In PRML by C.M. Bishop, chapter 9, (9.15) is as follows:
$$ \mathcal{N}(\mathbf{x}_n|\mathbf{x}_n, \sigma_j^2\mathbf{I}) = \frac{1}{(2\pi)^{1/2}}\frac{1}{\sigma_j} $$
I've tried to derive this myself:
$$ \mathcal{N}(\mathbf{x}_n|\mathbf{x}_n, \sigma_j^2\mathbf{I}) = \frac{1}{(2\pi)^{d/2}|\sigma_j^2\mathbf{I}|^{1/2}}e^{(\mathbf{x}_n-\mathbf{x}_n)\frac{1}{\sigma_j^2}\mathbf{I}(\mathbf{x}_n - \mathbf{x}_n)^T} $$ $$ = \frac{1}{(2\pi)^{d/2}|\sigma_j^2\mathbf{I}|^{1/2}}e^{0} $$ $$ = \frac{1}{(2\pi)^{d/2}(\prod_{i=1}^{d}\sigma_j^2)^{1/2}} $$ $$ = \frac{1}{(2\pi)^{d/2}(\sigma_j^{2d})^{1/2}} $$ $$ = \frac{1}{(2\pi)^{d/2}\sigma_j^{d}} = \frac{1}{(2\pi)^{d/2}}\frac{1}{\sigma_j^{d}} $$
which gives a clearly different result. What does Bishop mean? It seems to me like the input dimensionality $d$ is not 1, in his equation, otherwise, why have a multivariate distribution? And if it's not d, how can (9.15) be true?
I agree with you.$$\det(2\pi \sigma_j^2 I )^{-\frac12}=(2\pi\sigma_j^2)^{-\frac{d}{2}}$$
Here is an errata of PRML, equation $(9.15)$ is one of them.