Generally speaking the formula you wish to be true is not true.

In a simpler version, you're asking: $\text{Is }P(A|B,C) = P(A|B).P(B|C)\text{?}$

It's not generally so, not even for Gaussian variables.

You might be thinking of: 

$p(x_n, x_{n-1}, x_{n-2}, ... , x_1) = \\
\, p(x_n | x_{n-1}, x_{n-2}, ... , x_1).p( x_{n-1}| x_{n-2}, ... , x_1).p(x_{n-2}| x_{n-3}, ... , x_1)...p(x_2|x_1).p(x_1) $
- which is true.

You might even be thinking of the Markov property: $p(x_n, x_{n-1}, x_{n-2}, ... , x_1) = p(x_n | x_{n-1}).p( x_{n-1}| x_{n-2}).p(x_{n-2}| x_{n-3})...p(x_2|x_1).p(x_1)$, where each conditional can be replaced with conditioning on the previous value alone - which is sometimes true.

However, your actual problem sounds closer to a regression-updating problem.