Froolow comments on ‘Dissolving’ AI Risk – Parameter Uncertainty in AI Future Forecasting

[Froolow]

Hmm… I don’t see a contradiction here. I note you skimmed some of the methods, so it might perhaps help explain the contradiction to read the second half of section 3.3.2?

The bullet I bite is the first—most survey respondents are wrong, because they give point probabilities (which is what I asked for, in fairness) whereas in reality there will be uncertainty over those probabilities. Initiatively we might think that this uncertainty doesn’t matter because it will ‘cancel out’ (ie every time you are uncertain in a low direction relative to the truth I am uncertain in a high direction relative to the truth) but in reality—given specific structural assumptions in the Carlsmith Model—this is not true. In reality, the low-end uncertainty compounds and the high-end uncertainty is neutered, which is why you end up with an asymmetric distribution favouring very low-risk outcomes.

[harfe]

Thanks for biting a bullet, I think I am making progress in understanding your view.

I also realized that part of my “feeling of inconsistency” comes from not having realized that the table in section 3.2 reports geometric mean of odds instead of the average, and where the average would be lower.

Lets say we have a 2-parameter Carlsmith model, where we estimate probabilities $P\left(A\right)$ and $P\left(B|A\right)$, in order to get to a final estimate of the probability $P(A\cap B)$. Lets say we have uncertainty over our probability estimates, and we estimate $P\left(A\right)$ using a random variable $X$, and estimate $P\left(B|A\right)$ using a random variable $Y$. To make the math easier, I am going to assume that $X,Y$ are discrete (I can repeat it for a more general case, eg using densities if requested): We have $k$ possible estimates $a_i$ for $P\left(A\right)$, and $p_i:=P(X=a_i)$ is the probability that $X$ assigns the value $a_i$ for our estimate of $P\left(A\right)$. Similarly, $b_i$ are estimates for $P\left(B|A\right)$ that $Y$ outputs with probability $q_i:=P(Y=b_i)$. We also have ${\sum }_{i=1}^k{p}_i={\sum }_{i=1}^k{q}_i=1$.

Your view seems to be something like “To estimate $P(A\cap B)$, we should sample from $X$ and $Y$, and then compute the geometric mean of odds for our final estimate.”

Sampling from $X$ and $Y$, we get values $a_i\cdot b_i$ with probability $p_i\cdot q_i$, and then taking the geometric mean of odds would result in the formula

$$P(A\cap B)=\prod _{i=1}^k\prod _{j=1}^k{\left(\frac{a_i{b}_j}{1-a_i{b}_j}\right)}^{p_i{q}_j}.$$

Whereas my view is “We should first collapse the probabilities by taking the mean, and then multiply”, that is we first calculate $P\left(A\right)={\sum }_{i=1}^k{a}_i{p}_i$ and $P\left(B|A\right)={\sum }_{j=1}^k{b}_i{q}_i$, for a final formula of

$$P(A\cap B)=\left(\sum _{i=1}^k{a}_i{p}_i\right)\left(\sum _{j=1}^k{b}_j{q}_j\right).$$

And you are also saying ”$P(A\cap B)=P\left(A\right)\cdot P\left(B|A\right)$ is still true, but the above naive estimates for $P\left(A\right)$ and $P\left(B\right)$ are not good, and should actually be different (and lower than typical survey respondents in the case of AI xrisk estimates).” (I can’t derive a precise formula from your comments or my skim of the article, but I don’t think thats a crucial issue.)

Do I characterize your view roughly right? (Not saying that is your whole view, just parts of it).