R help - Apr 2011 - MLE where loglikelihood function is a function of numerical solutions

Hi there,

I'm trying to solve a ML problem where the likelihood function is a function
of two numerical procedures and I'm having some problems figuring out how to
do this.

The log-likelihood function is of the form L(c,psi) = 1/T sum [log (f(c,
psi)) - log(g(c,psi))], where c is a 2xT matrix of data and psi is the
parameter vector. f(c, psi) is the transition density which can be
approximated. The problem is that in order to approximate this we need to
first numerically solve 3 ODEs. Second, numerically solve 2 non-linear
equations in two unknowns wrt the data. The g(c,psi) function is known, but
dependent on the numerical solutions.
I have solved the ODEs using the deSolve package and the 2 non-linear
equations using the BB package, but the results are dependent on the
parameters.

How can I write a program that will maximise this log-likelihood function,
taking into account that the numerical procedures needs to be updated for
each iteration in the maximization procedure?

Any help will be much appreciated.


Kristian

	[[alternative HTML version deleted]]

Hi Kristian

The obvious approach is to treat it like any other MLE problem:  evaluation
of the log-likelihood is done as often as necessary for the optimizer  
you are using: eg a call to optim(psi,LL,...)  where LL(psi) evaluates  
the log likelihood at psi.  There may be computational shortcuts that  
would work if you knew that LL(psi+eps) were well approximated by  
LL(psi), for the values of eps used to evaluate numerical derivatives  
of LL.  Of course, then you might need to write your own custom  
optimizer.

albyn

Quoting Kristian Lind <kristian.langgaard.lind at gmail.com>:
> Hi there,
>
> I'm trying to solve a ML problem where the likelihood function is a
function
> of two numerical procedures and I'm having some problems figuring out
how to
> do this.
>
> The log-likelihood function is of the form L(c,psi) = 1/T sum [log (f(c,
> psi)) - log(g(c,psi))], where c is a 2xT matrix of data and psi is the
> parameter vector. f(c, psi) is the transition density which can be
> approximated. The problem is that in order to approximate this we need to
> first numerically solve 3 ODEs. Second, numerically solve 2 non-linear
> equations in two unknowns wrt the data. The g(c,psi) function is known, but
> dependent on the numerical solutions.
> I have solved the ODEs using the deSolve package and the 2 non-linear
> equations using the BB package, but the results are dependent on the
> parameters.
>
> How can I write a program that will maximise this log-likelihood function,
> taking into account that the numerical procedures needs to be updated for
> each iteration in the maximization procedure?
>
> Any help will be much appreciated.
>
>
> Kristian
>
> 	[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
>