[EbTech]

Fair enough! I'd like to point out that the Kolmogorov complexity approach can make sense of subjective probability too. Since you lack precise enough information to predict the dice roll, the most compressed way to write down your observations will involve a Shannon-style code with your subjective probabilities. If you have enough information but not enough computational power, the resource-bounded variants of Kolmogorov complexity may be more applicable.

[jtsuken]

Can you please define Shannon-style code? (Obviously, you don't have the shannon code, the compression method, in mind).

Also, isn't Kolmogorov complexity uncomputable and you run into multiple "who shaves the barber" issues, when trying to determine it?

[EbTech]

The compression code can be specified first. If you have a lot of data, the specification will be negligible in length, compared to the code itself. Together, the specification and the Shannon code give an upper bound on the Kolmogorov compmlexity. If this is the shortest known program, we may consider the Shannon code probabilities as our "best explanation" of the data.

You can also get posterior probabilities using a universal prior such as 2^-K(x), but of course, this can only be approximated in the limit of infinite runtime.