Simplicity Priors are Tautological

Any non-uniform prior inherently encodes a bias toward simplicity. This isn't an additional assumption we need to make - it falls directly out of the mathematics.

For any hypothesis h, the information content is $I(h) = -\log(P(h))$, which means probability and complexity have an exponential relationship: $P(h) = e^{-I(h)}$

This demonstrates that simpler hypotheses (those with lower information content) are automatically assigned higher probabilities. The exponential relationship creates a strong bias toward simplicity without requiring any special mechanisms.

The "simplicity prior" is essentially tautological - more probable things are simple by definition.

I would be interested in seeing those talks, can you maybe share links to these recordings?

Very good work, thank you for sharing!

Intuitively speaking, the connection between physics and computability arises because the coarse-grained dynamics of our Universe are believed to have computational capabilities equivalent to a universal Turing machine [19–22].

I can see how this is a reasonable and useful assumption, but the universe seems to be finite in both space and time and therefore not a UTM. What convinced you otherwise?

Thank you! I'll have a look!

Simplified the solomonoff prior is the distribution you get when you take a uniform distribution over all strings and feed them to a turing machine.

Since the outputs are also strings: What happens if we iterate this? What is the stationary distribution? Is there even one? The fixed points will be quines, programs that copy their source code to the output. But how are they weighted? By their length? Presumably you can also have quine-cycles of programs that generate each other in turn, in a manner reminiscent metagenesis. Do these quine cycles capture all probability mass or does some diverge?

Very grateful for answers and literature suggestions.

"Many parts of the real world we care about just turn out to be the efficiently predictable."

I had a dicussion about exactly these 'pockets of computational reducibility' today. Whether they are the same as the more vague 'natural abstractions', and if there is some observation selection effect going on here.

Very nice! Alexander and I were thinking about this after our talk as well. We thought of this in terms of the kolmogorov structure function and I struggled with what you call Claim 3, since the time requirements are only bounded by the busybeaver number. I think if you accept some small divergence it could work, I would be very interested to see.

Small addendum: The padding argument gives a lower bound of the multiplicity. Above it is bounded by the Kraft-McMillan inequality.

Interesting! I think the problem is dense/compressed information can be represented in ways in which it is not easily retrievable for a certain decoder. The standard model written in Chinese is a very compressed representation of human knowledge of the universe and completely inscrutable to me.
Or take some maximally compressed code and pass it through a permutation. The information content is obviously the same but it is illegible until you reverse the permutation.

In some ways it is uniquely easy to do this to codes with maximal entropy because per definition it will be impossible to detect a pattern and recover a readable explanation.

In some ways the compressibility of NNs is a proof that a simple model exists, without revealing a understandable explanation.

I think we can have (almost) minimal yet readable model without exponentially decreasing information density as required by LDCs.

Good points! I think we underestimate the role that brute force plays in our brains though.