I appreciate the thought, but I'm asking a different sort of question.

I'm not looking to calculate the deviance of a node. As you point out, the
deviance is a single value that you calculate from your likelihood
function. However, the likelihood function has many parameters, and these
parameters are nodes in the model. Thus, in order to calculate a deviance,
you need to choose a specific value for each parameter/node to use in that
calculation.  Dhat is the deviance you get when you calculate the
likelihood using the posterior means. My concern is that this is an awkward
definition. Because of the various nonlinear effects in most models, and
the alternate possible parameterizations, the value that you get for Dhat
is not unique. Rather it seems to depend on the specific way in which you
parameterize your model. There is a fuzziness in the definition of Dhat,
which results in a fuzziness in pD and DIC, which means that neither pD nor
DIC is the sort of authoritative fitness measure that I would have hoped
for.

I was hoping some of the software authors/experts might have insight into
the way that BUGS chooses to calculate Dhat, so that I could have more
confidence in the way that I use DIC.

Tim Handley
Research Assistant
Mediterranean Network, NPS
805-658-5759 (CHIS)
805-370-2396 (SAMO)




                                                                           
             "King, David B*"                                              
             <[log in to unmask]                                             
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                                                                   Subject 
                                       RE: [BUGS] on the meaning of pD     
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           




Hello Tim,

I don't think there is a value of deviance for every node because if there
were that would imply that every variable would have a likelihood function
associated with it.
Remember that the deviance is  -2 * the log likelihood and the log
likelihood is generally computed by taking the difference between the data
Y  and the prediction.

I believe that you have one log likelihood function for every model.

Does this help?


-----Original Message-----
From: (The BUGS software mailing list) [mailto:[log in to unmask]] On
Behalf Of Timothy Handley
Sent: Wednesday, February 15, 2012 3:08 PM
To: [log in to unmask]
Subject: [BUGS] on the meaning of pD

I'm comfortable with the intuitive meaning of pD, as the effective number
of model parameters. However, I'm unclear on how you go about calculating
this, and so I've become a little concerned about the meaning of reported
pD values.

Specifically, I looked at the Spiegelhalter et al 2002 article, but didn't
follow all the details. I did see where the article points out that, in a
hierarchical model, the number of parameters depends on perspective, on a
choice of focus. Is there an implicit choice of focus that is used in
Win/OpenBUGS when calculating pD? Are there consequences to this choice
that are important for the interpretation of pD and DIC values?

For example, consider the definition of pD:
  Dbar - Dhat
One can calculate the mean value of every node in a model, and then use
those values to calculate Dhat. Is this the way it is done? Is this the way
it ought to be done? For the case of a hierarchical model, it's not clear
to me that one ought to calculate Dhat in this manner. Suppose that you
have a hierarchical model like
  tau ~ dgamma(.01,.01)
  for(j in 1:N) { effect_size[j] ~ dlnorm(1, tau) } The above effect sizes
are drawn from an asymmetric distribution. And tau always feels like a
nonlinear parameter to me. In this situation, it seems likely that the
empirical distribution of the mean effects sizes will be different than the
distribution described by the log-normal with mean-tau.
Given this disconnect, I'm not clear that a simple calculation of mean
values would be the right way to calculate Dhat.

Any advice and clarification, particularly the intuitive kind, would be
greatly appreciated,

Tim Handley
Research Assistant
Mediterranean Network, NPS
805-658-5759 (CHIS)
805-370-2396 (SAMO)

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