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$X$ is a random real variable drawn from a distribution $F$ on the reals, $X \sim F$.

In a particular model, the density of $F$, $pdf_F$, is estimated using a collection of points $d$ and a free smoothing parameter $\sigma$, via Kernel density estimation (KDE):

$pdf_F(x) \propto \sum_i K(x, d_i, \sigma)$

where $K(x, y, \sigma)$ is some kernel with bandwidth $\sigma$, for instance $e^{-\frac{\| x-y \|^2}{\sigma^2}}$.

Is there a name for the distribution/process describing $F$? What is the correct notation to describe this? There is definitely a nonparametric Bayesian flavor to this approach (a distribution on distributions, which are indexed by $\sigma$), but it isn't described by a Dirichlet process.

Please forgive my ignorance, I'm teaching myself.

Here is my best guess:

$pdf_F(x) \propto \sum_i K(x, d_i, \sigma)$
$X \sim F$

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