[R] Hessian in box-constraint problem - concern OPTIM function

Spencer Graves spencer.graves at pdf.com
Tue Jun 24 11:23:44 CEST 2008

      This is an interesting question. 

      What is the problem you are trying to solve and how do the 
boundary conditions function as part of this system? 

      I ask because the asymptotic theory behind your formula for 's.e.' 
breaks down with parameters at boundaries.  It assumes that you are 
minimizing the negative log(likelihood) AND the optimum is in the 
interior of the region AND the log(likelihood) is sufficiently close to 
being parabolic that a reasonable approximation for the distribution of 
the maximum likelihood estimates (MLEs) has a density adequately 
approximated by a second-order Taylor series expansion about the MLEs.  
In this case, transforming the parameters will not solve the problem.  
If the maximum is at a boundary and if you send the boundary to Inf with 
a transformation, then a second-order Taylor series expansion of the 
log(likelihood) about the MLEs will be locally flat in some 
direction(s), so the hessian can not be inverted. 

      These days, the experts typically approach problems like this 
using Monte Carlo, often in the form of Markov Chain Monte Carlo 
(MCMC).  One example of an analysis of this type of problem appears in 
section 2.4 of Pinheiro and Bates (2000) Mixed-Effects Models in S and 
S-Plus (Springer). 

      You might get more help from this list if you provide a little 
more detail, preferably including commented, minimal, self-contained, 
reproducible code for a simplified version of your problem as described 
in the posting guide "http://www.R-project.org/posting-guide.html". 

      Hope this helps. 
      Spencer Graves

Cleber Nogueira Borges wrote:
> Hello all useRs,
> I am using the OPTIM function with particular interest in the method 
> because it is a box-constraint method.
> I have interest in the errors estimates too.
> I make:
> s.e. <- sqrt( diag( solve(  optim(...,method='L-BFGS-B', 
> hessian=TRUE)$hessian   )))
> but in help say:
> "Note that this is the Hessian of the unconstrained problem even if the
> box constraints are active."
> My doubts is:
> How to obtain a authentic hessian for a box-constraint problem?
> How I should make a interpretation of this result (concern the hessian) ?
> Is possible make some transformation or so can I considerate this 
> result a good approximation??
> I am grateful for some help!
> References are welcome! :-D
> Cleber Borges
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> and provide commented, minimal, self-contained, reproducible code.

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