posterior()#
- L2DecoupledSolver.posterior()[source]#
Solve the Bayesian operator inference regression, constructing the means and inverse covariances of probability distributions for the rows of an operator matrix posterior.
In this method, the \(i\)-th entry of
regularizer, denoted \(\lambda_i\), is interpreted as a prior variance for the \(i\)-th row of the operator matrix, which follows a multivariate normal distribution with the following mean and covariance.\[\begin{split}\bfmu_i &= \argmin_{\bfxi}\left\{ \|\D\bfxi - \z_i\|_2^2 + \lambda_i^2\|\bfxi\|_2^2 \right\}, \\ \bfSigma_i &= \sigma_i^2 \left( \D\trp\D + \lambda_i^2\I \right)^{-1}, \\ \sigma_i^2 &= \frac{1}{k}\left( \|\D\bfmu_i - \z_i\|_2^2 + \lambda_i^2\|\bfmu_i\|_2^2 \right),\end{split}\]where \(\z_i\in\RR^k\) is the \(i\)-th row of \(\Z\). See [GMW22] for details.
- Returns:
- meanslist of r (d,) ndarrays
Mean vectors.
- precisionslist of r (d, d) ndarrays
Inverse covariance matrices.
- Raises:
- RuntimeError
If any solver residual \(\sigma_i^2\) is zero (a perfect regression), meaning the resulting covariance should be zero. In this case, do a deterministic regression with
solve(), not a Bayesian regression.