by Sam Eisenstat 507 days ago | Abram Demski likes this | link | parent I at first didn’t understand your argument for claim (2), so I wrote an alternate proof that’s a bit more obvious/careful. I now see why it works, but I’ll give my version below for anyone interested. In any case, what you really mean is the probability of deciding a sentence outside of $$\Phi$$ by having it announced by nature; there may be a high probability of sentences being decided indirectly via sentences in $$\Phi$$. Instead of choosing $$\Phi$$ as you describe, pick $$\Phi$$ so that the probability $$\mu(\Phi)$$ of sampling something in $$\Phi$$ is greater than $$1 - \mu(\psi) \cdot \varepsilon / 2$$. Then, the probability of sampling something in $$\Phi - \{\psi\}$$ is at least $$1 - \mu(\psi) \cdot (1 + \varepsilon / 2)$$. Hence, no matter what sentences have been decided already, the probability that repeatedly sampling from $$\mu$$ selects $$\psi$$ before it selects any sentence outside of $$\Phi$$ is at least \begin{align*} \sum_{k = 0}^\infty (1 - \mu(\psi) \cdot (1 + \varepsilon / 2))^k \cdot \mu(\psi) & = \frac{\mu(\psi)}{\mu(\psi) \cdot (1 + \varepsilon / 2)} \\ & > 1 - \varepsilon / 2 \end{align*} as desired. Furthermore, this argument makes it clear that the probability distribution we converge to depends only on the set of sentences which the environment will eventually assert, not on their ordering! Oh, I didn’t notice that aspect of things. That’s pretty cool.

### NEW DISCUSSION POSTS

[Note: This comment is three
 by Ryan Carey on A brief note on factoring out certain variables | 0 likes

There should be a chat icon
 by Alex Mennen on Meta: IAFF vs LessWrong | 0 likes

Apparently "You must be
 by Jessica Taylor on Meta: IAFF vs LessWrong | 1 like

There is a replacement for
 by Alex Mennen on Meta: IAFF vs LessWrong | 1 like

Regarding the physical
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think that we should expect
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

I think I understand your
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

This seems like a hack. The
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

After thinking some more,
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yes, I think that we're
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

My intuition is that it must
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

To first approximation, a
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Actually, I *am* including
 by Vanessa Kosoy on The Learning-Theoretic AI Alignment Research Agend... | 0 likes

Yeah, when I went back and
 by Alex Appel on Optimal and Causal Counterfactual Worlds | 0 likes

> Well, we could give up on
 by Jessica Taylor on The Learning-Theoretic AI Alignment Research Agend... | 0 likes