This is a question about unit testing samplers.
1) I read the wiki page and I am a bit confused. Suppose that in a given model, I have a parameter
f, and I set it to a known value
f₀, simulate data, and using that data I ran Stan (using the model) and get posterior draws
fhat after burn-in. Then in
Δ = (fhat-E[f]) / MCMC-SE ~ N(0, 1)
should I use
E[f] = f₀, or the sample average of
fhat? Similarly, is
MCMC-SE the standard error of
fhat, calculated using the effective sample size?
2) Cook, Gelman and Rubin (2006) [Validation of Software for Bayesian Models...] propose using posterior quantiles of the true value, transformed into a
χ² via a unit normal. Is that related, or complementary to the above approach?