[ADMB Users] ADMB and hierarchical (multi-level) models

[Paul Conn]

Hi Saang-Yoon,

I agree that hierarchical models potentially pose problems for 
MLE/maximum a posteriori (MAP) estimation and inference (possibly 
leading to bias and overly precise estimates), but wouldn't MCMC 
estimates be okay because you're integrating over the plausible range of 
values for unobserved data (in a complete data sense)?  Have you tried 
fitting models with the 'mcmc' option in ADMB?

Papers by Mendelssohn (Fish Bull 1988) and DeValpine and Hilborn (CJFAS 
2005) pointed out problems with including latent states/missing data as 
'parameters' within maximum likelihood, but to my knowledge there hasn't 
been much follow up with regard to typical parameters of interest 
(abundance, biomass, etc.).  My sense is that MAP estimators still 
perform reasonably well with moderate amounts of process error 
(autocorrelated recruitment for instance) but it would be good to look 
into further. 

Paul

Saang-Yoon wrote:
> Dear ADMB users.
>
> I wonder about how people code multi-level models in ADMB.  I
> illustrate my question with a simple example.  Let's assume we have a
> simple regression model,
> Y = beta0 + beta1*X + error, where error ~ N(0, sigma2)
>
> Please think about two problems of (1) estimation of parameters
> (beta0, beta1, and sigma2), and then (2) prediction of unknown random
> variable (Y at a future time, given new X).  Strictly speaking,
> unknown Y at a future time (say, newY) is NOT a parameter but a random
> variable, although many fisheries papers treat the Y as a parameter.
> But I follow the incorrect treatment (i.e., newY as a parameter) at
> the moment to focus on my question about ADMB.  Also this is a simple
> “example” for showing my problem with ADMB when facing a hierarchical
> model.
>
> (1) Estimation of parameters, beta0, beta1, and sigma2
> L(beta0, beta1, sigma2 | observed Ys, observed Xs)
> This likelihood provides inference of these three parameters.  I call
> it L1
>
> (2) Calculation of new Y given new X.
> L(newY | beta0, beta1, sigma2, newX)
> I call this second likelihood function L2.  newX is a constant.
>
> These two steps can be viewed as a multi-level or hierarchical
> structure.  In ADMB, the objection function would be the sum of the
> respective negative loglikelihood functions: i.e.,
>  f = – logL1 – logL2;
> where beta0, beta1, sigma2, and newY are declared as free parameters
> in PARAMETER SECTION in ADMB.
>
> My problem with this above coding is that estimates of beta0, beta1,
> and sigma2 are affected by “newY” as well as “observed Ys” and
> “observed Xs”.  This is WRONG!!!   Estimation of beta0, and beta1, and
> sigma2 must depend ONLY on “observed Ys”, and “observed Xs”.
>
> I wonder about how ADMB experts do around this problem.  I would
> extremely appreciate your guidance and help.   Thank you,
>
> Saang-Yoon
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>   

-- 
Paul B. Conn, Ph.D.
Research Statistician
National Marine Fisheries Service
NOAA Fisheries Center for Coastal Fisheries and Habitat Research
Southeast Fisheries Science Center
101 Pivers Island Rd
Beaufort, NC 28516
phone 252.838.0807
fax 252.728.8619
Paul.Conn at noaa.gov

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