# Log likelihood

Shortcuts for log likelihood functions

## biogeme.loglikelihood module

Functions to calculate the log likelihood

author:

Michel Bierlaire

date:

Fri Mar 29 17:11:44 2019

biogeme.loglikelihood.likelihoodregression(meas, model, sigma)[source]

Computes likelihood function of a regression model.

Parameters:
• meas (Expression) – An expression providing the value $$y$$ of the measure for the current observation.

• model (Expression) – An expression providing the output $$m$$ of the model for the current observation.

• sigma (biogeme.expressions.Expression) – An expression (typically, a parameter) providing the standard error $$\sigma$$ of the error term.

Return type:

Expression

Returns:

The likelihood of the regression, assuming a normal distribution, that is

$\frac{1}{\sigma} \phi\left( \frac{y-m}{\sigma} \right)$

where $$\phi(\cdot)$$ is the pdf of the normal distribution.

biogeme.loglikelihood.loglikelihood(prob)[source]

Simply computes the log of the probability

Parameters:

prob (Expression) – An expression providing the value of the probability.

Return type:

Expression

Returns:

the logarithm of the probability.

biogeme.loglikelihood.loglikelihoodregression(meas, model, sigma)[source]

Computes log likelihood function of a regression model.

Parameters:
• meas (Expression) – An expression providing the value $$y$$ of the measure for the current observation.

• model (Expression) – An expression providing the output $$m$$ of the model for the current observation.

• sigma (Expression) – An expression (typically, a parameter) providing the standard error $$\sigma$$ of the error term.

Return type:

Expression

Returns:

the likelihood of the regression, assuming a normal distribution, that is

$-\left( \frac{(y-m)^2}{2\sigma^2} \right) - \frac{1}{2}\log(\sigma^2) - \frac{1}{2}\log(2\pi)$
biogeme.loglikelihood.mixedloglikelihood(prob)[source]

Compute a simulated loglikelihood function

Parameters:

prob (Expression) – 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

Return type:

Expression

$P(i|\xi_1,\ldots,\xi_L)$
Return type:

Expression

Returns:

the simulated loglikelihood, given by

$\ln\left(\sum_{r=1}^R P(i|\xi^r_1,\ldots,\xi^r_L) \right)$

where $$R$$ is the number of draws, and $$\xi_j^r$$ is the rth draw of the random variable $$\xi_j$$.

Parameters:

prob (Expression) –