Given a multivariate gaussian distribution of the form
$$\begin{pmatrix} y_{1}\\y_{2} \\y_{3}\end{pmatrix} \sim N\begin{pmatrix} \begin{pmatrix} 0\\0 \\0 \end{pmatrix}, &\begin{pmatrix} \sigma_{1}^{2} & 0 & \rho\sigma_{1}\sigma_{3}\\ 0& \sigma_{2}^{2} & \rho\sigma_{2}\sigma_{3}\\ \rho\sigma_{1}\sigma_{3} & \rho\sigma_{2}\sigma_{3} & \sigma_{3}^{2} \end{pmatrix} \end{pmatrix}$$ Since it is known in the gaussian distribution that zero covariance implies independence then is it safe to write the likelihood as: $$f(y_{1},y_{2},y_{3})=f(y_{3}|y_{1},y_{2})*f(y_{1})*f(y_{2})$$
If yes, then how can we write $f(y_{1},y_{2},y_{3})$
PS : A similar question has been asked in cross validated : Likelihood Factorization
