Multinomial probit

Key takeaways

Multinomial probit belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Multinomial probit to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Multinomial probit from memory before moving on to harder problems.

Reference excerpt

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 Yi, 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 Yi is a set of k observed values x1,i, ..., xk,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, 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 is a statistical model that can be used to predict the likely outcome of an 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 Yi are described as being categorically-distributed data, where each outcome value h for observation i occurs with an unobserved probability pi,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:

Y i | x 1 , i , … , x k , i ∼ Categorical ⁡ ( p i , 1 , … , p i , m ) , for i = 1 , … , n {\displaystyle Y_{i}|x_{1,i},\ldots ,x_{k,i}\ \sim \operatorname {Categorical} (p_{i,1},\ldots ,p_{i,m}),{\text{ for }}i=1,\dots ,n}

or equivalently

Pr [ Y i = h | x 1 , i , … , x k , i ] = p i , h , for i = 1 , … , n , {\displaystyle \Pr[Y_{i}=h|x_{1,i},\ldots ,x_{k,i}]=p_{i,h},{\text{ for }}i=1,\dots ,n,}

for each of m possible values of h.

Latent variable model Multinomial probit is often written in terms of a latent variable model:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Multinomial probit

Start with the simplest possible case. Write down what Multinomial probit claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Multinomial probit before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Multinomial probit ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Multinomial probit

In research: Multinomial probit appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Multinomial probit in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Multinomial probit is common in secondary-school and first-year university syllabi. It links to neighbouring topics Regression analysis, Statistical classification, so understanding it makes those chapters shorter.
In everyday life: Look for Multinomial probit outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Multinomial probit in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multinomial probit means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Multinomial probit out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multinomial probit in simple terms?

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.

Why does Multinomial probit matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Multinomial probit?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Multinomial probit.