# MatrixBasedLinearBeliefUpdate¶

Gaussian belief update in a matrix-based inference framework assuming linear information.

Updates the belief over the quantities of interest of a linear system $$Ax=b$$ given matrix-variate Gaussian beliefs with Kronecker covariance structure and linear observations $$y=As$$. The belief update computes $$p(M \mid y) = \mathcal{N}(M; M_{i+1}, V \otimes W_{i+1})$$, 1 2 such that

\begin{split}\begin{align} M_{i+1} &= M_i + (y - M_i s) (s^\top W_i s)^\dagger s^\top W_i,\\ W_{i+1} &= W_i - W_i s (s^\top W_i s)^\dagger s^\top W_i. \end{align}\end{split}

References

1

Hennig, P., Probabilistic Interpretation of Linear Solvers, SIAM Journal on Optimization, 2015, 25, 234-260

2

Wenger, J. and Hennig, P., Probabilistic Linear Solvers for Machine Learning, Advances in Neural Information Processing Systems (NeurIPS), 2020

Methods Summary

 __call__(solver_state) Update the belief about the quantities of interest of a linear system.

Methods Documentation

__call__(solver_state)[source]

Update the belief about the quantities of interest of a linear system.

Parameters

solver_state (LinearSolverState) – Current state of the linear solver.

Return type

LinearSystemBelief