# [R] MLE where loglikelihood function is a function of numerical solutions

Albyn Jones jones at reed.edu
Sun Apr 10 19:05:47 CEST 2011

```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]]
>
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