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I am currently reading Gaussian Process for Machine learning chapter 3.4 on Laplacian Approximation for binary Gaussian Process approximation (classification). I am stuck at section 3.4.1 (page 42) when the author applies Newton's method to approximate the posterior $p(f|X,y)$ where $f$ is the nuisance gaussian process with covariance matrix $K$ and $y$ is the class label for design matrix $X$. So far, I do understand how Newton's method is applied to iteratively find the maximum of the concave $\log p(f|X,y)$:

$$f^{new} = f - (\nabla \nabla \Psi)^{-1}(\nabla \Psi)=(K^{-1}+W)^{-1}(\nabla \log p(y|f) - K^{-1}f)\\ =(K^{-1}+W)^{-1}(Wf+\nabla \log p(y|f))$$ where $$\Psi(f) := \log p(y|f) + \log p(f|X)$$ is the log (unnormalized) posterior and $$W := - \nabla \nabla \log p(y|f).$$

But to gain more intuition about the iterative update, the author considered $f$ to contain two subvectors: $f_1$ which corresponds points that are NOT well explained, and $f_2$ that corresponds to points that are well explained (where $\partial \log p(y_i | f_i)/\partial f_i$ and $W_{ii}$ are close to zero). The author states in (3.19) that \begin{equation} f_1^{new} = K_{11}(I_{11}+W_{11} K_{11})^{-1}(W_{11} f_1+\nabla \log p(y_1|f_1)), \end{equation} \begin{equation} f_2^{new} = K_{21}K_{11}^{-1}f_1^{new}. \end{equation}

I do understand the first term but do not understand how $f_2^{new}$ came about. The author wrote that the second term could be obtained using the partitioned matrix inverse equations. Can someone please help me derive $f_2^{new}$ .

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After pondering on the problem, I caught my own mistake of treating $f^{new}_1$ and $f^{new}_2$ independently. I decided to leave my answer here on Cross Validated for anyone who have the same confusion as I did. In this answer I may make a few references to the book GPML, so check it out if anything is unclear.

Firstly, combine $f^{new}_1$ and $f^{new}_2$ into one vector $f$ and rewrite $(K^{-1} + W)^{-1}$ as $K(I+WK)^{-1}$ (which could be proven equivalent through factorization) in the updating formula. The process onward is rather lengthy so I'll screenshot my calculation and explain it afterward. calculation

In the calculation, $O$ represents matrix of all zero. (1) is due to Newton's method updating formula mentioned in the question. (2) is due to the fact that $W_{ii}$ and $\partial \log p(y_i|f_i)/\partial f_i$ are close to zero for well-explained $f_i$s and that $W$ is diagonal as mentioned in the GPML book. And (3) is due to the block matrix inversion formula.

We extract the first column of $f^{new}$ as $f^{new}_1$ and the second as $f^{new}_2$. The result of $f^{new}_2 = K_{21}K_{11}^{-1}f^{new}_1$ immediately follows as $K_{11}^{-1}$ and $K_{11}$ cancels out.

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