example (gp regressionsuppose the regression function is ( sin( px) [ we use gp regression to estimate gusing gaussian kernel of the form ( with bandwidth parameter the explanatory variables were drawn uniformly on the interval [ ]and the responses were obtained from ( )with noise level figure shows samples from the prior distribution for as well as the data points and the true sinusoidal regression function - - - - - - figure leftsamples drawn from the gp prior distribution rightthe true regression function with the data points again assuming that the variance is knownthe predictive distribution as determined by ( and ( is shown in figure for bandwidth (leftand (rightclearlydecreasing the bandwidth leads to the covariance between points and decreasing at faster rate with respect to the squared distance kx leading to predictive mean that is less smooth in the above expositionwe have taken the mean function for the prior distribution of to be identically zero if instead we have general mean function and write [ ( ) (xn )]then the predictive variance ( remains unchangedand the predictive mean ( is modified to read ( xm( xk( in )- ( ( typicallythe variance appearing in ( is not knownand the kernel itself depends on several parameters -for instance gaussian kernel ( with an unknown bandwidth parameter in the bayesian frameworkone typically specifies hierarchical model by introducing prior (thfor the vector th of such hyperparameters nowthe gp prior ( th(equivalentlyspecifying pg th)and the model for the likelihood of the data given ygthnamely ( gth)are both dependent on th the posterior distribution of ythis as before hyperparameters