Given unnormalized weights in logs, how do I compute normalized weights in logs?

by bamts   Last Updated November 08, 2018 23:19 PM

Basics

Given a set of nonnegative weights $\{w_i\}_{i=1}^n$, I can normalize them by computing $$W_i = \frac{w_i}{\sum_{j=1}^nw_j}\quad \forall i=1,...,n.$$ Easy enough. But for numerical reasons, it is good to work in logs.

Question

Given the set of unnormalized weights in logs $\{\ln w_i\}_{i=1}^n$, what's the best way to compute the set of normalized weights in logs $\{\ln W_i\}_{i=1}^n$?

Naive Approach

Given $\{\ln w_i\}_{i=1}^n$, I can obviously recover the $w_i$ by exponentiating the $\ln w_i$, and from there I can compute $$\ln W_i = \ln w_i - \ln\left[\sum_{j=1}^nw_j\right]\quad \forall i = 1,...,n.$$ Can I do better? Is it possible to compute the $\ln W_i$ without exponentiating the $\ln w_i$?



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