Inference

Inference routines for ODE filtering.

ode_filters.inference.sqr_inversion(A: Array, mu: Array, Sigma_sqr: Array, mu_z: Array, Sigma_z_sqr: Array, Q_sqr: Array | None = None) -> tuple[Array, Array, Array]

Numerically stable Bayesian inference using square-root representations.

Performs Bayesian update of a Gaussian given a linear observation model, using Cholesky factors and QR decomposition to maintain numerical stability.

Given p(x) ~ N(mu, Sigma) and p(z|x) ~ N(Ax + b, Q), computes the posterior p(x|z) in square-root form.

Parameters:

Name	Type	Description	Default
A	Array	Observation matrix (shape [n_obs, n_state]).	required
mu	Array	Prior mean (shape [n_state]).	required
Sigma_sqr	Array	Square root of prior covariance (shape [n_state, n_state]).	required
mu_z	Array	Marginal observation mean (shape [n_obs]).	required
Sigma_z_sqr	Array	Square root of marginal observation covariance (shape [n_obs, n_obs]).	required
Q_sqr	Array | None	Square root of measurement noise covariance (optional).	None

Returns:

Type	Description
Array	Tuple of (G, d, Lambda_sqr) where:
Array	G is the Kalman gain matrix (shape [n_state, n_obs])
Array	d is the posterior offset/mean correction (shape [n_state])
tuple[Array, Array, Array]	Lambda_sqr is the posterior covariance square root (shape [n_state, n_state])

ode_filters.inference.sqr_marginalization(A: Array, b: Array, Q_sqr: Array, mu: Array, Sigma_sqr: Array) -> tuple[Array, Array]

Marginalize out the linear transformation in a Gaussian model using square-root form.

Computes the marginal distribution of z = Ax + b given p(x) ~ N(mu, Sigma) and p(z|x) ~ N(Ax + b, Q). The result is p(z) = N(mu_z, Sigma_z) where: - mu_z = A @ mu + b - Sigma_z = A @ Sigma @ A.T + Q

The square-root form is preserved to maintain numerical stability.

Parameters:

Name	Type	Description	Default
A	Array	Linear transformation matrix (shape [n_obs, n_state]).	required
b	Array	Observation offset (shape [n_obs]).	required
Q_sqr	Array	Square root of observation noise covariance. Shape [n_obs, n_obs] or [n_obs]. If 1D array, will be converted to 2D.	required
mu	Array	Prior mean (shape [n_state]).	required
Sigma_sqr	Array	Square root of prior covariance (shape [n_state, n_state]).	required

Returns:

Type	Description
Array	Tuple of (mu_z, Sigma_z_sqr) where:
Array	mu_z is the marginal mean of z (shape [n_obs])
tuple[Array, Array]	Sigma_z_sqr is the square root of marginal covariance of z (shape [n_obs, n_obs])

Raises:

Type	Description
ValueError	If input shapes are incompatible or invalid.