Problem VI$.$ Bayesian Neural Network $(20$ points$)$

a$)$ Consider a neural network for regression, $t=y(w,x)+v,$ where $v$ is Gaussian, i$.$e$.,v(0,{}^{-1}|),$ and $w$ has a Gaussian priori, i$.$e$.,(m_0,{}^{-1}I|).$ Assume that $y(w,x)$ is the neural network output please derive the posterior distribution, $(D,,m_0,|),$ the posterior predictive distribution, $(x,D,,m_0,|),$ and the prior predictive distribution, $(,m_0,|),$ where $D=\{x_n,t_n\},n=1,$dots,$N,$ is the training data set.

b$)$ Consider a neural network for two$-$class classification, $y=(a(w,x))$ and a data set $D=\{x_n,t_n\},$ where $t_{n}in\{0,1\},w$ has a Gaussian priori, i$.$e$.,(0,{}^{-1}I|),$ and $a(w,x)$ is the neural network model. Please derive the posterior distribution, $(D,|),$ posterior predictive distribution, $(x,D,|),$ and the prior predictive distribution, $(|),$ respectively.