DALI tensors#

This section shows how to construct the DALI (Derivative Approximation for LIkelihoods) tensors using derivkit.forecast_kit.ForecastKit.

The DALI expansion extends the Fisher matrix by including higher-order derivative information, allowing non-Gaussian structure in the likelihood to be approximated locally around a fiducial parameter point theta0.

In DerivKit, the doublet-DALI expansion returns two tensors, commonly denoted G and H.

The model must map parameters theta to a 1D data vector, and the observable covariance matrix must have shape (n, n), where n is the length of the data vector.

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

If you want to visualize parameter contours from the DALI tensors, see the example DALI contours, which shows how to generate GetDist samples and plot confidence regions from (G, H).

DALI tensor shapes#

For p model parameters:

the third-order tensor G has shape (p, p, p)
the fourth-order tensor H has shape (p, p, p, p)

These tensors are evaluated at the fiducial parameter point theta0.

Basic usage#

The example below constructs the DALI tensors around a fiducial parameter point theta0 using a simple toy model.

>>> 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 DALI tensors
>>> fk = ForecastKit(function=model, theta0=theta0, cov=cov)
>>> g_tensor, h_tensor = fk.dali()
>>> print(g_tensor.shape)
(2, 2, 2)
>>> print(h_tensor.shape)
(2, 2, 2, 2)

Choosing a derivative backend#

As with Fisher forecasting, you can control how 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)
>>> # Fiducial setup
>>> theta0 = np.array([1.0, 2.0])
>>> cov = np.eye(3)
>>> fk = ForecastKit(function=model, theta0=theta0, cov=cov)
>>> # Use a finite-difference derivative backend
>>> g_tensor, h_tensor = fk.dali(
...     method="finite",
...     n_workers=2,
...     stepsize=1e-2,
...     num_points=5,
...     extrapolation="ridders",
...     levels=4,
... )
>>> print(g_tensor.shape)
(2, 2, 2)
>>> print(h_tensor.shape)
(2, 2, 2, 2)

Parallel execution#

DALI tensor components can be computed in parallel using n_workers.

This parallelizes derivative evaluations across parameters and tensor entries.

>>> import numpy as np
>>> from derivkit.forecast_kit import ForecastKit
>>> # Define the model
>>> 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))
>>> g_tensor, h_tensor = fk.dali(n_workers=4)
>>> print(g_tensor.shape)
(2, 2, 2)
>>> print(h_tensor.shape)
(2, 2, 2, 2)

Notes#

DALI tensors are evaluated locally at theta0.
The expansion captures non-Gaussian structure in parameter space through higher-order derivatives of the model.
In the current implementation, the likelihood is assumed to be Gaussian in the data with fixed covariance.
Higher-order derivatives increase computational cost relative to Fisher.
Changing the derivative backend affects numerical accuracy but not the structure of the DALI expansion.