# Averaging Log-Likelihood Values:Numerical Stability

## 1. Weighted log-likelihood values

The problem addressed in this page is the computation of:

It’s very advisable to employ **log-likelihoods** (and **log-weights**) for a extended dynamic range. This avoids the problem of weights becoming zero.

Then, if we have as inputs the log-weights () and the individual log-likelihoods (), the problem becomes:

The problem: the exponential of very large/small numbers will produce under/overflow.

The solution: to shift all the exponents and next correct it in the log sum:

This method is implemented in the C++ function mrpt::math::averageLogLikelihood.

## 2. Unweighted log-likelihood values (Arithmetic mean)

If all the samples have equal weights, the formula simplifies to: