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You should update by +-1% on AI doom surprisingly frequently
This is just a fact about how stochastic processes work. If your p(doom) is Brownian motion in 1% steps starting at 50% and stopping once it reaches 0 or 1, then there will be about 50^2=2500 steps of size 1%. This is a lot! If we get all the evidence for whether humanity survives or not uniformly over the next 10 years, then you should make a 1% update 4-5 times per week. In practice there won't be as many due to heavy-tailedness in the distribution concentrating the updates in fewer events, and the fact you don't start at 50%. But I do believe that evidence is coming in every week such that ideal market prices should move by 1% on maybe half of weeks, and it is not crazy for your probabilities to shift by 1% during many weeks if you think about it often enough. [Edit: I'm not claiming that you should try to make more 1% updates, just that if you're calibrated and think about AI enough, your forecast graph will tend to have lots of >=1% week-to-week changes.]
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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 co...
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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.
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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 t
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Oops, you're correct about the typo and also about how this doesn't restrict belief change to Brownian motion. Fixing the typo.