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From this discussion, I know when to use the log, but now I am wondering how to guess the trends of log graph of accumulative data based on linear accumulative data graph?

For example, this table is linear data of the cumulative number of deaths per capita

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And this table is about the log of data of the cumulative number of deaths per capita

enter image description here

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    $\begingroup$ Do you mean you want to know what the bottom graph looks like by only looking at the top graph? The same proportional increase in the confirmed cases at the top is represented by the same vertical distance at the bottom. However, doing this reliably jn your head would be too difficult for my brain. In some cases, like all countries but the US, it's actually impossible because you don't even see what happens to them. The reason is that the US numbers are so much higher that the other countries look like there is no change. However, a change from 1 case to 2 is the same as 1mio to 2 Mio. $\endgroup$
    – AKdemy
    Aug 18, 2021 at 6:46
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    $\begingroup$ The world economic forum offers a good explanation for the reasons you see log scales used in pandemics. $\endgroup$
    – AKdemy
    Aug 18, 2021 at 6:47
  • $\begingroup$ a great source, thank you @AKdemy $\endgroup$
    – Phil Nguyen
    Aug 18, 2021 at 6:53

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The answer will depend on the estimation method.

Classical Pearson and Neyman Frequentist statistics are not invariant to the log transformation. The estimator under the logarithmic transform, when taken as the power of Euler's number, will not match the estimator under the raw data. That is an artifact of the log function itself.

The maximum likelihood estimator will be invariant under the logarithmic transformation but it will generally not be unbiased.

The Bayesian estimator may or may not be invariant under the logarithmic transformation. However, if you did a good, professional job in building the prior distribution, it will not be invariant and taking the log will give you a different answer than if you left it with raw data. Nonetheless, if you were careful in constructing your prior distribution, it is quite possible they will be close enough that you do not care.

Because Bayesian models are generative instead of sampling based, the choice of using the log or not would depend upon your model and not on what would be convenient or mentally helpful. If you believe that nature uses logs, then you use logs. If you believe that nature does not use logs, then you do not use logs.

The use of logarithms to make it mentally easier to understand is different from asking how to project it mathematically. The logarithm compresses data allowing ease of comparison. It can do something other than what you planned it to do in some statistical paradigms.

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