3
$\begingroup$

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.

$\endgroup$

2 Answers 2

6
$\begingroup$

You can have a look at Kernel density estimation

$\endgroup$
0
1
$\begingroup$

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) $

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.