The problem

The solution

The categorical distribution is analagous to a Bernoulli distribution with k > 2 outcomes:

The Binomial, Bernoulli, and categorical distributions are each special cases of the multinomial distribution.

Binomial distribution

Bin(n, p) = Multinom(n, (1–p, p))

Bernoulli distribution

Bernoulli(p) = Multinom(1, (1–p, p))

Categorical distribution

Cat(p₁, p₂, … p_k) =
Multinom(1, (p₁, p₂, … p_k))

But we still need a link function

Putting it all together gives us the multinomial logistic regression model (a.k.a. categorical regression model)

Note that with only two categories, the multinomial logistic regression reduces to the standard logistic regression:

Interpreting these cofficients on their own is complex — the results are analagous to a series of logistic regressions against the reference category, conditional on the other outcomes.

E.g. α_m and β_m determine the probability of a person being married rather than single, assuming that they are neither divorced nor widowed.

Assessing the sign of the α and β estimates is the only straightforward interpretation.