First, let me quickly remind you of the two operations: convolution and cross-correlation between 2 function $f$ and $g$, assuming continuous domain.

Cross-Correlation $f \star g$ : $\int f(\tau)g(\tau-t)d\tau$

Convolution: $f \otimes g$: $\int f(\tau)g(t-\tau)d\tau$

The difference being in convolution, besides shifting one function, say $g$, like cross-correlation, you reverse it first. I have read works which take convolution of 2 distributions. My question is, why would you take convolution when you can take the cross-correlation? It looks to me as though cross-correlation of probability functions has very obvious interpretative sense while convolution after the reverse of one function, there is no particular interpretive meaning.

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    $\begingroup$ What is the context of this question? What is the very obvious interpretation that you talk about? $\endgroup$ – Sextus Empiricus Mar 26 '19 at 14:37
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    $\begingroup$ Note the explanatory text in the convolution tag: Convolution is a function-valued operation on two functions $f$ and $g$: $ \int_{-\infty}^{\infty} f(\tau)g/(t-\tau) d \tau$. Often used for obtaining the density of a sum of independent random variables. This tag should also be used for the inverse operation of deconvolution. DO NOT use this tag for convolutional neural networks Also see this question: stats.stackexchange.com/questions/331973/… $\endgroup$ – Sextus Empiricus Mar 26 '19 at 15:02

A typical example where you use convolution of density functions is when finding the density of sum of independent RVs, e.g. $Z=X+Y$, $$f_Z(z)=\int f_X(x)*f_Y(z-x)$$ Intuitively, it's like multiplying $P(X=x)$ and $P(Y=z-x)$ (since $x+(z-x)=z$) . and summing all such situations to obtain $P(Z=z)$. Of course, there is more than this behind the scenes, but it's a very intuitional way of grasping the idea. Cross-correlation doesn't make this much sense here (i.e. density of sum of independent RVs for example).

A Note: In some libraries (e.g. tensorflow), convolution is often calculated without reversing the signal with good reason.


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