= Gaussian posterior in the zero-noise limit
{c}
For $m=Au+\eta$, with independent $u\sim\mathcal N(0,I_d)$ and $\eta\sim\mathcal N(0,\delta^2I_k)$, the <multivariate normal distribution> of the <Bayesian posterior> has <posterior mean> $b_\delta=(A^TA+\delta^2I_d)^{-1}A^Tm$ and <covariance matrix> $C_\delta=\delta^2(A^TA+\delta^2I_d)^{-1}$. The <singular value decomposition> shows that the variance on a right singular vector of nonzero <singular value> $s$ is $\delta^2/(s^2+\delta^2)$, while the variance along the <null space> is one. Thus, for fixed data, the mean tends to the <minimum-norm least-squares solution> and the covariance tends to the <orthogonal projection> onto the <null space>. Unobserved components retain their <prior distribution> even as the noise vanishes.
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