Question $2$ : $[12\mathrm{.}5$ Points$]$ Python Exercise on Gaussian Process Regression $($GPR$)$ for posterior prediction.

Generate $10$ data points $($these points will serve as training data points$)$ with negligible noise $($corresponds to noiseless GP regression$).$ Use the following python function with default noise variance.

import numpy as $np$

def generate$\_$noisy$\_$points$(n=10,$ noise$\_$variance$=1$e$-6)$:

$,$np$.$random.seed$(777)$

$,x=$ np$.$random.uniform$(-3.,3.,($n$,1))$

$,y=np*sin(x)+np*random*randn(n,1)**$ noise$\_$variance ${?}^{****}0\mathrm{.}5$

$,$return $x,y$

Write Python script for the following questions :

a$)[1\mathrm{.}5$ Points$]$ Plot the variation of $x$ wrt $y$

b$)[4$ Points$]$ Generate $100$ test data points and Draw $10$ function samples from the GP prior distribution. Show the mathematical representation of GP prior distribution.

c$)[7$ Points$]$ Compute and Plot the GP posterior distribution given the original $($training$)10$ data points. Use the RBF kernel function. Please also show the mathematical representation of posterior distribution.