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Say I have a dataset $X \subset{\mathbb R}^d$ (assumed to be iid samples) and I want to estimate the probability density of some unseen point y under that (unknown distribution). One way of doing that is by placing a Gaussian with small variance $\sigma$ at every datapoint and averaging their densities: $$ p(y) = \frac{1}{|X|}\sum_{x \in X} N(y \mid x, \sigma I) $$ Where $N(y|x, \sigma I)$ is the probability density of y under a multivariate Gaussian distribution with mean $y$ and covariance matrix $\sigma I$ (ie. the identity matrix scaled by \sigma).

This seems like such a basic method that there should be a name for it. Is there? I thought it might be convolution, but that's not getting me anywhere.

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You can have a look at Kernel density estimation

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Actually you were on the right track. The KDE you wrote there is actually obtained by convolving a distribution, which you get by placing a Dirac-delta function over each observed datapoint, by a Gaussian kernel :). Something like:

$p_0(x)={1 \over |X|}\sum\limits_{x_i \in X} \delta_{x_i}(x)$

$p(x) = p_0(x) * \mathcal{N}(x| 0,\sigma) $

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