generalized linear models the normal linear model in section deals with continuous response variables -such as height and crop yield -and continuous or discrete explanatory variables given the feature vectors {xi }the responses {yi are independent of each otherand each has normal distribution with mean > bwhere > is the -th row of the model matrix generalized linear models allow for arbitrary response distributionsincluding discrete ones definition generalized linear model generalized linear model in generalized linear model (glmthe expected response for given feature vector [ ]is of the form [ xh(xbactivation function link function ( for some function hwhich is called the activation function the distribution of (for given xmay depend on additional dispersion parameters that model the randomness in the data that is not explained by the inverse of function is called the link function as for the linear model( is model for single pair (xyusing the model simplification introduced at the end of section the corresponding model for whole training set {(xi yi )is that the {xi are fixed and that the {yi are independenteach yi satisfying ( with xi writing [ yn ]and defining as the multivalued function with components hwe have ex (xb)where is the (modelmatrix with rows > > common assumption is that yn come from the same family of distributionse normalbernoullior poisson the central focus is the parameter vector bwhich summarizes how the matrix of explanatory variables affects the response vector the class of generalized linear models can encompass wide variety of models obviously the normal linear model ( is generalized linear modelwith [ xxbso that is the identity function in this casey (xbs ) nwhere is dispersion parameter logistic regression logistic distribution example (logistic regressionin logistic regression or logit modelwe assume that the response variables yn are independent and distributed according to yi ber( ( > ))where here is defined as the cdf of the logistic distribution - large values of > thus lead to high probability that yi and small (negativevalues of > cause yi to be with high probability estimation of the parameter vector from the observed data is not as straightforward as for the ordinary linear modelbut can be accomplished via the minimization of suitable training lossas explained below as the {yi are independentthe pdf of [ yn ]is ( bx[ ( > )]yi [ ( > )] -yi (xi=