Brutal Garcon

• Brutal Garcon 24 May 2026 20:56 UTC

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  in reply to: frontier64’s comment on: A rel­a­tively brief ex­pla­na­tion of Boltz­mann Brains

  The original post makes the (true) claim that any coherent observations are evidence against being a BB. I’m making the pedantic point that if you are almost certain that the real universe is BB dominated, then observing that that is indeed true suggests you might not be a BB.
  (Conversely: a person convinced that the universe is BB dominated would take observations suggesting it wasn’t as increasing their likelihood of being a BB. (and also the likelihood that they’re wrong about cosmology, but I’m assuming that starts tiny))

• Brutal Garcon 23 May 2026 14:03 UTC

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  on: A rel­a­tively brief ex­pla­na­tion of Boltz­mann Brains

  As an aside: if you have strong enough priors that the real cosmology will be dominated by Boltzmann Brains, then observing that you indeed seem to be in a cosmology that will be BB dominated should increase your probability of being a real observer.

Bru­tal Gar­con’s Shortform

• Brutal Garcon 17 Jul 2025 19:26 UTC

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  on: Bru­tal Gar­con’s Shortform

  I think the Kelly betting criterion always gives “sensible” results. By which I mean: there’s no hyper-st-petersburg-lottery for which maximizing expected log wealth means investing infinity times your current wealth, even if diverges; the Kelly criterion should always give you a finite fraction of your wealth (maybe >1) you ought to bet.

  (Sorry if this isn’t a novel idea, just noticed this and needed to put it down somewhere)

  Sketch proof for a toy model, I think this generalizes.

  Assume we are deciding what fraction, , of our wealth to wager on a bet that will return dollars, where is a random variable that takes values with probability . The fraction of our wealth that we don’t wager is unaffected.

  We assume , and to ensure the question is interesting, at least one and at least one (otherwise one should obviously invest as much/​little (respectively) as one can).

  Our expected log wealth (as a multiple of what we stated with), having invested is

  It is very easy to get this to diverge, e.g for all positive integers .

  The Kelly criterion says we should look for maxima of. Formally, we have

  and we want to solve .

  The first observation to make is that f’(q) converges for almost all values of : even if grows rapidly with , the summands above will tend to , whose sum must converge if the are probabilities.

  The exceptions are the simple poles at each , and a possible pole at if diverges—for this argument, we will assume it does, otherwise everything converges and we can do this the normal way.

  The second is that is negative everywhere, except at the poles mentioned above, so any stationary point of is a local maximum.

  Finally, consider the smallest ^[1]. There is an associated pole . is negative to the left of this pole, and positive to the right (as is negative everywhere); this is also true for the pole at . As there are no poles between and

  must be zero at some value of in that interval.

  1. ^

     I’m not sure what would happen if there were no smallest x_i < 1