# Is this a typo/error in Bishop's book

I am currently going through the chapter 9 - Mixture Models and EM from Bishop's book - Pattern Recognition and Machine Learning (2006).

I could not understand the maximization step with respect to the mean-parameters. I am particularly referring to the equation (9.16).

$$0= -\sum_{1}^{n} \gamma(z_{nk} ) \Sigma_k (x_n - \mu_k)$$

Shouldn't that be $\Sigma_k^{-1}$, instead of $\Sigma_k$ ?

Also, a follow-up question in the same line of thought.

When we differentiate the likelihood with respect to $\Sigma$, how does he take care of the ${\lvert \Sigma\rvert} ^ {p/2}$ (where $p$ is the number of dimensions) that is present in the multivariate normal distribution? Looks like, from eq(9.19), he is simply ignoring the determinant term.

• This error, and more, are corrected in the erratum May 20, 2018 at 10:45
• @Nadiels I was surprised by the number of typos present in this book. Several per page!
– a06e
Dec 30, 2022 at 9:05

Indeed, indeed, there is a typo in (9.16) and it should be $$0=\sum_{n=1}^N \gamma(z_{nk}) {\mathbf \Sigma}_k^{-1} ({\mathbf x}_n-{\mathbf \mu}_k)$$Fortunately, this does not impact the next equation (9.17).

As for deriving the conditional MLE of the covariance matrix ${\mathbf \Sigma}_k$, the result and the method are correct. The determinant is accounted for when taking the derivative in ${\mathbf \Sigma}_k$. (If it was ignored, the lhs of (9.19) would be zero.) The steps are the same as in the Normal case, except for the weights $\gamma(z_{nk})$.