By "Grain of Ignorance" I mean that the semimeasure loss is nonzero at every string, that is the conditionals of M are never a proper measure. Since this gap is not computable, it cannot be (easily) removed, though to be fair the conditional distribution is only limit computable anyway (same as the normalized M). However, it is not clear that there is any natural/forced choice of normalization, so I usually think of the set of possible normalizations as a credal set (and I mean ignorance in that sense). I will soon put an updated version of my "Value under Ignorance" paper (about this) on arXiv.
Vovk's trick refers to predicting like the mixture - a "specialist expert" can opt out of offering a prediction by matching the Bayesian mixture's prediction, so that its weight is not updated (assuming that it has access to the Bayesian mixture). I think the usual citation is "Prediction with Expert Evaluators Advice" (referring to section 6) which is with Chernov. I believe this was an influence on logical induction.
Chaitin's Number of Wisdom. Knowledge looks like noise from outside.
I express this by saying "sufficiently advanced probabilistic reasoning is indistinguishable from prophetic intuition".
2Dovetailing. Every meta-cognition enthusiast reinvents Levin/Hutter search, usually with added epicycles.
To frame it in a very different way, learning math and generally gaining lots of abstractions and getting good wieldy names for them is super important for thinking. Doing so increases your "algorithmic range", within your very constrained cognition.
1Grain of Truth (Reflective Oracles). Understanding an opponent perfectly requires greater intelligence or something in common.
And understanding yourself. Of course, you have plenty in common with yourself. But, you don't have everything in common with yourself, if you're growing.
1Chaitin's Number of Wisdom. Knowledge looks like noise from outside.
To a large extent, but not quite exactly (which you probably weren't trying to say), because of "thinking longer should make you less surprised". From outside, a big chunk of alien knowledge looks like noise (for now), true. But there's a "thick interface" where just seeing stuff from the alien knowledgebase will "make things click into place" (i.e. will make you think a bit more / make you have new hypotheses (and hypothesis bits)). You can tell that the alien knowledgebase is talking about Things even if you aren't very familiar with those Things.
Lower Semicomputability of M. Thinking longer should make you less surprised.
I'd go even farther and say that in "most" situations in real life, if you feel like you want to think about X more, then the top priority (do it first, and do it often ongoingly) is to think of more hypotheses.
Very nice!
Conversely, it may be possible to identify practical situations where some of these aphorisms are sub-optimal, which could help point out the limitations of applying AIT to real agents?
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Epistemic status: Compressed aphorisms.
This post contains no algorithmic information theory (AIT) exposition, only the rationality lessons that I (think I've) learned from studying AIT / AIXI for the last few years. Many of these are not direct translations of AIT theorems, but rather frames suggested by AIT. In some cases, they even fall outside of the subject entirely (particularly when the crisp perspective of AIT allows me to see the essentials of related areas).
Prequential Problem. The posterior predictive distribution screens off the posterior for sequence prediction, therefore it is easier to build a strong predictive model than to understand its ontology.
Reward Hypothesis (or Curse). Simple first-person objectives incentivize sophisticated but not-necessarily-intended intelligent behavior, therefore it is easier to build an agent than it is to align one.
Coding Theorem. A multiplicity of good explanations implies a better (ensemble) explanation.
Gacs' Separation. Prediction is close but not identical to compression.
Limit Computability. Algorithms for intelligence can always be improved.
Lower Semicomputability of M. Thinking longer should make you less surprised.
Chaitin's Number of Wisdom. Knowledge looks like noise from outside.
Dovetailing. Every meta-cognition enthusiast reinvents Levin/Hutter search, usually with added epicycles.
Grain of Uncertainty (Cromwell's Rule). Anything with a finite description gets nonzero probability.
Grain of Truth (Reflective Oracles). Understanding an opponent perfectly requires greater intelligence or something in common.
Grain of Ignorance (Semimeasure Loss). You cannot think long enough to know that you do not need to think for longer.
Solomonoff Bound. Bayesian sequence prediction has frequentist guarantees for log loss.
Information Distance. There are no opposites.
Prediction of Selected Bits. Updating on the unpredictable can damage your beliefs about the predictable.
Vovk's Trick. Self-reflection permits partial models.