Problem IV$.$ Logistics Regression $(10$ points$)$

Consider a two$-$class classification problem with the logistic sigmoid function, $y=(w^T(x)),$ for a given data set $D=\{{}_n,t_n\},$ where $t_{n}in\{0,1\},{}_n=(x_n),n=1,$dots,$N,$ and the likelihood function is given by

$(w|)=pro{d}_n=1^N{y}_n^{t_n}(1-y_n{)}^{1-t_n}$

where $w$ has a Gaussian priori, i$.$e$.,(m_0,{}^{-1}I|).$ Please derive the posterior distribution, $(D,m_0,|),$ the posterior predictive distribution, $(x,D,m_0,|),$ and the prior predictive distribution, and $(m_0,|),$ respectively. $($Hint: using Delta approximation and Laplace approximation properly$).$