But… Why would p(doom) move like Brownian motion until stopping at 0 or 1?
I don’t disagree with your conclusions, there’s a lot of evidence coming in, and if you’re spending full time or even part time thinking about alignment, a lot of important updates on the inference. But assuming a random walk seems wrong.
Is there a reason that a complex, structured unfolding of reality would look like a random walk?
Although I’ve seen it mentioned that technically the change in the belief on a Bayesian should follow a Martingale, and Brownian motion is a martingale.
I’m not super technically strong on this particular part of the math. Intuitively it could be that in a bounded reasoner which can only evaluate programs in P, any pattern in its beliefs that can be described by an algorithm in P is detected and the predicted future belief from that pattern is incorporated into current beliefs. On the other hand, any pattern described by an algorithm in EXPTIME∖P can’t be in the class of hypotheses of the agent, including hypotheses about its own beliefs, so EXPTIME patterns persist.
Technically, the probability assigned to a hypothesis over time should be the martingale (i.e. have expected change zero); this is just a restatement of the conservation of expected evidence/law of total expectation.
The random walk model that Thomas proposes is a simple model that illustrates a more general fact. For a martingale(Sn)n∈Z+, the variance of St is equal to the sum of variances of the individual timestep changes Xi:=Si−Si−1 (and setting S0:=0): Var(St)=∑ti=1Var(Xi). Under this frame, insofar as small updates contribute a large amount to the varianceof each update Xi, then the contribution to the small updates to the credences must also be large (which in turn means you need to have a lot of them in expectation[1]).
Note that this does not require any strong assumption besides that the the distribution of likely updates is such that the small updates contribute substantially to the variance. If the structure of the problem you’re trying to address allows for enough small updates (relative to large ones) at each timestep, then it must allow for “enough” of these small updates in the sequence, in expectation.
While the specific +1/-1 random walk he picks is probably not what most realistic credences over time actually look like, playing around with it still helps give a sense of what exactly “conservation of expected evidence” might look/feel like. (In fact, in the dath ilan of Swimmer’s medical dath ilan glowfics, people do use a binary random walk to illustrate how calibrated beliefs typically evolve over time.)
Now, in terms of if it’s reasonable to model beliefs as Brownian motion (in the standard mathematical sense, not in the colloquial sense): if you suppose that there are many, many tiny independent additive updates to your credence in a hypothesis, your credence over time “should” look like Brownian motion at a large enough scale (again in the standard mathematical sense), for similar reasons as to why the sum of a bunch of independent random variables converges to a Gaussian. This doesn’t imply that your belief in practice should always look like Brownian motion, any more than the CLT implies that real world observables are always Gaussian. But again, the claim Thomas makes carries thorough
I also make the following analogy in my head: Bernouli:Gaussian ~= Simple Random Walk:Brownian Motion, which I found somewhat helpful. Things irl are rarely independent/time-invarying Bernoulli or Gaussian processes, but they’re mathematically convenient to work with, and are often ‘good enough’ for deriving qualitative insights.
Note that you need to apply something like the optional stopping theorem to go from the case of ST for fixed T, to the case of Sτ where τ is the time you reach 0 or 1 credence and the updates stop.
I get conservation of expected evidence. But the distribution of belief changes is completely unconstrained.
Going from the class martingale to the subclass Brownian motion is arbitrary, and the choice of 1% update steps is another unjustified arbitrary choice.
I think asking about the likely possible evidence paths would improve our predictions.
You spelled it conversation of expected evidence. I was hoping there was another term by that name :)
To be honest, I would’ve preferred if Thomas’s post started from empirical evidence (e.g. it sure seems like superforecasters and markets change a lot week on week) and then explained it in terms of the random walk/Brownian motion setup. I think the specific math details (a lot of which don’t affect the qualitative result of “you do lots and lots of little updates, if there exists lots of evidence that might update you a little”) are a distraction from the qualitative takeaway.
A fancier way of putting it is: the math of “your belief should satisfy conservation of expected evidence” is a description of how the beliefs of an efficient and calibrated agent should look, and examples like his suggest it’s quite reasonable for these agents to do a lot of updating. But the example is not by itself necessarily a prescription for how your belief updating should feel like from the inside (as a human who is far from efficient or perfectly calibrated). I find the empirical questions of “does the math seem to apply in practice” and “therefore, should you try to update more often” (e.g., what do the best forecasters seem to do?) to be larger and more interesting than the “a priori, is this a 100% correct model” question.
But… Why would p(doom) move like Brownian motion until stopping at 0 or 1?
