[R] Prediction interval of Y using BMA

Spencer Graves spencer.graves at pdf.com
Sun Jul 16 06:23:27 CEST 2006

	  From what I've heard, the primary reason for using Bayesian Model 
Averaging (BMA) is to get better predictions, both better point 
estimates and better estimates of the uncertainties.  I could not find a 
function to compute that.

	  However, if you've got point estimates of predictions, I can give you 
a formula for the variance of the prediction error:

	  var(Y|x) = var(E(Y|x,i)over i)+E(var(Y|x,i)over i),

where Y is the response variable, and 'i' indicates which model.  This 
is a companion to the formula for the predictions:

	  E(Y|x) = E(E(Y|x,i)over i).

	  If you've got a function to compute E(Y|x,i) and compute their 
expectation over i, it should not be too difficult to modify that 
function to also compute var(Y|x,i) and then to compute both 
E(var(Y|x,i)over i) and var(E(Y|x,i)over i).  Then if distributions are 
roughly normal, you've got what you need.

	  Does this make sense?
	  Hope this helps.
	  Spencer Graves
p.s.  I've copied the maintainer for the BMA package on this email.  I 
hope he will correct any misstatement in these comments and update us on 
any relevant capabilities we may have missed.
Nicolas.Meyer at chru-strasbourg.fr wrote:
> Hello everybody, 
> In order to predict income for different time points, I fitted a linear
> model with polynomial effects using BMA (bicreg(...)). It works fine, the
> results are consistent with what we are looking for. 
> Now, we would like to predict income for a future time point t_next and of
> course draw the prediction interval around the estimated value for this
> point t_next. I've found the formulae for the ponctual estimation of t_next
> but I cannot succeed in finding the formula to compute the prediction
> interval based on BMA sets of models, which requires the adequate variance.
> The paper of Hoeting et al. on BMA gives the posterior variance of a
> parameter but what I want is the posterior variance of a predicted Y and not
> the posterior variance of a beta, since the goal here is not to build an
> explanatory model but an accurately predictive model. 
> Is there a way to get around it with the function of the BMA package or does
> anybody has indication on where to find the formulae ? I searched the web
> for hours but nothing like this is to be found. 
> Any help will be appreciated. 
> Thank's in advance,  
> Nicolas Meyer
> Département de Santé Publique
> Unité de Biostatistique et Méthodologie
> 03 88 11 63 58
> nicolas.meyer at chru-strasbourg.fr <mailto:nicolas.meyer at chru-strasbourg.fr> 
> La conjecture de Feynman : 
> "To report a significant result and reject the null in favor of an
> alternative hypothesis is meaningless unless the alternative hypothesis has
> been stated before the data was obtained"
> "The meaning of it all", 1998. 
> 	[[alternative HTML version deleted]]
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