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Up to now, I believed that k-dimensional normal distribution has probability function: $\frac{1}{\sqrt{(2 \pi)^k |\Sigma|}}e^{-\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}}$

Recently I have read an article http://image.ntua.gr/iva/research/agm/. I've created my own implementation of EM described in that article. However, I was confused about different results. I've found that authors of article have normal distribution probability function defined as: $\frac{1}{\sigma^k}e^{-\frac{||x-\mu||^2}{2 \sigma^2}}$

Yes, one thing is that they do not consider axis sizes / rotations, so their covariance matrix reduces to $\sigma^2 I$. But, what about that $\frac{1}{\sigma^k}$ factor, why there isn't $\frac{1}{\sqrt{(2 \pi)^k \sigma^2}}$? I can't understand to this.

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  • $\begingroup$ In general, if $A$ is an $n\times n$ matrix and $\alpha$ is a scalar, then $|\alpha A|=\alpha^n |A|$. $\endgroup$ – Stefan Hansen Mar 17 '14 at 14:20
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Note that if the covariance matrix is assumed to be $ \sigma^2 I$, the determinant $|\Sigma|$ indeed reduces to $\sigma^k$. As for the $\sqrt{2\pi^k}$ factor, my best guess would be that it's dropped from likelihood maximization procedures to simplify computations. This term is constant w.r.t. distribution parameters and therefore has no effect on likelihood value. The first formula you provided is correct.

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  • $\begingroup$ Thank you very much for explanation $\endgroup$ – Michal Mar 17 '14 at 14:28

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