Optimal Subsampling for Multinomial Logistic Models With Big Data

Abstract

This section is dedicated to presenting the explicit forms of πij(β)’s and their derivatives, which are important parts in searching the maximum likelihood estimator and in the theoretical proofs. The categorical probability πij(β) for Models (2.1)-(2.4) can be calculated directly, and the first derivative of πij(β) with respect to β can be gotten through ∂πij(β) ∂β = πij(β) ∂ log πij(β) ∂β , (S1.1)