I'm referencing this Wikipedia article.

I understand that a Taylor expansion of a function $f(x)$ around $x = a$ can be given by

$f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$

According to the linked section of the article, we can approximate the model function as $f(x_i, \beta)$ for some data point $x_i$ and a vector of weights $\beta$.

So, looking a the Taylor expansion above, and evaluating up to $n = 1$, I get this...

$f(x_i, \beta) = f^k(x_i, \beta) + \frac{\partial{f(x_i, \beta)}}{\partial \beta_j}(\beta _j - \beta _j^k)$

as the first order Taylor expansion of $f$ evaluated around $\beta _j$.

I'm confused about why we can just sum over all the $j$ when expressing the Taylor expansion wrt to the vector $\beta ^k$.

Thanks for any assistance.

  • $\begingroup$ The Jacobian of a function is a matrix, and the derivative approximation term can be considered as a dot product of two vectors for fixed $x_i$. Think about the case of a single variable function and extend it to multi-variables. $\endgroup$ – Tab1e May 24 at 0:53
  • $\begingroup$ @Tab1e sorry I don't quite see the dot product that you mentioned $\endgroup$ – Carpetfizz May 24 at 22:05

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