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In searching for an answer I came across this 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.

pdf annotated with 3 r values

(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.)

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    $\begingroup$ In some contects, a ratio of pdf's is a likelihood ratio. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Feb 14, 2017 at 11:15
  • $\begingroup$ @kjetilbhalvorsen Yes. I have seen likelihoods for parameters of a distribution, so it seems this is a different context. Are you saying this ratio could be defined as "the likelihood ratio for the observation relative to the mode" ? $\endgroup$
    – PM.
    Commented Feb 14, 2017 at 11:50
  • $\begingroup$ You just rescaled the probability density of $f$ by the magnitude of the mode. So the resulting function $r(x)$ is still a density for the random variable in question, albeit not a probability density. This could of course be alleviated by multiplying $r(x)$ by the magnitude of the mode. So you are actually just using a rescaled version of the probability density. $\endgroup$
    – Jeremias K
    Commented Feb 14, 2017 at 11:52
  • $\begingroup$ @JeremiasK Indeed and that rescaling adds some interpretability I think. If I say $r=1$ I know this is a likely observation but if I say $f=1$ then I can't conclude anything? $\endgroup$
    – PM.
    Commented Feb 14, 2017 at 13:43
  • $\begingroup$ any more on this? and whether the pdf/cdf ratio is actually adopted in any applications $\endgroup$
    – develarist
    Commented Jul 19, 2020 at 12:25

1 Answer 1

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Assuming the density function $f$ is continuous, if we look at the density at two points $x_1, x_2$, and the ratio $f(x_1)/f(x_2)=\lambda$, say. We can then infer that the probability of observations in a short interval around $x_1$ is $\lambda$ times the probability of observations in a short interval (same length) around $x_2$.

So the density ratio can be interpreted, in this way, as an (approximate) relative probability. This is also called a relative distribution. For some examples see https://stats.stackexchange.com/a/274058/11887

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