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Super basic question about the BIC — is it defined in terms of log base ten or the natural logarithm? I see the latter on Wikipedia; but see ‘log’ not ‘ln’ in the original paper (though am aware that ‘log’ can mean ‘ln’...)

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Note that $\log_b(x)=\frac{\log_a(x)}{\log_a(b)} =: c \log_a(x)$. In that sense, all logs are proportional. So they all have the same order. In statistics, the logarithms are often not distinguished for this reason. I saw people use “log” for cases where they don’t care about the base (as it just leads to a different constant).

In this specific case, using a different base will change the result. You’re minimizing something $+ c \ln(d)$, where $c$ is a constant which is determined by the basis of choice. So, if you choose a different basis, the penalty becomes larger (or smaller).

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If one isn't clarifying the base when calling "log" normally the natural logarithm is meant. And so it is the case here as well.

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    $\begingroup$ At least in statistics that's the convention. Other fields may be different! $\endgroup$ Feb 19 at 16:29
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    $\begingroup$ I'd say that introductory statistics texts and courses might mean log base 10 when they say log, but once past that stage the default surely changes to natural logarithms. Halmos called ln “a textbook vulgarization", Not for me to say he was wrong. I use it all the time -- when among geographers and kin -- to disambiguate. $\endgroup$
    – Nick Cox
    Feb 19 at 16:35
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    $\begingroup$ Some authors in some fields e.g. in information theory use base 2 as a default. If you're talking about bits, that is a natural convention. $\endgroup$
    – Nick Cox
    Feb 20 at 10:24

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