### log-sum-exp trick

when I implement models with discrete variables (which actually happens more than one can think), I always end up estimating this value:

\[ V = \log \left( \sum_i e^{b_i} \right) \]

Why ? This usually happens at the denominator of a Bayes formula for example. I try to keep \(log\)-probabilities all the time so that not to have to deal with very small numbers and to do additions instead of multiplications. By the way, I was looking at the time and latency of floating-point instructions in the latest processors (like Intel Core i7 for example), and I realized that still in 2010, additions are faster than multiplications (even with SSEx and the like).

Therefore, use \(log\)

In this expression, \(b_i\) are the log-probabilities and therefore \(e^{b_i}\) are very small or very big yielding to overflow or underflow sometimes. A scaling trick can help using numbers in a better range without loss of accuracy and for a little extra cost as follows:

\[ \begin{array}{rcl} \log \left( \sum_i e^{…

\[ V = \log \left( \sum_i e^{b_i} \right) \]

Why ? This usually happens at the denominator of a Bayes formula for example. I try to keep \(log\)-probabilities all the time so that not to have to deal with very small numbers and to do additions instead of multiplications. By the way, I was looking at the time and latency of floating-point instructions in the latest processors (like Intel Core i7 for example), and I realized that still in 2010, additions are faster than multiplications (even with SSEx and the like).

Therefore, use \(log\)

In this expression, \(b_i\) are the log-probabilities and therefore \(e^{b_i}\) are very small or very big yielding to overflow or underflow sometimes. A scaling trick can help using numbers in a better range without loss of accuracy and for a little extra cost as follows:

\[ \begin{array}{rcl} \log \left( \sum_i e^{…