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Here we analyze the performance of the mean field method. Specifically, we are interested in when the mean field method gives the exact solution, and when it yields a terrible estimate.

In mean field method, we approximate the posterior probability \(p(\beta|\textbf y,\textbf X)\) using family of functions \(q(\beta)\) s.t. \[ q(\beta)=\prod_{j=1}^pq(\beta_j). \]

In ridge regression, where we assume the prior

\[ \beta\sim N(0,\sigma^2\sigma_b^2I), \]

and the likelihood

\[ L(\beta;\textbf X,\textbf y)\propto N(\textbf y-\textbf X\beta;0,\sigma^2I), \]

where \(N(u;\mu,\Sigma)\) denotes the density at point \(U=u\) for random variable \(U\sim N(\mu,\Sigma)\).

Then we can solve the posterior probability \(p(\beta|\textbf y,\textbf X)\) analytically as follows

\[ \begin{aligned} p(\beta|y,X)\propto &N(\beta;0,\sigma^2\sigma_b^2I)\cdot N(\textbf y-\textbf X\beta;0,\sigma^2I)\\ \propto&\exp\left\{-\frac1{2\sigma^2\sigma^2_b}\|\beta\|^2_2\right\}\cdot \exp\left\{-\frac1{2\sigma^2}\|\textbf y-\textbf X\beta\|^2_2\right\}\\ \propto &\exp\left\{-\frac1{2\sigma^2}[\beta^T(\textbf X^T\textbf X+I/\sigma^2_b)\beta-2\textbf y^T\textbf X\beta]\right\}\\ \propto&\exp\left\{-\frac1{2\sigma^2}(\beta-\hat\beta_{ridge})^T\left(\textbf X^T\textbf X+I/\sigma_b^2\right)(\beta-\hat\beta_{ridge})\right\}, \end{aligned} \]

where \(\hat\beta_{ridge}=(\textbf X^T\textbf X+I/\sigma^2_\beta)^{-1}\textbf X^T \textbf y\).

Thus \[ \beta|\textbf y,\textbf X\sim N\left(\hat\beta_{ridge},\sigma^2\left(\textbf X^T\textbf X+I/\sigma_b^2\right)^{-1}\right). \]

If the mean field approximation is exact, i.e. this Gaussian distribution is in the mean field function family, then the covariance matrix \(\sigma^2\left(\textbf X^T\textbf X+I/\sigma_b^2\right)^{-1}\) should be diagonal. It is equivalent to \((\textbf X^T\textbf X)\) is diagonal, i.e. the columns of \(\textbf X\) are orthogonal.

In the opposite, we expect the performance of the mean field method to be really bad if the directions of columns of \(\textbf X\) are very close to each other. We demonstrate this with some simulation. See the last part of simulation result.