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Commonly used in diffusion models (https://arxiv.org/pdf/2006.11239.pdf), what does the notation, $\mathcal{N}(x_{t};\mu_{\theta},\Sigma_{\theta})$ mean? I get the parameterized by $\mu$ and $\sigma$ part, but I'm unclear exactly on the $x_t$ part.

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It's Gaussian $\mathcal{N}$ probability density function parametrized by $\mu_\theta$ and $\Sigma_\theta$ evaluated at $x_t$ point.

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  • $\begingroup$ thanks ! so in the case where $x$ is an image, what is the corresponding dimensionality of the mu and sigma ? $\endgroup$
    – bdfy
    Commented Oct 18, 2022 at 22:12
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    $\begingroup$ @bdfy I didn't read the paper but if $x_t$ has a length $k$ then the same is $\mu_\theta$ and $\Sigma_\theta$ is $k \times k$. $\endgroup$
    – Tim
    Commented Oct 19, 2022 at 5:18

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