Log-linear model

A log-linear model is a mathematical model that takes the form of a function whose logarithm is a first-degree polynomial function of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form

\exp \left(c+\sum _{i}w_{i}f_{i}(X)\right)\,,

in which the $f i (X)$ are quantities that are functions of the variables $X$ , in general a vector of values, while $c$ and the $w i$ stand for the model parameters.

The term may specifically be used for:

A log-linear plot or graph, which is a type of semi-log plot.
Poisson regression for contingency tables, a type of generalized linear model.
A multinomial logit model (another type of generalized linear model), especially in the context of machine learning and natural language processing, where it is often termed a "maximum entropy model". In this case (unlike as in Poisson regression) the log-probability by itself is not linear, but has a similar form, with a linear term and a term consisting of the logarithm of a "normalizing constant" (which is constant with respect to the possible outcomes, but not constant with respect to the parameters, and hence must be reckoned with when finding the best parameters for a given set of training data).
Markov random field#Log-linear model, a Markov random field with a log-linear distribution. The same issue applies here as in the multinomial logit model.

See also