Fisher matrix#

This section shows how to construct a Fisher information matrix using derivkit.forecast_kit.ForecastKit.

The Fisher matrix describes the local curvature of a Gaussian likelihood with respect to the model parameters. It is commonly used for forecasting parameter constraints and estimating parameter covariances.

In DerivKit, the Fisher matrix is built from:

a model function mapping parameters theta to a 1D data vector
a fixed observable covariance matrix cov

The primary interface for Fisher forecasting is derivkit.forecast_kit.ForecastKit.fisher().

If you want to generate parameter samples or visualize confidence contours from the Fisher matrix, see the example Fisher contours, which shows how to construct GetDist samples and plot Fisher ellipses.

For a conceptual overview of Fisher forecasting, its interpretation, and other forecasting frameworks in DerivKit see ForecastKit.

Basic usage#

The model must return a 1D array of length n for an input parameter vector theta of length p. The covariance must have shape (n, n).

>>> import numpy as np
>>> from derivkit.forecast_kit import ForecastKit
>>> np.set_printoptions(precision=8, suppress=True)
>>> # Define a simple toy model: R^2 -> R^3
>>> def model(theta):
...     a, b = theta
...     return np.array([a, b, a + 2.0 * b], dtype=float)
>>> # Fiducial parameters and covariance
>>> theta0 = np.array([1.0, 2.0])
>>> cov = np.eye(3)
>>> # Build ForecastKit and compute Fisher matrix
>>> fk = ForecastKit(function=model, theta0=theta0, cov=cov)
>>> fisher = fk.fisher()
>>> print(fisher)
[[2. 2.]
 [2. 5.]]
>>> print(fisher.shape)
(2, 2)

Interpreting the result#

For p parameters, the Fisher matrix has shape (p, p).

The diagonal elements correspond to inverse variances (up to correlations).
The inverse Fisher matrix approximates the parameter covariance near theta0.

>>> import numpy as np
>>> from derivkit.forecast_kit import ForecastKit
>>> # Same toy model as above
>>> def model(theta):
...     a, b = theta
...     return np.array([a, b, a + 2.0 * b], dtype=float)
>>> theta0 = np.array([1.0, 2.0])
>>> cov = np.eye(3)
>>> # Compute Fisher matrix and invert it
>>> fk = ForecastKit(function=model, theta0=theta0, cov=cov)
>>> fisher = fk.fisher()
>>> cov_theta = np.linalg.inv(fisher)
>>> print(cov_theta.shape)
(2, 2)
>>> print(np.all(np.isfinite(cov_theta)))
True

Choosing a derivative backend#

The Fisher matrix is constructed from derivatives of the model with respect to the parameters. You can control how these derivatives are computed by passing method and backend-specific options.

All keyword arguments are forwarded to derivkit.derivative_kit.DerivativeKit.differentiate().

>>> import numpy as np
>>> from derivkit.forecast_kit import ForecastKit
>>> np.set_printoptions(precision=8, suppress=True)
>>> # Define the model
>>> def model(theta):
...     a, b = theta
...     return np.array([a, b, a + 2.0 * b], dtype=float)
>>> theta0 = np.array([1.0, 2.0])
>>> cov = np.eye(3)
>>> # Use a finite-difference derivative backend
>>> fk = ForecastKit(function=model, theta0=theta0, cov=cov)
>>> fisher = fk.fisher(
...     method="finite",
...     n_workers=2,
...     stepsize=1e-2,
...     num_points=5,
...     extrapolation="ridders",
...     levels=4,
... )
>>> print(fisher)
[[2. 2.]
 [2. 5.]]

Parallel execution#

Fisher matrix elements can be computed in parallel using n_workers.

This becomes important in practical forecasting applications with many parameters and costly model evaluations.

>>> import numpy as np
>>> from derivkit.forecast_kit import ForecastKit
>>> # Model definition
>>> def model(theta):
...     return np.array([theta[0], theta[1], theta[0] + 2.0 * theta[1]])
>>> # Enable parallel execution
>>> fk = ForecastKit(function=model, theta0=np.array([1.0, 2.0]), cov=np.eye(3))
>>> fisher = fk.fisher(n_workers=4)
>>> print(fisher.shape)
(2, 2)

Notes#

The Fisher matrix assumes a Gaussian likelihood with fixed covariance.
Derivatives are evaluated at theta0.
The choice of derivative backend can affect numerical accuracy and cost.
For likelihoods with parameter-dependent covariance, see the generalized Fisher matrix utilities.