On Likelihood Fitting

The Likelihood object is a subclass of Maximize. The error_func() and eval_jacobian() definitions have been changed to facilitate what one would expect from Likelihood fitting:

error_func 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 \(\vec{x}\).