Curiously, this means there’s no object for which we have defined both “complexity” and “prior probability”, which makes all arguments of the form “complexity is high ⇒ probability is low” automatically suspect.
The above is not as strong as it seems. It can be shown that on any given Turing machine, the prior probability of an equivalence class of programs is at most an O(1) factor greater than the part of that probability due to just the simplest program in the class.
Arguments can still be suspect if you don’t know which program is the simplest. However, it is often possible to show that a program is probably the simplest; since there are many more complex programs than simple ones, a randomly drawn complex hypothesis is extremely unlikely to have a shorter equivalent.
This is an interesting pattern to keep in mind.
bump
The above is not as strong as it seems. It can be shown that on any given Turing machine, the prior probability of an equivalence class of programs is at most an O(1) factor greater than the part of that probability due to just the simplest program in the class.
Arguments can still be suspect if you don’t know which program is the simplest. However, it is often possible to show that a program is probably the simplest; since there are many more complex programs than simple ones, a randomly drawn complex hypothesis is extremely unlikely to have a shorter equivalent.