def fit_polynomial_bayes(x, t, M, alpha, beta):
    phi = np.vander(x,M+1)
    Snpart1 =(alpha*np.identity(M+1))
    Snpart2 = (beta*phi.T.dot(phi))
    Sn = inv(Snpart1+Snpart2)
    Mn = beta*Sn.dot(phi.T).dot(t)
    return Mn, Sn

def predict_polynomial_bayes(x, m, S, beta):
    N = S.shape[0]
    phi = np.vander(x, N=N)
    sigma =  1/beta + np.dot(np.dot(phi.T, S), phi)
    predM = np.dot(m.T, phi)
    return predM, sigma


x, t = gen_sinusoidal2(7)
M= 5
alpha = 0.5
beta = 1/(0.2**2)
Mn, Sn = fit_polynomial_bayes(x,t,M,alpha, beta)

for  i in range(len(x)):
    predict_polynomial_bayes(x[i], Mn, Sn, beta)