iid where {ei ( to simplify the analysislet us assume that is knownso no prior needs to be specified for let [ ( ) (xn )]be the (unknownvector of regression values placing gp prior on the function is equivalent to placing multivariate gaussian prior on the vector gg ( )( where the covariance matrix of is gram matrix (implicitly associated with feature map through the kernel )given byk( ( ( xn ( ( ( xn ( (xn (xn (xn xn the likelihood of our data given gdenoted ( )is obtained directly from the model ( )( gngs in ( solving this bayesian problem involves deriving the posterior distribution of yto do sowe first note that since has covariance matrix in (which can be seen from ( ))the joint distribution of and is again normalwith mean and covariance matrixk in , ( the posterior can then be found by conditioning on yvia theorem giving yn ( in )- yk ( in )- this only gives information about at the observed points xn it is more interesting to consider the posterior predictive distribution of : ( xfor new input we can find the corresponding posterior predictive pdf ( yby integrating out the joint posterior pdf ( gg )which is equivalent to taking the expectation of ( gwhen is distributed according to the posterior pdf ( )that isz ( yp( gp( ydg to do so more easily than direct evaluation via the above integral representation of ( )we can begin with the joint distribution of [ , ]which is multivariate normal with mean and covariance matrix ek ( kk( xe xwhere [ ( xx ) ( xxn )]it now followsagain by using theorem that ( yhas normal distribution with mean and variance given respectively by and predictive ( xk( in )- ( ( xk( xe xk( in )- ( these are sometimes called the predictive mean and variance it is important to note that ( we are predicting the expected response ey xhereand not the actual response