Multinomial Probit - General Specification

General Specification

It is assumed that we have a series of observations Y_i, for i=1...n, of the outcomes of multi-way choices from a categorical distribution of size m (there are m possible choices). Along with each observation Y_i is a set of observed explanatory variables x_1,i, ..., x_k,i (aka independent variables, predictor variables, features, etc.). Some examples:

• The observed outcomes might be "has disease A, has disease B, has disease C, has none of the diseases" for a set of rare diseases with similar symptoms, and the explanatory variables might be characteristics of the patients thought to be pertinent (sex, race, age, blood pressure, body-mass index, presence or absence of various symptoms, etc.).
• The observed outcomes are the votes of people for a given party or candidate in a multi-way election, and the explanatory variables are the demographic characteristics of each person (e.g. sex, race, age, income, etc.).

The multinomial probit model seeks to construct a statistical model that can be used to predict the likely outcome of another unobserved multi-way trial given the associated explanatory variables. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes.

Formally, the outcomes Y_i are described as being categorically-distributed data, where each outcome h is determined by an unobserved probability p_i,h that is specific to the outcome at hand, but related to the explanatory variables. That is:

or equivalently

where denotes the expected value of Y_i.