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What do we mean by the notation $\mathbf{x}_{p} \in \mathbb{R}^{N \times\left(P^{2} \cdot C\right)}$?

This itself isn't really an expression but a description of what $x_p$ looks like. Specifically, $x_p$ is a real-valued vector with the shape [N, P^2 * C]. Of ...
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How does Bishop derive $\ln p\left(\mathbf{x} \mid \mu, \sigma^{2}\right)$, when $p$ is a Gaussian?

This is not so difficult (just a bit verbose if you do all steps). Just replace $\mathcal{N}\left(x_{n} \mid \mu, \sigma^{2}\right)$ with $\frac{1}{\left(2 \pi \sigma^{2}\right)^{1 / 2}} \exp \left\{-\...
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