statsmodels.discrete.discrete_model.MNLogit.hessian¶

MNLogit.hessian(params)[source]¶

Multinomial logit Hessian matrix of the log-likelihood

Parameters:¶

paramsarray_like

The parameters of the model

Returns:¶

hessndarray, (J*K, J*K)

The Hessian, second derivative of loglikelihood function with respect to the flattened parameters, evaluated at params

Notes

$$\frac{{\partial }^2\ln L}{\partial {\beta }_j\partial {\beta }_l}=-\sum _{i=1}^n\frac{\exp \left({\beta }_j^{\prime }{x}_i\right)}{\sum _{k=0}^J\exp \left({\beta }_k^{\prime }{x}_i\right)}[\mathbf{1}(j=l)-\frac{\exp \left({\beta }_l^{\prime }{x}_i\right)}{\sum _{k=0}^J\exp \left({\beta }_k^{\prime }{x}_i\right)}]x_i{x}_l^{\prime }$$

where $\mathbf{1}(j=l)$ equals 1 if j = l and 0 otherwise.

The actual Hessian matrix has J**2 * K x K elements. Our Hessian is reshaped to be square (J*K, J*K) so that the solvers can use it.

This implementation does not take advantage of the symmetry of the Hessian and could probably be refactored for speed.