[R-sig-eco] correlated parameters, absolute model fit, and MCMC

[Nicholas Lewin-Koh]

Hi Dan,
Since you are fitting a fully Bayesian model, the output from your MCMC
sample
is a sample from the joint posterior distribution. So given that your
chain is mixing
properly and converged (a whole nother kettle of fish) I don't see why
the correlation
is a problem. You can assess the model fit using posterior predictive
checks,
ie from the joint posterior distribution you can generate the posterior
predictive
distribution. There are may papers on this a good place to start is
Gelman et al. (2004) "Bayesian Data Analysis"

Hope this helps
Nicholas 
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Hello -

I'm investigating model adequacy for several stochastic models; I've  
obtained posterior distributions for all parameters using approximate  
MCMC. I'd now like to know, in an absolute sense, how good these  
parameters actually are. In other words, given my data D, and a set  
of summary statistics on those data S(D), I would like to simulate  
date D' under the model parameters and look at the distribution of S 
(D') and see how well it matches up with S(D). This should be a  
straightforward evaluation of absolute model fit via parametric  
simulation.

However, because some of the parameters are correlated, I cannot  
simply use the parameter set with the maximum posterior probability.  
Nor can I use maximum likelihood estimates for the parameters (I am  
using approximate MCMC because no likelihood function can be  
specified, but the simulations are easy, and the data are well- 
characterized by several summary statistics).

Does anyone have any suggestions on how to deal with correlated  
parameters like this, or alternative approaches for evaluating model  
fit?

One solution would be simply to run simulations with parameters  
sampled at random from the complete converged chain (or set of  
chains), with the constraint that I would have to sample all  
parameters from a given generation simultaneously. Parameter sets  
would thus be sampled roughly in proportion to their joint posterior  
probability. But I wonder whether there  may be other approaches out  
there that I should investigate.

This may be more stats than R, but it is an ecological analysis I'm  
running in R!

Thanks-
~Dan Rabosky