leaderbot.models.BradleyTerry.fisher#

BradleyTerry.fisher(w=None, epsilon=1e-08, order=2)#

Observed Fisher information matrix.

Parameters:

warray_like, default=None

Parameters \(\boldsymbol{\theta}\). If None, the pre-trained parameters are used, provided is already trained.

epsilonfloat, default=1e-8

The step size in finite differencing method in estimating derivative.

order{2, 4}, default=2

Order of Finite differencing:

2: Second order central difference.
4: Fourth order central difference.

Returns:

Jnumpy.ndarray: The observed Fisher information matrix of size \(m \times m\) where \(m\) is the number of parameters.

Raises:

RuntimeWarning: If loss is nan.
RuntimeError: If the model is not trained and the input w is set to None.

See also

loss: Log-likelihood (loss) function.

Notes

The observed Fisher information matrix is the negative of the Hessian of the log likelihood function. Namely, if \(\boldsymbol{\theta}\) is the array of all parameters of the size \(m\), then the observed Fisher information is the matrix \(\mathcal{J}\) of size \(m \times m\)

\[\mathcal{J}(\boldsymbol{\theta}) = - \nabla \nabla^{\intercal} \ell(\boldsymbol{\theta}),\]

where \(\ell(\boldsymbol{\theta})\) is the log-likelihood function (see loss()).

Examples

>>> from leaderbot.data import load
>>> from leaderbot.models import Davidson

>>> # Create a model
>>> data = load()
>>> model = Davidson(data)

>>> # Generate an array of parameters
>>> import numpy as np
>>> w = np.random.randn(model.n_param)

>>> # Fisher information for the given input parameters
>>> J = model.fisher(w)

>>> # Fisher information for the trained parameters
>>> model.train()
>>> J = model.fisher()