fips.estimators.BayesianSolver

class fips.estimators.BayesianSolver(z, x_0, H, S_0, S_z, c=None, gamma=1.0)

Bayesian inversion estimator class.

This class implements a Bayesian inversion framework for solving inverse problems, also known as the batch method.

The regularization factor (gamma) controls the balance between fitting observations and staying close to the prior. In the cost function, gamma (γ) multiplies the observation term: J(x) = (x - x_0)^T S_0^{-1} (x - x_0) + gamma*(z - Hx)^T S_z^{-1} (z - Hx)

This amounts to scaling S_z by 1/gamma.

• gamma > 1: Increases weight on data fitting (less regularization, fits observations more closely)

• gamma = 1: Standard Bayesian inversion (default)

• gamma < 1: Decreases weight on data fitting (more regularization, stays closer to prior)

Parameters:

• z (np.ndarray) – Observed data

• x_0 (np.ndarray) – Prior model estimate

• H (np.ndarray) – Forward operator

• S_0 (np.ndarray) – Prior error covariance

• S_z (np.ndarray) – Model-data mismatch covariance

• c (np.ndarray | float, optional) – Constant data, defaults to 0.0

• gamma (float, optional) – Regularization factor (γ) that multiplies the observation term in the cost function, by default 1.0

Attributes

S_hat	Posterior Error Covariance Matrix with regularization.
U_red	Uncertainty reduction vector.
desroziers	Desroziers et al. (2005) diagnosed observation error covariance.
reduced_chi2	Reduced Chi-squared statistic.
uncertainty_reduction	Uncertainty reduction metric.
x_hat	Posterior Mean Model Estimate (solution).
y_0	Prior mean data estimate.
y_hat	Posterior mean observation estimate.
gamma	Regularization factor (γ) that multiplies the observation term in the cost function.
z	Observed data.
x_0	Prior model state estimate.
S_z	Model-data mismatch covariance.
c	Constant data, which can be a scalar or an array matching the shape of z.
n_z	Number of observations (length of z).
n_x	Number of state variables (length of x_0).

Methods

cost(x)	Cost function with regularization.
forward(x)	Forward model calculation.
leverage(x)	Calculate the leverage matrix.
residual(x)	Forward model residual.