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$\newcommand{\cost}{\operatorname{cost}}$My cost-metric is in following form

\begin{equation} \cost(x,y) = A(x,y_1) \times \sum_{i}b_i B_i(x,y_i)\tag{1} \end{equation}

where $A$ and $B$'s follow normal distribution. For my computer implementation, I am thinking of taking $\log(\cdot)$ to avoid computing exponential. By taking log of (1), I get

\begin{equation} \log(\cost(x,y)) = \log(A(x,y_1)) + \log\left(\sum_i b_i B_i(x,y_i)\right) \end{equation}

which simplifies first term but second term remains unchanged. Can I bring $\log(\cdot)$ inside the summation?

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    $\begingroup$ No, you cannot. $\endgroup$
    – Jay Zha
    May 24, 2017 at 23:12
  • $\begingroup$ Are $A$ and $B_i$ all independent of each other? In what what do they depend on $x$ and $y$? Are those supposed to be parameters identifying which normal distribution it is? $\endgroup$ May 24, 2017 at 23:42

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No, you can't bring the logarithm inside the summation because that's equivalent to saying that the logarithm of a sum is the sum of logarithms.

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