Multinomial probit
Part of a series on 
Regression analysis 

Models 
Estimation 
Background 

In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confused with the multivariate probit model, which is used to model correlated binary outcomes for more than one independent variable.
General specification
It is assumed that we have a series of observations Y_{i}, for i = 1...n, of the outcomes of multiway choices from a categorical distribution of size m (there are m possible choices). Along with each observation Y_{i} is a set of k observed values x_{1,i}, ..., x_{k,i} of explanatory variables (also known as 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, bodymass index, presence or absence of various symptoms, etc.).
 The observed outcomes are the votes of people for a given party or candidate in a multiway election, and the explanatory variables are the demographic characteristics of each person (e.g. sex, race, age, income, etc.).
The multinomial probit model is a statistical model that can be used to predict the likely outcome of an unobserved multiway 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 categoricallydistributed data, where each outcome value h for observation i occurs with an unobserved probability p_{i,h} that is specific to the observation i at hand because it is determined by the values of the explanatory variables associated with that observation. That is:
 [math]\displaystyle{ Y_ix_{1,i},\ldots,x_{k,i} \ \sim \operatorname{Categorical}(p_{i,1},\ldots,p_{i,m}),\text{ for }i = 1, \dots , n }[/math]
or equivalently
 [math]\displaystyle{ \Pr[Y_i=hx_{1,i},\ldots,x_{k,i}] = p_{i,h},\text{ for }i = 1, \dots , n, }[/math]
for each of m possible values of h.
Latent variable model
Multinomial probit is often written in terms of a latent variable model:
 [math]\displaystyle{ \begin{align} Y_i^{1\ast} &= \boldsymbol\beta_1 \cdot \mathbf{X}_i + \varepsilon_1 \, \\ Y_i^{2\ast} &= \boldsymbol\beta_2 \cdot \mathbf{X}_i + \varepsilon_2 \, \\ \ldots & \ldots \\ Y_i^{m\ast} &= \boldsymbol\beta_m \cdot \mathbf{X}_i + \varepsilon_m \, \\ \end{align} }[/math]
where
 [math]\displaystyle{ \boldsymbol\varepsilon \sim \mathcal{N}(0,\boldsymbol\Sigma) }[/math]
Then
 [math]\displaystyle{ Y_i = \begin{cases} 1 & \text{if }Y_i^{1\ast} \gt Y_i^{2\ast},\ldots,Y_i^{m\ast} \\ 2 & \text{if }Y_i^{2\ast} \gt Y_i^{1\ast},Y_i^{3\ast},\ldots,Y_i^{m\ast} \\ \ldots & \ldots \\ m &\text{otherwise.} \end{cases} }[/math]
That is,
 [math]\displaystyle{ Y_i = \arg\max_{h=1}^m Y_i^{h\ast} }[/math]
Note that this model allows for arbitrary correlation between the error variables, so that it doesn't necessarily respect independence of irrelevant alternatives.
When [math]\displaystyle{ \scriptstyle\boldsymbol\Sigma }[/math] is the identity matrix (such that there is no correlation or heteroscedasticity), the model is called independent probit.
Estimation
For details on how the equations are estimated, see the article Probit model.
References
 Greene, William H. (2012). Econometric Analysis (Seventh ed.). Boston: Pearson Education. pp. 810–811. ISBN 9780273753568.
Original source: https://en.wikipedia.org/wiki/Multinomial probit.
Read more 