scipy.stats.Normal.

llf#

Normal.llf(sample, /, *, axis=-1)[source]#

Log-likelihood function

Given a sample \(x\), the log-likelihood function (LLF) is the logarithm of the joint probability density of the observed data. It is typically viewed as a function of the parameters \(\theta\) of a statistical distribution:

\[\mathcal{L}(\theta | x) = \log \left( \prod_i f_\theta(x_i) \right) = \sum_{i} \log(f_\theta(x_i))\]

where \(f_\theta\) is the probability density function with parameters \(\theta\).

As a method of ContinuousDistribution, the parameter values are specified during instantiation; llf accepts only the sample \(x\) as sample.

Parameters:

samplearray_like: The given sample for which to calculate the LLF.
axisint or tuple of ints: The axis over which the reducing operation (sum of logarithms) is performed.

Notes

The LLF is often viewed as a function of the parameters with the sample fixed; see the Notes for an example of a function with this signature.

References

[1]

Likelihood function, Wikipedia, https://en.wikipedia.org/wiki/Likelihood_function

Examples

Instantiate a distribution with the desired parameters:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> X = stats.Normal(mu=0., sigma=1.)

Evaluate the LLF with the given sample:

>>> sample = [1., 2., 3.]
>>> X.llf(sample)
-9.756815599614018
>>> np.allclose(X.llf(sample), np.sum(X.logpdf(sample)))
True

To generate a function that accepts only the parameters and holds the data fixed:

>>> def llf(mu, sigma):
...     return stats.Normal(mu=mu, sigma=sigma).llf(sample)
>>> llf(0., 1.)
-9.756815599614018