## PRIOR statement in mixed procedure

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# PRIOR statement in mixed procedure

Dear Everybody,

I need some help about the PRIOR statement in the mixed procedure.

Is thsi option allow bayesian estimation?

The different estimates are exactly the same with and without the option and the posterior distribution mean and 5% and 95% percentiles are very close to the frequentist mean and the 90% confidence interval.

I use the non informative JEFFREYS prior.

SAS doesn't mention this procedure as a procedure able to make bayesian estimation.

Samir

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Solution
‎01-26-2016 04:56 AM
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Posts: 490

## Re: PRIOR statement in mixed procedure

Posted in reply to Sam28041977

What PROC MIXED provide is falling under what is called "Empirical Bayes". Because the parameters or hyperparameters of the prior distributions of the random effects are actually estimated from the data via the likelihood functions.

Not only PROC MIXED offeres that but other procedures also see: Usage Note 23407: Available Bayesian methods

But if we talk about fully Bayesian treatment then you should consider using one of the procedures mentioned as in here: Bayesian Analysis Using SAS/STAT Software

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‎01-26-2016 04:56 AM
Super Contributor
Posts: 490

## Re: PRIOR statement in mixed procedure

Posted in reply to Sam28041977

What PROC MIXED provide is falling under what is called "Empirical Bayes". Because the parameters or hyperparameters of the prior distributions of the random effects are actually estimated from the data via the likelihood functions.

Not only PROC MIXED offeres that but other procedures also see: Usage Note 23407: Available Bayesian methods

But if we talk about fully Bayesian treatment then you should consider using one of the procedures mentioned as in here: Bayesian Analysis Using SAS/STAT Software

Occasional Contributor
Posts: 12

## Re: PRIOR statement in mixed procedure

Posted in reply to mohamed_zaki

Thanks a lot mohamed-zaki for your answer.

It confirms what I suspect.

Have a nice day,

Samir

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Posts: 684

## Re: PRIOR statement in mixed procedure

Posted in reply to Sam28041977

I somewhat disagree with the 'solution' post. With he PRIOR statement, one is getting true Bayesian analysis with PROC MIXED. See the very nice presentation in chapter 13 of the book SAS for Mixed Models, 2nd edition (2006). One just does not have much control over the priors. A Jeffrey's prior is used for the variance parameters (these are based on the data, but are considered to be non-informative). A flat (constant) improper prior is used for the fixed effects. There is direct sampling from the posteriors (using a independence chain). With these restrictions, the posterior is conditionally multivariate normal. The statistics paper describing the approach in MIXED is: Wolfinger and Kass (2000). Non-conjugate Bayesian Analysis of Variance Component Models. BIOMETRICS 56:768-774.

For more flexible approaches to Bayesian anlaysis, one has to use the other listed procedures.

Posts: 2,655

## Re: PRIOR statement in mixed procedure

[ Edited ]

I would like to add to Larry's comment.  C.R. Henderson showed in the 80s, based on a report from Laird and Ware, that using maximum likelihood estimates of the G and R matrices yields an empirical Bayes estimate of the random effects.  So even without sampling, mixed model estimates of the variance components are Bayesian, given the data.  This traces back to Harville's work that REML is related to Bayesian marginal inference.{Harville, D. A. (1974). Bayesian inference for variance components using only error contrasts. Biometrika 61, 383–385.}

I think I just proved I'm old...

Steve Denham

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