   3.  An Untrollable Mathematician   post by Abram Demski 393 days ago  Alex Appel, Sam Eisenstat, Vadim Kosoy, Jack Gallagher, Jessica Taylor, Paul Christiano, Scott Garrabrant and Vladimir Slepnev like this  1 comment  
 Followup to All Mathematicians are Trollable.
It is relatively easy to see that no computable Bayesian prior on logic can converge to a single coherent probability distribution as we update it on logical statements. Furthermore, the nonconvergence behavior is about as bad as could be: someone selecting the ordering of provable statements to update on can drive the Bayesian’s beliefs arbitrarily up or down, arbitrarily many times, despite only saying true things. I called this wild nonconvergence behavior “trollability”. Previously, I showed that if the Bayesian updates on the provabilily of a sentence rather than updating on the sentence itself, it is still trollable. I left open the question of whether some other side information could save us. Sam Eisenstat has closed this question, providing a simple logical prior and a way of doing a Bayesian update on it which (1) cannot be trolled, and (2) converges to a coherent distribution.
 
   5.  The Happy Dance Problem   post by Abram Demski 461 days ago  Scott Garrabrant and Stuart Armstrong like this  1 comment  
 Since the invention of logical induction, people have been trying to figure out what logically updateless reasoning could be. This is motivated by the idea that, in the realm of Bayesian uncertainty (IE, empirical uncertainty), updateless decision theory is the simple solution to the problem of reflective consistency. Naturally, we’d like to import this success to logically uncertain decision theory.
At a research retreat during the summer, we realized that updateless decision theory wasn’t so easy to define even in the seemingly simple Bayesian case. A possible solution was written up in Conditioning on Conditionals. However, that didn’t end up being especially satisfying.
Here, I introduce the happy dance problem, which more clearly illustrates the difficulty in defining updateless reasoning in the Bayesian case. I also outline Scott’s current thoughts about the correct way of reasoning about this problem.
 
  6.  Policy Selection Solves Most Problems   post by Abram Demski 449 days ago  Alex Appel and Vladimir Slepnev like this  4 comments  
 It seems like logically updateless reasoning is what we would want in order to solve many decisiontheory problems. I show that several of the problems which seem to require updateless reasoning can instead be solved by selecting a policy with a logical inductor that’s run a small amount of time. The policy specifies how to make use of knowledge from a logical inductor which is run longer. This addresses the difficulties which seem to block logically updateless decision theory in a fairly direct manner. On the other hand, it doesn’t seem to hold much promise for the kind of insights which we would want from a real solution.
 
      11.  Smoking Lesion Steelman II   post by Abram Demski 508 days ago  Tom Everitt and Scott Garrabrant like this  1 comment  
 After Johannes Treutlein’s comment on Smoking Lesion Steelman, and a number of other considerations, I had almost entirely given up on CDT. However, there were still nagging questions about whether the kind of selfignorance needed in Smoking Lesion Steelman could arise naturally, how it should be dealt with if so, and what role counterfactuals ought to play in decision theory if CDTlike behavior is incorrect. Today I sat down to collect all the arguments which have been rolling around in my head on this and related issues, and arrived at a place much closer to CDT than I expected.
 
  12.  Comparing LICDT and LIEDT   post by Abram Demski 487 days ago  Alex Appel likes this  discuss  
 Attempted versions of CDT and EDT can be constructed using logical inductors, called LICDT and LIEDT. It is shown, however, that LICDT fails XOR Blackmail, and LIEDT fails Newcomb. One interpretation of this is that LICDT and LIEDT do not implement CDT and EDT very well. I argue that they are indeed forms of CDT and EDT, but stray from expectations because they also implement the ratifiability condition I discussed previously. Continuing the line of thinking from that post, I discuss conditions in which LICDT=LIEDT, and try to draw out broader implications for decision theory.
 
   14.  Smoking Lesion Steelman   post by Abram Demski 598 days ago  Tom Everitt, Sam Eisenstat, Vadim Kosoy, Paul Christiano and Scott Garrabrant like this  10 comments  
 It seems plausible to me that any example I’ve seen so far which seems to require causal/counterfactual reasoning is more properly solved by taking the right updateless perspective, and taking the action or policy which achieves maximum expected utility from that perspective. If this were the right view, then the aim would be to construct something like updateless EDT.
I give a variant of the smoking lesion problem which overcomes an objection to the classic smoking lesion, and which is solved correctly by CDT, but which is not solved by updateless EDT.
 
  15.  Futarchy Fix   post by Abram Demski 631 days ago  Scott Garrabrant and Stuart Armstrong like this  9 comments  
 Robin Hanson’s Futarchy is a proposal to let prediction markets make governmental decisions. We can view an operating Futarchy as an agent, and ask if it is aligned with the interests of its constituents. I am aware of two main failures of alignment: (1) since predicting rare events is rewarded in proportion to their rareness, prediction markets heavily incentivise causing rare events to happen (I’ll call this the entropymarket problem); (2) it seems prediction markets would not be able to assign probability to existential risk, since you can’t collect on bets after everyone’s dead (I’ll call this the existential risk problem). I provide three formulations of (1) and solve two of them, and make some comments on (2). (Thanks to Scott for pointing out the second of these problems to me; I don’t remember who originally told me about the first problem, but also thanks.)
 
        22.  All Mathematicians are Trollable: Divergence of Naturalistic Logical Updates   post by Abram Demski 1022 days ago  Jessica Taylor, Patrick LaVictoire, Scott Garrabrant and Vladimir Slepnev like this  1 comment  
 The post on naturalistic logical updates left open the question of whether the probability distribution converges as we condition on more logical information. Here, I show that this cannot always be the case: for any computable probability distribution with naturalistic logical updates, we can show it proofs in an order which will prevent convergence. In fact, at any time, we can drive the probability of \(x\) up or down as much as we like, for a wide variety of sentences \(x\).
As an aid to intuition, I describe the theorem informally as “all mathematicians are trollable”. I was once told that there was an “all mathematicians go to Bayesian hell” theorem, based on the fact that a computable probability distribution must suffer arbitrarily large logloss when trying to model mathematics. The idea here is similar. We are representing the belief state of a mathematician with a computable probability distribution, and trying to manipulate that belief state by proving carefullyselected theorems to the mathematician.
 
    
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