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Suppose $f$ and $g$ are two probability density functions. I have seen economists use $\int f(x)g(x) dx$ as some kind of similarity measure. For example, Jaffe (1986) uses sum of product of two proportions of budget in each area as a measure of similarity of two firms. https://www.nber.org/system/files/working_papers/w1815/w1815.pdf.

Specifically, $F_{i}\in\mathbb{R}^d$ is defined as the proportion of budget that firm $i$ devotes to in $d$ areas so $\sum_j F_{ij}=1$ and $0 \leq F_{ij} \leq 1$ for all $j = 1, 2, \ldots, d$. The similarity measure between $F_i$ and $F_j$ is defined as $P_{ij} = \frac{F_i^\top F_j}{\|F_i\|_2\|F_j\|_2}$.

If we think of $F_i$ as the probability mass function of a multinomial distribution or more generally the probability density function of a distribution, what is $P_{ij}$ measuring? It has the form of (uncentered) correlation of two p.m.f.'s/p.d.f.'s but is there any justification? Any statistician has used it as some sort of distance/angle between two measures? What is the relationship with correlation of the random variables that are endowed with these two p.d.f.s?

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    $\begingroup$ The integral $\int f(x)g(x)\mathrm{d}x$ doesn't make a whole lot of sense generally because it changes with the unit of measure of $x.$ The integral $\int\sqrt{f(x)g(x)}\mathrm{d}x$ would have an invariant meaning and could be interpretable as a cosine dissimlarity (between the $L^2$ functions $\sqrt{f}$ and $\sqrt{g}$). But the stuff you write after "specifically" seems to have little to do with this. Are you trying to ask about the general formulation you began with or about the discrete distributions you wind up discussing? $\endgroup$
    – whuber
    Nov 4, 2020 at 21:07
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    $\begingroup$ See stats.stackexchange.com/questions/296361/…. $\endgroup$ Nov 4, 2020 at 23:17
  • $\begingroup$ @whuber I think the second paragraph is a discrete version of the general question where $f(x)$ and $g(x)$ are probability mass functions of multinomial distributions. But i think the Bhattacharyya distance kjetil b halvorsen mentioned with his post is what I am looking for. So thanks both! $\endgroup$
    – cccfran
    Nov 5, 2020 at 2:40

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