# SolutionBasedProjectedRHSBeliefUpdate¶

Gaussian belief update in a solution-based inference framework assuming projected right-hand-side information.

Updates the belief over the quantities of interest of a linear system $$Ax=b$$ given a Gaussian belief over the solution $$x$$ and information of the form $$y = s\^top b=s^\top Ax$$. The belief update computes the posterior belief about the solution, given by $$p(x \mid y) = \mathcal{N}(x; x_{i+1}, \Sigma_{i+1})$$, 1 such that

\begin{split}\begin{align} x_{i+1} &= x_i + \Sigma_i A^\top s (s^\top A \Sigma_i A^\top s + \lambda)^\dagger s^\top (b - Ax_i),\\ \Sigma_{i+1} &= \Sigma_i - \Sigma_i A^\top s (s^\top A \Sigma_i A s + \lambda)^\dagger s^\top A \Sigma_i, \end{align}\end{split}

where $$\lambda$$ is the noise variance.

Parameters

noise_var (Union[float, Real, floating]) – Variance of the scalar observation noise.

References

1

Cockayne, J. et al., A Bayesian Conjugate Gradient Method, Bayesian Analysis, 2019, 14, 937-1012

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