I wrote this post after struggling for a few hours with the derivation of the negative log-likelihood function , and finding out that the derivative didn’t match what was expected, only to find out that it was indeed matching and that instead, there’s a discrepancy on the semantics of the “**log**” notation.

Let’s go step by step:

Two of the most common logarithms are the logarithm of base $e$, also called natural logarithm which is written in two forms:

$$\ln{x} = \log_{e}{x}$$

and the logarithm of base 10 also called common logarithm or decimal logarithm:

$$\log_{10}{x}$$

Additionally scientific calculators have a “**log**” and a “**ln**” function,
which apply the logarithm of base 10 and the logarithm of base $e$ respectively:

By Waifer X - 100502-1150494Uploaded by Pieter Kuiper, CC BY 2.0, link

But something that I learned while checking for errors doing the derivative is that mathematicians and calculator designers
have different things in mind when writing “**log**":

From Wikipedia’s Common Logarithm:

On calculators it is usually “log”, but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write “log”. To mitigate this ambiguity the ISO 80000 specification recommends that $log_{10}{x}$ should be written $lg{x}$ and $log_{e}{x}$ should be $ln{x}$.

And from Wikipedia’s Natural Logarithm:

The natural logarithm of x is generally written as $ln x$, $log_{e} x$, or sometimes, if the base $e$ is implicit, simply $log x$.

Therefore in the case of the negative log-likelihood function:

$$ L(\boldsymbol{y}) = - \log(\boldsymbol{y}) $$

Although it’s not implicit that it refers to a logarithm of base $e$, *it is
indeed the natural logarithm*, it’s just being used as mentioned in the given
excerpt of common logarithm (i.e: mathematicians usually mean natural logarithm when they write “log”), and therefore its derivative is:

$$ \dfrac{\partial L_{\boldsymbol{y}}}{\partial \boldsymbol{y}} = -\dfrac{1}{\boldsymbol{y}} $$

Mathematicians be crazy!

Last modified on 2018-05-07

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