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I'm reading PRML and I have a question,don't solve by myself.
So, please anyone help me.

PRML


Q:Why following third time is not multiplied with $E(\boldsymbol{\hat{w}})$

$\cfrac{1}{2} (\boldsymbol{w} - \boldsymbol{\hat{w}} )^\mathrm{T} E(\boldsymbol{\hat{w}}) \boldsymbol{H}(\boldsymbol{w} - \boldsymbol{\hat{w}})$

$E(\boldsymbol{w}) \simeq E(\boldsymbol{\hat{w}}) + ( \boldsymbol{w} - \boldsymbol{\hat{w}} )^\mathrm{T} \boldsymbol{b} + \cfrac{1}{2} (\boldsymbol{w} - \boldsymbol{\hat{w}} )^\mathrm{T} \boldsymbol{H}(\boldsymbol{w} - \boldsymbol{\hat{w}}) \cdots(5.28) $

$ \boldsymbol{b} \equiv \nabla E \,|_{w=\hat{w}} \, \cdots(5.29)$



Taylor series

$f(\boldsymbol{x}) \approx f(\boldsymbol{a}) + Df(\boldsymbol{a}) (\boldsymbol{x}-\boldsymbol{a}) + \frac{1}{2} (\boldsymbol{x}-\boldsymbol{a})^T Hf(\boldsymbol{a}) (\boldsymbol{x}-\boldsymbol{a}).$

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  • $\begingroup$ A) For all we know $E(\hat w)$ might be zero, rendering that term useless. B) Why should it be multiplied? The only effect would be a rescaling of the matrix $H$. $\endgroup$ Dec 22, 2018 at 8:00
  • $\begingroup$ Are you familiar with the usual (=single variable) Taylor series? With several variable Taylor series the vector $\mathbf{b}$ consists of the first order partial derivatives, the matrix $H$ contains the second order partial derivatives etc. $\endgroup$ Dec 22, 2018 at 8:01
  • $\begingroup$ Thank you for your comment. $\endgroup$
    – Eiji
    Dec 22, 2018 at 8:45
  • $\begingroup$ That $Hf(a)$ is NOT a matrix $H$ multiplied by $f(a)$. It is the Hessian of the function $f$ evaluated at the point $a$. It is probably better to typeset it differently. Like $H(f)(a)$ or $H_f(a)$. Possibly $(Hf)(a)$. $\endgroup$ Dec 22, 2018 at 8:48
  • $\begingroup$ Anyway, it looks like the author of the text abbreviated $H(E)$ to just $H$. $\endgroup$ Dec 22, 2018 at 8:51

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