# When to use the raw dataset as opposed to a transformed dataset for computing divergence?

Let us assume I have a set of observations:

# Dataset 1: Raw

$A = [a_1,a_2,a_3,a_4,...]$

$B = [b_1,b_2,b_3,b_4,...]$

One assumptions before I proceed: Range of the values that the random variable can take is $[0,N]$ and $N$ can be $>>$ any of the values present in the datasets. Therefore, $A$ can contain a max value $\approx N$ but $B$ can contain a max values $<< N$. For example,

$A = [1, 5, 100, 1000]$

$B = [100, 200, 10K, 10M]$

Notice how the max value is different in both the set of observations. I want to test the hypothesis that both the datasets have similar distributions.

To compare the distribution of these two datasets, I can use something like KL-Divergence or JS-Divergence which take input the pdfs of the datasets.

Now, I can also create another set of data:

# Dataset 2: Difference

$A^{diff} = [a_2 - a_1, a_3 - a_2, a_4 - a_3, ...]$

$B^{diff} = [b_2 - b_1, b_3 - b_2, b_4 - b_3, ...]$

# Dataset 3: Normalized

$A^{norm} = [a_1/N,a_2/N,a_3/N,a_4/N,...]$

$B^{norm} = [b_1/N,b_2/N,b_3/N,b_4/N,...]$

# Dataset 4: Difference-Normalized

$A^{diff-norm} = [(a_2 - a_1)/N, (a_3 - a_2)/N, (a_4 - a_3, ...]$

$B^{diff-norm} = [(b_2 - b_1)/N, (b_3 - b_2)/N, (b_4 - b_3)/N, ...]$

And I think after looking at this question, there are two other datasets:

# Dataset 6: Difference of Logs perhaps?

Is there an argument for which set of data to pick for computing the divergence? That is, what exactly are the divergences computed in each case revealing?

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