Consider some finite string $x=(x\_1,x\_2,...,x\_{n-1},x\_n)$ that is drawn from a *non-stationary* process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as,
$$
P\_M(x)=\sum\_{i=1}^{\infty}2^{-|s\_{i}(x)|}
$$
to predict $x\_n$, by computing the probability $$P\_M(x\_n | x\_1,...x\_{n-1})=P\_M(x\_1,x\_2,...,x\_{n-1},x\_n)/P\_M(x\_1,x\_2,...,x\_{n-1})$$? Or is it a necessity that the process that $x$ is drawn from be stationary?


Edit: Here $s\_i(x)$ refers to the $i$-th program which generated $x=(x\_1,x\_2,...,x\_{n-1})$ and we want to predict the probability that $x\_n$ is either a $1$ or a $0$.



Find probability of non-stationary inputs into Turing machine?
The equation you have for algorithmic probability states how to compute (or, more specifically, lower semicompute) $P\_M(x)$ for any finite string $x$, so it can also be used to predict $x\_n$ by evaluating $P\_M(x\_n \vert x\_1, \dots , x\_{n-1})$. It holds for *any* finite string $x$ -- it does not matter what distribution (if any) $x$ is drawn from.


