Recursive definition of conditional normal distribution

Question

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 $p(x_n | x_{n-1})p(x_{n-1} | p_{n-2})p(x_{n-2} | x_{n-3}) \cdots p(x_2 | x_1)$ ?

Answer 1

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.

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Followup: when does the Markov property hold?

It will hold when the first equation is the same thing as the second equation.

i.e. when

$p(x_t | x_{t-1}, x_{t-2}, ... , x_1)= p(x_t | x_{t-1})$ at each value of $t$.

That is, when the distribution at time $t$ only depends on the value at time $t-1$.

With Gaussian distributions, and models that can be written in state space form, it's common to use one of the forms of the Kalman filter for this situation.