I don’t disagree with your conclusions, there’s a lot of evidence coming in, and if you’re spending full time or even part time thinking about alignment, a lot of important updates on the inference. But assuming a random walk seems wrong.
Is there a reason that a complex, structured unfolding of reality would look like a random walk?
Because[1] for a Bayesian reasoner, there is conservation of expected evidence.
Although I’ve seen it mentioned that technically the change in the belief on a Bayesian should follow a Martingale, and Brownian motion is a martingale.
I’m not super technically strong on this particular part of the math. Intuitively it could be that in a bounded reasoner which can only evaluate programs in P, any pattern in its beliefs that can be described by an algorithm in P is detected and the predicted future belief from that pattern is incorporated into current beliefs. On the other hand, any pattern described by an algorithm in EXPTIME∖P can’t be in the class of hypotheses of the agent, including hypotheses about its own beliefs, so EXPTIME patterns persist.
Technically, the probability assigned to a hypothesis over time should be the martingale (i.e. have expected change zero); this is just a restatement of the conservation of expected evidence/law of total expectation.
The random walk model that Thomas proposes is a simple model that illustrates a more general fact. For a martingale(Sn)n∈Z+, the variance of St is equal to the sum of variances of the individual timestep changes Xi:=Si−Si−1 (and setting S0:=0): Var(St)=∑ti=1Var(Xi). Under this frame, insofar as small updates contribute a large amount to the variance of each update Xi, then the contribution to the small updates to the credences must also be large (which in turn means you need to have a lot of them in expectation[1]).
Note that this does not require any strong assumption besides that the the distribution of likely updates is such that the small updates contribute substantially to the variance. If the structure of the problem you’re trying to address allows for enough small updates (relative to large ones) at each timestep, then it must allow for “enough” of these small updates in the sequence, in expectation.
While the specific +1/-1 random walk he picks is probably not what most realistic credences over time actually look like, playing around with it still helps give a sense of what exactly “conservation of expected evidence” might look/feel like. (In fact, in the dath ilan of Swimmer’s medical dath ilan glowfics, people do use a binary random walk to illustrate how calibrated beliefs typically evolve over time.)
Now, in terms of if it’s reasonable to model beliefs as Brownian motion (in the standard mathematical sense, not in the colloquial sense): if you suppose that there are many, many tiny independent additive updates to your credence in a hypothesis, your credence over time “should” look like Brownian motion at a large enough scale (again in the standard mathematical sense), for similar reasons as to why the sum of a bunch of independent random variables converges to a Gaussian. This doesn’t imply that your belief in practice should always look like Brownian motion, any more than the CLT implies that real world observables are always Gaussian. But again, the claim Thomas makes carries thorough
I also make the following analogy in my head: Bernouli:Gaussian ~= Simple Random Walk:Brownian Motion, which I found somewhat helpful. Things irl are rarely independent/time-invarying Bernoulli or Gaussian processes, but they’re mathematically convenient to work with, and are often ‘good enough’ for deriving qualitative insights.
Note that you need to apply something like the optional stopping theorem to go from the case of ST for fixed T, to the case of Sτ where τ is the time you reach 0 or 1 credence and the updates stop.
I get conservation of expected evidence. But the distribution of belief changes is completely unconstrained.
Going from the class martingale to the subclass Brownian motion is arbitrary, and the choice of 1% update steps is another unjustified arbitrary choice.
I think asking about the likely possible evidence paths would improve our predictions.
You spelled it conversation of expected evidence. I was hoping there was another term by that name :)
To be honest, I would’ve preferred if Thomas’s post started from empirical evidence (e.g. it sure seems like superforecasters and markets change a lot week on week) and then explained it in terms of the random walk/Brownian motion setup. I think the specific math details (a lot of which don’t affect the qualitative result of “you do lots and lots of little updates, if there exists lots of evidence that might update you a little”) are a distraction from the qualitative takeaway.
A fancier way of putting it is: the math of “your belief should satisfy conservation of expected evidence” is a description of how the beliefs of an efficient and calibrated agent should look, and examples like his suggest it’s quite reasonable for these agents to do a lot of updating. But the example is not by itself necessarily a prescription for how your belief updating should feel like from the inside (as a human who is far from efficient or perfectly calibrated). I find the empirical questions of “does the math seem to apply in practice” and “therefore, should you try to update more often” (e.g., what do the best forecasters seem to do?) to be larger and more interesting than the “a priori, is this a 100% correct model” question.
Oops, you’re correct about the typo and also about how this doesn’t restrict belief change to Brownian motion. Fixing the typo.