I used Stan to simulate the posterior sample from a model with five parameters and obtained a very good convergence for 4 parallel chains. I calculated the posterior sample means as approximate Bayes estimates for the model parameters. The results of Bayes estimates and MLEs are almost the same. Is it necessary to prove the posterior mean does exist for all the parameters? My model is some kind of modified Weibull model and there is a paper, Sun and Speckman (2005), shows that the theoretical posterior mean for a Weibull distribution parameter fails to exist.
Thanks for any comment in advance.
not an expert on this, but maybe the simple solution is to use median instead of mean? But I fear that if posterior mean doesn’t exist, the posterior might turn out to be difficult to explore for the sampler - are the fit diagnostics OK? (no divergences, Rhat < 1.01, n_eff > ~100)?
I also find it quite plausible, that the paper speaks about a specific case that may not apply to your particular model. Could you provide the link to the paper and code for your model?
Hope that helps!
The posterior mean will exist as long as the distribution is not too wide tailed. For instance, a Cauchy distribution doesn’t have a mean. But we can use a Cauchy prior, then when we combine with data, the posterior will have a mean. Median can work for Cauchy.
The thing to do to test your model is to simulate data from it and see if you can recover it within posterior intervals. If the sampling works, this should work.
P.S. I moved this to the “modeling” category.
Dear Martin and Bob,
Thank you very much for your kind comment. The diagnostics test is perfect.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat alpha 0.0118 0.0000 0.0000 0.0117 0.0117 0.0118 0.0118 0.0118 4325 0.9998 beta 0.0773 0.0002 0.0100 0.0587 0.0704 0.0770 0.0835 0.0986 2948 0.9995 gamma 0.4525 0.0010 0.0421 0.3717 0.4243 0.4518 0.4803 0.5365 1745 0.9996 theta 88.7129 0.1865 10.9784 68.3228 81.2060 88.2621 95.6059 111.8775 3465 0.9995 lambda 0.0105 0.0000 0.0013 0.0079 0.0096 0.0104 0.0114 0.0132 3388 1.0000 lp__ -848.5131 0.0393 1.6280 -852.4742 -849.3797 -848.1620 -847.3125 -846.3607 1719 1.0013
I also did one conference paper on using stan for Weibull model and the results are also good too. Please find the link for our paper.
PS: My post related to my current submitted paper and a young researcher from China who just starting to learn Bayes (no expertise) asking me to prove that. There are so many paper using Bayes for modified Weibull models without proving for the existence of posterior mean, provided that the MCMC methods produce good posterior sample.
Thank you for your comment and for moving my post to the right category.
I am not sure whether your last post expected a reaction from us or if you are just providing more detail without expecting an answer. If you still need help, let us know!
Best of luck with your model!