is a K-vector of probabilities, with the further restriction that all probabilities must add up to 1. Each probability indicates the likelihood of occurrence of one of the K possible values. For the multinomial distribution, and for the vector form of the categorical distribution, the expected values of the elements of the vector can be related to the predicted probabilities similarly to the binomial and Bernoulli distributions. Fitting Maximum likelihood The maximum likelihood estimates can be found using an iteratively reweighted least squares algorithm or a Newton's method with updates of the form: where is the observed information matrix (the negative of the Hessian matrix) and is the score function; or a Fisher's scoring method: where is the Fisher information matrix. Note that if the canonical link function is used, then they are the same. Bayesian methods In general, the posterior distribution cannot be found in closed form and so must be approximated, usually using Laplace approximations or some type of Markov chain Monte Carlo method such as Gibbs sampling. Examples General linear models A possible point of confusion has to do with the distinction between generalized linear models and general linear models, two broad statistical models. Co-originator John Nelder has expressed regret over this terminology. The general linear model may be viewed as a special case of the generalized linear model with identity link and responses normally distributed. As most exact results of interest are obtained only for the general linear model, the general linear model has undergone a somewhat longer historical development. Results for the generalized linear model with non-identity link are asymptotic (tending to work well with large samples). Linear regression A simple, very important example of a generalized linear model (also an example of a general linear model) is linear regression. In linear regression, the use of the least-squares estimator is justified by the Gauss–Markov theorem, which does not assume that the distribution is normal. From the perspective of generalized linear models, however, it is useful to suppose that the distribution function is the normal distribution with constant variance and the link function is the identity, which is the canonical link if the variance is known. Under these assumptions, the least-squares estimator is obtained as the maximum-likelihood parameter estimate. For the normal distribution, the generalized linear model has a closed form expression for the maximum-likelihood estimates, which is convenient. Most other GLMs lack closed form estimates.