Чи слід вважати, що в статистиці

Я вивчаю статистику і часто натрапляю на формули, що містять, logі я завжди плутаюся, якщо мені слід тлумачити це як стандартне значенняlog , тобто базу 10, або якщо в статистиці цей символ log зазвичай вважається природним журналом ln.

Зокрема, я вивчаю оцінку частоти хорошого Тюрінга як приклад, але моє питання є більш загальним.

It's safe to assume that without explicit base $\log=\ln$ in statistics, because base 10 log is not used very often in statistics. However, other posters bring up a point that $\log_{10}$ or other bases can be common in some other fields, where statistics is applied, e.g. information theory. So, when you read papers in other fields, it gets confusing at times.

Wikipedia's entropy page is a good example of confusing usage of $\log$. In the same page they mean base 2, $e$ and any base. You can figure out by the context which one is meant, but it requires reading the text. This is not a good way to present the material. Compare it to Logarithm page where the base is clearly shown in every formula or $\ln$ is used. I personally think this is the way to go: always show the base when $\log$ sign is used. This would also be ISO compliant for the standard doesn't define usage of unspecified base with $\log$ symbol as @Henry pointed out.

Finally, ISO 31-11 standard prescribes $\text{lb}$ and $\lg$ signs for base 2 and 10 logarithms. Both are rarely used these days. I remember that we used $\lg$ in high school, but that was in another century in another world. I have never seen it since used in a statistical context. There isn't even the tag for $\text{lb}$ in LaTeX.

Це залежить.

Поза кількома контекстами, наприклад перетворенням значення в децибели, логарифми базових 10 є досить рідкісними рівняннями. Однак графіки розміщення журналів часто знаходяться в базі-10, хоча це має бути досить легко перевірити за допомогою міток на осях.

У математичному контексті невдалекий , ймовірно, буде природним журналом (тобто log e або ln ). З іншого боку, інформатика часто використовує логарифми base-2 ( log 2 ), і вони не завжди чітко позначені як такі. Хороша новина полягає в тому, що ви можете конвертувати між базами тривіально, а використання "неправильної" бази дозволить виключити вашу відповідь постійним фактором.$\log$$\log_{e}$$\ln$$\log_2$

У папері "Доброзичливе без сліз" у Гейлі 1995 р. Логарифми в тексті насправді - це (так написано на стор. 5), але код R / S + у додатку використовує функцію, яка є фактично log e або $\log_{10}$log$\log_e$$\ln$. As @Henry points out below, this makes no practical difference.

If I were forced to guess, here are some heuristics:

• If powers of 2, $e$, or 10 are also present, the logs are likely to have the corresponding base.

• If it arises from integrating $1/x$ (or, more generally, involves calculus), it's likely to be a natural log.

• $\log_2$$n$$\log_n$

• $\log_2$$\textrm{bits} \rightarrow \log_2$$\textrm{nats} \rightarrow \ln$$\textrm{bans} \rightarrow \log_{10}$

• $\frac{1}{e} \textrm{ or } 1- \frac{1}{e}$, (37% and 63%, respectively) of an initial value suggests a natural log.

To answer your question: no, you cannot assume a general fixed notation for the logarithm.

A similar question was recently discussed in SE.Math: What is the difference between the three types of logarithms? from a mathematical point of view. Generally, there are different notations that depend on habits ($\log_{10}$ seems of use in medical research) or language (for instance in German, Russian, French). Unfortunately, the same notation sometimes ends up representing different definitions. Quoting from the above SE.Math link:

Notation $\ln x$ (almost) unambiguously denotes the natural logarithm $\log_e x$ (latin: logarithmus naturalis), or logarithm in base $e$. The notation $\log x$ should be the adopted notation for the natural logarithm, and it is so in mathematics. However, it often represents the "most natural" depending on the field: I learned it as the base-$*10$ logarithm ($\log_{10}$) at school, and it is often used this way in engineering (for instance in the definition of decibels)

Quite often, if you are not concerned with the meaning of physical units (like decibels @Matt Krause), nor interested in specific rates of change (in biostatistics, the $\log$-ratio for fold-change often denotes the base-$2$ logarithm $\log_2$), it is likely that the natural logarithm ($\log_e$) is used.

For instance, in power or Box-Cox transforms (for variance stabilization), the natural logarithm appears as a limit when the exponent tends to $0$.

Going back to your initial motivation, the Good-Turing Frequency Estimation, it is interesting to read The Population Frequencies of Species and the Estimation of Population Parameters, I. J. Good, Biometrika, 1953. Here, he used logarithmms in different contexts: variable transformation for variance stabilisation (mentioning Bartlett and Anscombe), sum of harmonic series, entropy. We see that he generally uses $\log$ as the natural logarithm, and once in a while in the paper specifies $\log_e$ or $\log_{10}$, when the context requires it. For variance stabilization, or basic entropy estimation, a factor on the logarithm does not change much the result, as the outcome allows a linear change.

In the Akaike Information Criterion the base is $e$, and $\ln(\hat L)$ of the maximum likelihood $\hat L$ is being compared additively to the number of parameters $k$:

Thus it seems that if you use any other base for the logarithm in the AIC, you may end up drawing the wrong conclusion and selecting the wrong model.