the main reason why the multivariate normal distribution plays an important role in data science and machine learning is that it satisfies the following propertiesthe details and proofs of which can be found in appendix affine combinations are normal marginal distributions are normal conditional distributions are normal normal linear models normal linear models combine the simplicity of the linear model with the tractability of the gaussian distribution they are the principal model for traditional statisticsand include the classic linear regression and analysis of variance models definition normal linear model normal linear model in normal linear model the response depends on -dimensional explanatory variable [ ]via the linear relationship xb ( where ( thusa normal linear model is linear model (in the sense of definition with normal error terms similar to ( )the corresponding normal linear model for the whole training set {(xi yi )has the form xb ( where is the model matrix comprised of rows > > and ( in consequentlyy can be written as xb szwhere ( in )so that (xbs in it follows from ( that its joint density is given by ( bs ( ps ) || -xb| ( estimation of the parameter can be performed via the least-squares methodas discussed in example an estimate can also be obtained via the maximum likelihood method this simply means finding the parameters and that maximize the likelihood of the outcome ygiven by the right-hand side of ( it is clear that for every value of the likelihood is maximal when ky xbk is minimal as consequencethe maximum likelihood estimate for is the same as the least-squares estimate ( we leave it as an exercise (see exercise to show that the maximum likelihood estimate of is equal to ky xb bk ( where is the maximum likelihood estimate (least squares estimate in this caseof