The Matern covariance function is a popular choice in the machine learning community due to its flexibility and ability to generalise to other well-known kernels such as the Radial Basis Function (RBF) class (i.e. Gaussian kernel). In fact, as i approaches infinity, the Matern covariance function converges to the RBF covariance function. The Matern covariance function is characterised by two positive parameters: & and f. The parameter v determines the smoothness of the function, while / controls the length-scale over which the covariance function varies. I denotes the Gamma function, and the modified Bessel function K, is used to ensure that the covariance function is non-negative. Here, we present the mathematical definition of the Matern covariance function, which is given by equation 2.5: 21-2 V2ud V2ud kMatern (2, y) = ] - T(v) (2.5) with positive parameters & and f. K, is a modified Bessel function, d = dist(x, y) denotes the pairwise Euclidean distance, and of being the signal variance. Question 2.1: Matern Class Covariance Decomposition (5 Points) For machine learning applications, it is customary to opt for half-integer values for the smoothness parameter r in order to achieve further simplification of the Matern covariance function. Show that for v = p+ 1/2, where pe N. the Matern covariance function can be expressed as a product of an exponential and a p-polynomial term: KMatern (r, y) = of exp V2p + Id [(p+1) \\ (pti)! (2v/2p+ Id\\ P-i [(2p + 1) _ i!(p - i)! (2.6) Hint: You may utilise any Gamma function identities without proof from here (although you should include the identities you have used in each step of your proof). Additionally, you may use without proof the following formula: where (n + 7, i) = In-+1); and K.() is the modified Bessel function