Equivalence of random coinflips view and minimal programs view

Two of the several ways of viewing the deterministic variant of Solomonoff induction are:

• The "random coinflips view": Given a monotone Turing machine, we feed the machine with random data (such as from unbiased coinflips). The probability of string $x$ is then the probability that the machine outputs something starting with $x$. This one is somewhat hard to think about, because one usually thinks of a Turing machine as e.g. a Python interpreter that takes Python programs. If one feeds a Python interpreter with randomness, one is likely to end up running a horrible mess that isn't even syntactically valid. I guess the reply here is something like (1) if a program isn't even syntactically valid, it just results in a program with no output; (2) using an encoding, one can filter out all invalid programs so that each input into the UTM does something; (3) more generally, thinking of "coinflips" is meant to suggest the idea that one has no information about what the program looks like.
• The "minimal programs view": Given a monotone Turing machine $T$, the minimal programs producing $x$ are those programs $p$ such that $T(p)=x*$ (i.e. $T$ when run with $p$ produces an output starting with $x$) and such that no proper prefix of $p$ produces $x$ (i.e. if $q\prec p$ then $T(q)\ne x*$). This view then says that to get the probability of $x$, we add up all the minimal programs weighted by length as follows: for each program $p$ we assign the weight $2^{-\left|p\right|}$, where $\left|p\right|$ is the length of $p$ in bits. Thus the measure is ${\sum }_p{2}^{-\left|p\right|}$, where the sum is taken over all minimal programs $p$ producing $x$. The idea with the $2^{-\left|p\right|}$ is that there are $2^n$ programs of length $n$, so given just the information that $p$ has length $n$, the probability that we have a specific program is $1/2^n=2^{-n}$.

These two views turn out to be equivalent in the sense that .