In searching for an answer I came across thisthis about the pdf/cdf ratio but I would like to know if there is any meaning, name or supporting theory relating to the ratio of two pdf values, the numerator and denominator evaluated with the same pdf e.g. $f(x_1)/f(x_2)$ where $f(x)$ is the pdf.
I am specifically interested in using and interpreting ratios where the denominator pdf is evaluated at the highest valued mode so the ratio, $r$, is always $ \in [0,1]$ i.e. $$r_i=\frac{f(x_i)}{f(x_{mode})}$$
The plot below shows a skewed distribution. The further the observation is from the mode, the lower the $r$ value due to the pdf being lower in value relative to the value of the pdf at the mode. I interpret this as indicating that such observations are more rare than observations closer to the mode. The ratio provides a quantitative measure of this. I believe this measure also functions in a similar way for multi-modal distributions.
(For context I am investigating if this is a useful measure of how unusual or rare an observation is relative to a mode in a cheminformatics setting, though it would not necessarily be restricted to this setting.)
