I agree.
The issue arises in genome-enabled prediction, where p>>>>n. In this case, the maximum likelihood estimator admits an infinite number of solutions, so inferences are driven by the prior. And technology is such that n grows slower than p. For example, going from a 3K SNP chip to a 1 million SNP chip, took a few years only, and the rate of phenotyping has been much lower.
Hence, one people make inferences from posteriors (perfectly identified), such inferences are prior driven. Not much Bayesian learning here and hard to defend the solution with the standard argument that the likelihood will eventually dominate, asymptotically. If YOUR prior is A and MY prior is B, then how do you convince people of your (mine) theory in the absence of learning from data?
Regards,
Daniel
________________________________________
From: JI Denis Jean-Baptiste [[log in to unmask]]
Sent: Friday, March 15, 2013 2:13 AM
To: Daniel Gianola
Cc: [log in to unmask]
Subject: Re: [BUGS] Bayesian asymptotics
Hi Daniel,
On Wed, 13 Mar 2013, Daniel Gianola wrote:
> With genomic data (e.g., SNPs), is is common that Bayesian regression
> models have more regression coefficients than data points, ie, n<<p.
> Does Bayesian asymptotics work in this case? Under regularity
> conditions, for a model with p<n, the posterior distribution has an
> asymptotic mean that coincides with a maximum likelihood estimator, but
> in the p>>n situation, the ML estimator does not exist.
(*) What do you mean by 'asymptotic behavior' when 'n<<p'? Keeping n/p
constant?
(*) My feeling is that there is no identifiability point with Bayesian
statistics because we are working in the well defined joint prior
distribution of the parameters (using a proper prior, of course); even if
there is no data, you can make correct inferences, just based on the prior
(since prior and posterior distributions are identical).
regards,
Jean-Baptiste DENIS
m'el : [log in to unmask] unit'e de recherche MIA
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