# On Likelihood Fitting¶

The LogLikelihood objective function should be used to perform log-likelihood maximization. The __call__() and eval_jacobian() definitions have been changed to facilitate what one would expect from Likelihood fitting:

__call__ gives the value of log-likelihood at the given values of $$\vec{p}$$ and $$\vec{x}_i$$, where $$\vec{p}$$ is a shorthand notation for all parameter, and $$\vec{x}_i$$ the same shorthand for all independent variables.

$\log{L(\vec{p}|\vec{x}_i)} = \sum_{i=1}^{N} \log{f(\vec{p}|\vec{x}_i)}$

eval_jacobian() gives the derivative with respect to every parameter of the log-likelihood:

$\nabla_{\vec{p}} \log{L(\vec{p}|\vec{x}_i)} = \sum_{i=1}^{N} \frac{1}{f(\vec{p}|\vec{x}_i)} \nabla_{\vec{p}} f(\vec{p}|\vec{x}_i)$

Where $$\nabla_{\vec{p}}$$ is the derivative with respect to all parameters $$\vec{p}$$. The function therefore returns a vector of length len(p) containing the Jacobian evaluated at the given values of $$\vec{p}$$ and $$\vec{x}$$.