as the squared-error loss is the most widely-used loss function for regressionwe will adopt this loss function in most of this the optimal prediction function ghas to be learned from the training set by minimizing the training loss (yi (xi )) ( ` (gn = learner over suitable class of functions note that in the above definitionthe training set is assumed to be fixed for random training set we will write the training loss as ` (gthe function ggt that minimizes the training loss is the function we use for prediction -the so-called learner when the function class is clear from the contextwe drop the superscript in the notation as we already saw in ( )conditional on xthe response can be written as (xe( )where ( this motivates standard modeling assumption in supervised learningin which the responses yn conditional on the explanatory variables xn xn are assumed to be of the form yi (xi ei nwhere the {ei are independent with ei and var ei for some function and variance the above model is usually further specified by assuming that is completely known up to an unknown parameter vectorthat isyi (xi bei ( while the model ( is described conditional on the explanatory variablesit will be convenient to make one further model simplificationand view ( as if the {xi were fixedwhile the {yi are random for the remainder of this we assume that the training feature vectors {xi are fixed and only the responses are randomthat ist {( )(xn yn )the advantage of the model ( is that the problem of estimating the function from the training data is reduced to the (much simplerproblem of estimating the parameter vector an obvious disadvantage is that functions of the form (bmay not accurately approximate the true unknown gthe remainder of this deals with the analysis of models of the form ( in the important case where the function (bis linearthe analysis proceeds through the class of linear models ifin additionthe error terms {ei are assumed to be gaussianthis analysis can be carried out using the rich theory of normal linear models