I am acquiring samples that are Gaussian distributed and I need to calculate

p(x_n | x_n-1, x_n-2, ... , x_1)

for each sample x_n as it comes in. I am trying to break this expression down so that I can calculate this recursively. I can not store past variables indefinitely as the system will run for long time periods and the calculations will become progressively more expensive.

Is it valid to express this as p(x_n | x_n-1)p(x_n-1 | p_n-2)p(x_n-2 | x_n-3)...p(x_2 | x_1) ?

I am acquiring samples that are Gaussian distributed and I need to calculate

$$ p(x_n | x_{n-1}, x_{n-2}, \ldots , x_1) $$

for each sample $x_n$ as it comes in. I am trying to break this expression down so that I can calculate this recursively. I can not store past variables indefinitely as the system will run for long time periods and the calculations will become progressively more expensive.

Is it valid to express this as $p(x_n | x_{n-1})p(x_{n-1} | p_{n-2})p(x_{n-2} | x_{n-3}) \cdots p(x_2 | x_1)$ ?