclid.org/euclid.ba/1340371048 which talks about this, and there are other articles out there too. $\endgroup$ – jbowman May 25 '17 at 15:47
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    $\begingroup$ Correct (but note the warning by @jbowman ) if $\mu$ and $\sigma$ are independent (as you surely know, just mentioning it for someone who may not be aware of this). $\endgroup$ – DeltaIV May 25 '17 at 16:14

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