Jacobian matrix for multinomial logit model log-likelihood

Parameters:params (array) – The parameters of the multinomial logit model.
Returns:jac – The derivative of the loglikelihood for each observation evaluated at params .
Return type:array-like


\[\frac{\partial\ln L_{i}}{\partial\beta_{j}}=\left(d_{ij}-\frac{\exp\left(\beta_{j}^{\prime}x_{i}\right)}{\sum_{k=0}^{J}\exp\left(\beta_{k}^{\prime}x_{i}\right)}\right)x_{i}\]

for \(j=1,...,J\), for observations \(i=1,...,n\)

In the multinomial model the score vector is K x (J-1) but is returned as a flattened array. The Jacobian has the observations in rows and the flatteded array of derivatives in columns.