The existence of posterior mean

Modeling
1

Dear all,
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.
Tien

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2

Hi,
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!

3

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.

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4

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.
http://itekcmsonline.com/rps2prod/esrel2019/e-proceedings/html/0494.xml
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.
Best regards
Tien

5

Dear Bob,
Thank you for your comment and for moving my post to the right category.
Best regards
Tien

6

Dear Tien,
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!

7

Dear Martin,
Thank you for your kind response. I got my the answer from your (and Bob’s) comment.

Tien

PS: I am sorry for my late reply.

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