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I am reading PRML, Chapter 3.5.3, screen shot attached. I can understand the derivation and maths but hard to understand the meaning of "The eigenvalues of data co-variance, $\Phi^T\Phi$ matrix measure the curvature of the likelihood function.". Can you please help me understand by providing an example which can give me the intuitive meaning of the following statement. "The eigenvalues of data co-variance matrix, $\Phi^T\Phi$ measure the curvature of the likelihood function." __________________________________________________________PRML

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    $\begingroup$ He doesn't actually say the quote that is being put in quotation marks. He says that the eigenvalues of $\beta \Phi^T \Phi$ measure the curvature of the likelihood function. This is a result of the fact that $\beta \Phi^T \Phi$ is the Hessian of the likelihood function (in this context), and the eigenvalues of the Hessian of a smooth function $f$ tell you something about the "curvature" or shape of the graph of $f$. This can be understood by thinking about the second-order Taylor series approximation of $f$. $\endgroup$ – littleO Mar 14 at 10:50
  • $\begingroup$ Thanks but thats what my question is; $\Phi^T\Phi$ is the measure of data scatter (in data space), then how it playing a role in deciding the likelihood function (in parameter space) $\endgroup$ – Arun Chauhan Mar 14 at 12:42
  • $\begingroup$ I would start by looking at Fig 3.15 and equation 3.87 :) $\endgroup$ – Neal Mar 14 at 13:12
  • $\begingroup$ What I can see, they are some what orthogonal. But is it always true? $\endgroup$ – Arun Chauhan Mar 14 at 23:25
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The marginal likelihood here is $$ p(t|\alpha,\beta) = \int p(t|w,\beta) p(w|\alpha) \text{d}w = c\int \exp(-E(w))\, \text{d}w $$ So the energy or error $E(w)$ basically determines the likelihood, where \begin{align} E(w) &= \frac{\beta}{2}||t - \Phi w||^2 + \frac{\alpha}{2} w^Tw \\ &= E(m_N) + \frac{1}{2}(w-m_N)^TA(w-m_N) \end{align} where $\Phi$ is the design matrix and $$ A = \alpha I + \beta \Phi^T\Phi = \mathcal{H}[E(w)] = \nabla\nabla E(w) $$ is the Hessian of the error. Notice that the eigenvalues $\lambda_A$ of $A$ are real and positive since it is symmetric positive definite. Notice also that the eigenvalues of $A$ are directly related to those ($\lambda$) of $\Phi^T\Phi$ by a constant scale and shift: $$ Av = \lambda_Av\;\implies\; \Phi^T\Phi v = \frac{1}{\beta}(\lambda_A - \alpha)v = \lambda v $$ In other words, $\Phi^T\Phi$ controls the eigenvalues of the Hessian.

Informally speaking, we view the second derivative (in this case, Hessian) as the curvature of the function. In this case, since $A$ is the Hessian of the error $E(w)$, and the error determines the likelihood, we can reasonably say that $\Phi^T\Phi$ determines the curvature of the likelihood through $A$.

But this can be made more precise. Consider the eigendecomposition of $\Phi^T\Phi = U\Lambda U^T$, where $U$ is orthogonal. Working in the orthonormal basis (rotation of weight space) defined by $U=(u_1,\ldots,u_M)$, we can consider how $E(w)$ looks.

Notice that a level curve of the error in weight space forms an ellipse, with the width of the ellipse in each direction (aligned to some $u_i$) controlled by (proportional to) $\lambda_i$. A wide ellipse (small eigenvalue for that axis) means that a large range of the parameter space spanned by that axis $u$ have very little effect on the error. In other words, the error surface is not very curved (along that axis).

Essentially, smaller eigenvalues mean more contour elongation, which means less curvature (smaller Hessian).

He says:

a smaller curvature corresponds to a greater elongation of the contours of the likelihood function

The relation to the covariance of the (embedded) data somewhat makes sense. Recall that the posterior here is written: \begin{align*} p(w|t) &= \mathcal{N}(w|m_N,S_N) \\ m_N &= \beta S_N \Phi^T t \\ S_N^{-1} &= \alpha I + \beta \Phi^T\Phi = A \end{align*} So in this case, the covariance-like quantity $\Phi^T\Phi$ inversely controls the posterior covariance of the weights. As the eigenvalues $\lambda$ get larger, the covariance of the posterior over weights gets smaller. This means that such parameters are tightly constrained to their mean (as straying too far will lead to a huge increase in error).

Why?! Suppose all your data is clustered around one point. The eigenvalues (data covariance) will then be small. But the posterior will be under constrained then! I.e., there are many parameter values that could let us explain that little cluster. We want many points, spread all over the space, in order to nail down a good function, with low posterior covariance.

So, greater (embedded) data covariance means larger eigenvalues, which means (1) smaller posterior covariance and (2) less elongation, which means larger error surface curvature (i.e., larger Hessian eigenvalues).

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  • $\begingroup$ Thank you for such a great insight. I am still trying to understand it thoroughly :) Meanwhile I still have few doubts Q1. Can I always say that Error function, E(w) is -log likelihood function? which makes the error function a mirror image of -log likelihood function (this is what brings the inverse relation between Hessian and Covariane); Or it the case with generalized linear models only. $\endgroup$ – Arun Chauhan Mar 18 at 13:10
  • $\begingroup$ @ArunChauhan Glad to help. At least in ML, it is often fine to call the error a negative log-likelihood. The VAE literature for instance does this. $\endgroup$ – user3658307 Mar 18 at 16:04

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