# Error in Equation 6.91 in Pattern Recognition and Machine Learning

In Pattern Recognition and Machine Learning Chapter 6.4.6 the author applies the Laplace approximation to Gaussian process regression and in doing so has to maximise the log likelihood $$ln p(t_N | \theta)$$ in order to obtain the parameters $$\theta$$ of the covariance function.

I am trying to derive expression 6.91 for the derivative of the terms involving $$C_N$$ by following along with the tutors solution manual for the book. Following the solution manual if you gather the terms that involve $$C_N$$ you get:

$$-\frac{1}{2}a_N^{*T}C_N^{-1}a_N^*-\frac{1}{2}ln |C_N W_N +I|$$

as given in the solution manual. The author then states to apply C.21 and C.22 to the above to get the derivative wrt $$\theta_j$$:

$$\frac{\partial}{\partial x}A^{-1}=-A^{-1}\frac{\partial A}{\partial x}A^{-1}$$

and C.22:

$$\frac{\partial }{\partial x} ln |A| = Tr(A^{-1} \frac{\partial A}{\partial x})$$

If you apply these you get:

$$\frac{1}{2}a_N^{*T}C_N^{-1}\frac{\partial C_N}{\partial \theta_j}C_N^{-1}a_N^*-\frac{1}{2}Tr((C_N W_N + I)^{-1}\frac{\partial C_N}{\partial \theta_j}W_N)$$

Since $$W_N$$ is constant wrt $$\theta_j$$. Whereas the formula in the book has the following for the trace term:

$$-\frac{1}{2}Tr((C_N W_N + I)^{-1}W_N\frac{\partial C_N}{\partial \theta_j})$$

i.e. the order of the last two matrices is swapped.

I'm fairly sure this is a typo since $$Tr(ABC) \neq Tr(ACB)$$

I couldn't find this listed in any errata online. I found an unofficial solutions manual which says it thinks 6.91 is incorrect but it wasn't stated what they thought the error was so I wanted to check that my reasoning is sound and that this is indeed an error.

Note $$C_N$$ is symmetric.

EDIT: There's a similar typo listed in this unofficial errata which is one of the top search results on Google for 'PRML errata', but it concerns equation 6.93 and there is nothing listed in there or the official errata about 6.91.

The derivative for $$\ln |A|$$ is notation agnostic and is correct. But, C.22 is in numerator notation, and the rest should be in the same notation for consistency. Therefore, following numerator notation, your formula is the correct one.