Removing undefined (NaN) values during log-likelihood maximization

Question

I am trying to maximize a Poisson likelihood function for an array of values, of the form

llh_ij = -m_ij + k_ij*ln(m_ij/k_ij).

(where I use the numerical approximation ln(k!) ~ k*ln(k)).

For my purposes, the m_ij are the number of events predicted by a model, and the k_ij are the number of events observed. For the total log-likelihood, I sum the likelihoods of all elements of the array.

In some cases, the m = k = 0, and the log likelihood becomes (-Inf - -Inf) = NaN.

Is it an acceptable solution to deal with this by replacing the NaN values with zero? The way that I have implemented this solution in my (Matlab) code (following an example from a similar analysis) is

if(any(isnan(llh(:))))
llh(isnan(llh)) = 0;
end

...

m = sum(sum(llh, 1),2);

If I do not reset the NaN values of the likelihood, the entire llh is returned as NaN and the maximization fails.

By setting the NaN elements of the likelihood array to zero, this procedure effectively causes the elements that would be NaN to be ignored in the analysis, but I am having difficulty justifying this procedure, and am concerned that it introduces or ignores errors later in the analysis (the point is to calculate a set of confidence intervals in addition to the MLE, and the intervals are quite large where they should be essentially zero).

Is this a conceptually acceptable way to deal with these NaN elements, and if not, is there a more formal solution?

Answer 1

I guess the $m_{ij}$'s in your model are functions of a smaller number of parameters, and you then get into trouble when you evaluate the likelihood at some point in the parameterspace where some of these $m_{ij}$'s become numerically equal to zero? This is quite likely to happen near or at the MLEs. At such points in the parameter space, however, these observations have probability approaching 1 and so a contribution approaching $\ln(1)=0$ to the log-likelihood, so the way you handle this is in effect correct. An alternative general solution would be to always skip computing $\ln m_{ij}$ whenever you have observations $k_{ij}=0$ since $\ln m_{ij}$ is never needed for such observations.

Btw, I believe your expression $-m_{ij} + k_{ij}\ln(m_{ij}/k_{ij})$ is incorrect and should be $-m_{ij} + k_{ij}\ln m_{ij}$ (the log of the Poisson probability $e^{-m_{ij}}m_{ij}^{k_{ij}}/k_{ij}!$ omitting the constant term $-\ln k_{ij}!$)?