""" Functions to calculate the log likelihood
:author: Michel Bierlaire
:date: Fri Mar 29 17:11:44 2019
"""
# Too constraining
# pylint: disable=invalid-name,
from biogeme.expressions import Expression, exp, log, MonteCarlo
[docs]
def loglikelihood(prob: Expression) -> Expression:
"""
Simply computes the log of the probability
:param prob: An expression providing the value of the probability.
:return: the logarithm of the probability.
"""
return log(prob)
[docs]
def mixedloglikelihood(prob: Expression) -> Expression:
"""Compute a simulated loglikelihood function
:param prob: An expression providing the value of the
probability. Although it is not formally necessary,
the expression should contain one or more random
variables of a given distribution, and therefore
is defined as
.. math:: P(i|\\xi_1,\\ldots,\\xi_L)
:return: the simulated loglikelihood, given by
.. math:: \\ln\\left(\\sum_{r=1}^R
P(i|\\xi^r_1,\\ldots,\\xi^r_L) \\right)
where :math:`R` is the number of draws, and :math:`\\xi_j^r`
is the rth draw of the random variable :math:`\\xi_j`.
"""
ell = MonteCarlo(prob)
return log(ell)
[docs]
def likelihoodregression(
meas: Expression, model: Expression, sigma: Expression
) -> Expression:
"""Computes likelihood function of a regression model.
:param meas: An expression providing the value :math:`y` of the measure
for the current observation.
:param model: An expression providing the output :math:`m` of the model
for the current observation.
:param sigma: An expression (typically, a parameter) providing the
standard error :math:`\\sigma` of the error term.
:type sigma: biogeme.expressions.Expression
:return: The likelihood of the regression, assuming a normal distribution,
that is
.. math:: \\frac{1}{\\sigma} \\phi\\left( \\frac{y-m}{\\sigma} \\right)
where :math:`\\phi(\\cdot)` is the pdf of the normal distribution.
"""
return exp(loglikelihoodregression(meas, model, sigma))
[docs]
def loglikelihoodregression(
meas: Expression, model: Expression, sigma: Expression
) -> Expression:
"""Computes log likelihood function of a regression model.
:param meas: An expression providing the value :math:`y` of the
measure for the current observation.
:param model: An expression providing the output :math:`m` of the
model for the current observation.
:param sigma: An expression (typically, a parameter) providing
the standard error :math:`\\sigma` of the error term.
:return: the likelihood of the regression, assuming a normal distribution,
that is
.. math:: -\\left( \\frac{(y-m)^2}{2\\sigma^2} \\right) -
\\frac{1}{2}\\log(\\sigma^2) - \\frac{1}{2}\\log(2\\pi)
"""
t = (meas - model) / sigma
f = -(t**2) / 2 - log(sigma**2) / 2 - 0.9189385332
return f