or Bayesian techniques. Model components The GLM consists of three elements: 1. A particular distribution for modeling from among those which are considered exponential families of probability distributions, 2. A linear predictor , and 3. A link function such that . Probability distribution An overdispersed exponential family of distributions is a generalization of an exponential family and the exponential dispersion model of distributions and includes those families of probability distributions, parameterized by and , whose density functions f (or probability mass function, for the case of a discrete distribution) can be expressed in the form The dispersion parameter, , typically is known and is usually related to the variance of the distribution. The functions , , , , and are known. Many common distributions are in this family, including the normal, exponential, gamma, Poisson, Bernoulli, and (for fixed number of trials) binomial, multinomial, and negative binomial. For scalar and (denoted and in this case), this reduces to is related to the mean of the distribution. If is the identity function, then the distribution is said to be in canonical form (or natural form). Note that any distribution can be converted to canonical form by rewriting as and then applying the transformation . It is always possible to convert in terms of the new parametrization, even if is not a one-to-one function; see comments in the page on exponential families. If, in addition, is the identity and is known, then is called the canonical parameter (or natural parameter) and is related to the mean through For scalar and , this reduces to Under this scenario, the variance of the distribution can be shown to be For scalar and , this reduces to Linear predictor The linear predictor is the quantity which incorporates the information about the independent variables into the model. The symbol η (Greek "eta") denotes a linear predictor. It is related to the expected value of the data through the link function. η is expressed as linear combinations (thus, "linear") of unknown parameters β. The coefficients of the linear combination are represented as the matrix of independent variables X. η can thus be expressed as Link function The link function provides the relationship between the linear predictor and the mean of the distribution function. There are many commonly used link functions, and their choice is informed by several considerations. There is always a well-defined canonical link function which is