Prelimineries Gaussian Processes Stochastic Processes and Gaussian Processes oStochastic processes: A stochastic process (or random process)y(x)is specified by giving the joint distribution for any finite set of instances {x1,...,xn in a consistent manner. ●Gaussian processes: A Gaussian process is a distribution over functions y(x)s.t.the values of y(x)evaluated at an arbitrary set of points {x1,...,xn jointly have a Gaussian distribution. Assuming y(x)has zero mean,the specification of a Gaussian process is completed by giving the covariance function of y(x)evaluated at any two values of x,given by the kernel function K(,): Ey(x)y(x】=K(x,x): 4口4日+1艺4至卡三及0 Li,Zhang and Yeung (CSE.HKUST) LWP AISTATS 2009 5/23Preliminaries Gaussian Processes Stochastic Processes and Gaussian Processes Stochastic processes: A stochastic process (or random process) y(x) is specified by giving the joint distribution for any finite set of instances {x1, . . . , xn} in a consistent manner. Gaussian processes: A Gaussian process is a distribution over functions y(x) s.t. the values of y(x) evaluated at an arbitrary set of points {x1, . . . , xn} jointly have a Gaussian distribution. Assuming y(x) has zero mean, the specification of a Gaussian process is completed by giving the covariance function of y(x) evaluated at any two values of x, given by the kernel function K(·, ·): E[y(xi)y(xj)] = K(xi , xj). Li, Zhang and Yeung (CSE, HKUST) LWP AISTATS 2009 5 / 23