How to extend Solomonoff Induction to continuous domain

Question

Informally speaking, Solomonoff Induction provides an Bayesian optimal way to predict the next symbol from observed symbol sequences $S_1S_2S_3...S_n$ where each $S_i$ belongs to a finite symbol set $\Sigma$. To get the posterior probability distribution of the next symbol, we make a weighted average prediction of all Turing Machines that produces first $n$ symbols. For each Turing Machine $M$ within consideration, the weight would be inverse exponential of "complexity" of that Turing Machine.

$$p(M) = 2^{-L(M)}$$

Where $L(M)$ is the description length of $M$ in context of an universal Turing Machine $U$.

If we replace:

• finite symbol set $\Sigma$ to continuous domain $D$, such as $R^n$.
• Symbol sequence $S_i$ to discrete time series, for example $x_1x_2x_3...x_n$ where $x_i \in D$
• Turing Machine $M$ to discrete time dynamical system, for example $x_{n+1} = f(x_n)$, which is mapping $R^n \to R^n$

The Bayesian inference would still work in this case, however we have to redefine the complexity measure $p(f) \to [0, \infty)$, which is probability density function of $f$.

Question: Is there any work or related theory that extends the complexity measure elegantly to dynamical systems?

I would also be interested in results for continuous dynamical systems, in terms of differential equations.

Thanks.

Answer 1

The set of computable real functions is also recursively enumerable. Assume it is enumerated by $(f_i)_{i\in\omega}$. Then the Solomonoff prior of $f_i$ is $p(f_i)=2^{-K(i)}$. So there is no difference whether the function is defined over natural numbers or real numbers, as long as it is computable.