Frequentist vs Bayesian Linear Gaussian Models

Consider the following linear Gaussian system: where $p(x)$ is our prior. The Bayesian inference problem can be expressed in closed-form as1: Where can I find an equivalent "frequentist" estimation procedure? Can they also be expressed in closed form?

1 "Machine Learning - A Probabilistic Perspective", 2012, pp. 119. Kevin Murphy.

This problem is regression with a known covariance matrix where $$Y-b \sim N(Ax,\Sigma_y).$$
The only unknown here is $x$, so the maximum likelihood estimator is $$\hat{x}_{MLE} = \text{argmax}_{x} p(y|x) = [A^T\Sigma_y^{-1}A]^{-1}A^T \Sigma_y^{-1}(y-b)$$ which is the posterior mode in the limit as $\Sigma_x^{-1}\to 0$, i.e. the improper uniform distribution.
The sampling distribution for $\hat{x}_{MLE}$ is normal with mean $$[A^T\Sigma_y^{-1}A]^{-1}A^T \Sigma_y^{-1}A x$$ and covariance matrix $$A[A^T\Sigma_y^{-1}A]^{-1}[A^T\Sigma_y^{-1}A]^{-1}A^T.$$ The sampling distribution can be used to construct confidence, i.e. frequentist, intervals.