These results from my conversations with Charlie Steiner at the May 2931 MIRI Workshop on Logical Uncertainty will primarily be of interest to people who’ve read section 2.4 of Paul Christiano’s NonOmniscience paper.
If we write a reasoner that keeps track of probabilities of a collection of sentences \(\varphi_1,\dots,\varphi_n\) (that grows and shrinks as the reasoner explores), we need some way of tracking known relationships between the sentences. One way of doing this is to store the pairwise probability distributions, ie not only \(\Pr(\varphi_i)\) for all \(i\) but also \(\Pr(\varphi_i\wedge\varphi_j)\) for all \(i,j\).
If we do this, a natural question to ask is: how can we update this data structure if we learn that eg \(\varphi_1\) is true?
We’ll refer to the updated probabilities as \(\Pr(\cdot\varphi_1)\).
It’s fairly reasonable for us to want to set \(\Pr(\varphi_i\varphi_1):=\Pr(\varphi_i\wedge\varphi_1)/\Pr(\varphi_1)\); however, it’s less clear what values to assign to \(\Pr(\varphi_i\wedge\varphi_j\varphi_1)\), because we haven’t stored \(\Pr(\varphi_i\wedge\varphi_j\wedge\varphi_1)\).
One option would be to find the maximum entropy distribution over truth assignments to \(\varphi_1,\dots,\varphi_n\) under the constraint that the stored pairwise distributions are correct. This seems intractable for large \(n\); however, in the spirit of locality, we could restrict our attention to the joint truth value distribution of \(\varphi_1,\varphi_i,\varphi_j\). Maximizing its entropy is simple (it boils down to either convex optimization or solving a cubic), and yields a plausible candidate for \(\Pr(\varphi_i\wedge\varphi_j\wedge\varphi_1)\) that we can derive \(\Pr(\varphi_i\wedge\varphi_j\varphi_1)\) from. I’m not sure what global properties this has, for example whether it yields a positive semidefinite matrix \((\Pr(\varphi_i\wedge\varphi_j))_{i,j}\).
A different option, as noted in section 2.4.2, is to observe that the matrix \((\Pr(\varphi_i\wedge\varphi_j))_{i,j}\) must be positive semidefinite under any joint distribution for the truth values. This means we can consider a zeromean multivariate normal distribution with this matrix as its covariance; then there’s a closedform expression for the KullbackLeibler divergence of two such distributions, and this can be used to define a sort of conditional distribution, as is done in section 2.4.3.
However, as the paper remarks, this isn’t a very familiar way of defining these updated probabilities. For example, it lacks the desirable property that \(\Pr(\varphi_i\varphi_1)=\Pr(\varphi_i\wedge\varphi_1)/\Pr(\varphi_1)\).
Fortunately, there is a natural construction that combines these ideas: namely, if we consider the maximumentropy distribution for the truth assignment vector \((1_{\varphi_1},\dots,1_{\varphi_n})\) with the given second moments \(\operatorname{E}(1_{\varphi_i}1_{\varphi_j})\), but relax the requirement that their values be in \(\{0,1\}\), then we find a multivariate normal distribution \[\mathcal{N}\left(\left(\Pr(\varphi_i)\right)_i,\left(\Pr(\varphi_i\wedge\varphi_j)\Pr(\varphi_i)\Pr(\varphi_j)\right)_{i,j}\right)\mbox{.}\] If we wish to update this distribution after observing \(\varphi_1\) by finding the candidate distribution \((1_{\varphi_1},\dots,1_{\varphi_n}\varphi_1)\) of highest relative entropy with \(\Pr(1_{\varphi_1}=1\varphi_1)=1\), as proposed in the paper, then we will get the multivariate normal conditional distribution \[\mathcal{N}\left(\left(\Pr(\varphi_1\wedge\varphi_i)/\Pr(\varphi_1)\right)_i,\left(\Pr(\varphi_i\wedge\varphi_j)\Pr(\varphi_i)\Pr(\varphi_j)\frac{(\Pr(\varphi_1\wedge\varphi_i)\Pr(\varphi_1)\Pr(\varphi_i))(\Pr(\varphi_1\wedge\varphi_j)\Pr(\varphi_1)\Pr(\varphi_j))}{\Pr(\varphi_1)\Pr(\varphi_1)^2}\right)_{ij}\right)\mbox{.}\]
Note that this generally has \(\operatorname{Var}\left(1_{\varphi_i}\middle\varphi_1\right)\not=\operatorname{E}\left(1_{\varphi_i}\middle\varphi_1\right)\left(1\operatorname{E}\left(1_{\varphi_i}\middle\varphi_1\right)\right)\), which is a mismatch; this is related to the fact that a conditional variance in a multivariate normal is never higher than the marginal variance, which is an undesirable feature for a distribution over truthvalues.
This is also related to other undesirable features; for example, if we condition on more than one sentence, we can arrive at conditional probabilities outside of \([0,1]\). (For example if 3 sentences have \(\Pr(\varphi_1)=\Pr(\varphi_2)=\Pr(\varphi_3)=\frac{1}{3},\Pr(\varphi_1\wedge\varphi_2)=\Pr(\varphi_1\wedge\varphi_3)=\Pr(\varphi_2\wedge\varphi_3)=\varepsilon\) then this yields \(\Pr(\varphi_3\varphi_1,\varphi_2)=\frac{1+15\varepsilon}{1+9\varepsilon}\approx 1\); this makes sense because this prior is very confident that \(1_{\varphi_1}+1_{\varphi_2}+1_{\varphi_3}\approx 1\), with standard deviation \(\sqrt{6\varepsilon}\).)
Intermediate relaxations that lack these particular shortcomings are possible, such as the ones that restrict the relaxed \(1_{\varphi_1},\dots,1_{\varphi_n}\) to the sphere \(\sum_i (2x_i1)^2=n\) or ball \(\sum_i (2x_i1)^2\leq n\). Then the maximum entropy distribution, similarly to a multivariate normal distribution, has quadratic logdensity, though the Hessian of the quadratic may have nonnegative eigenvalues (unlike in the normal case). In the spherical case, this is known as a FisherBingham distribution.
Both of these relaxations seem difficult to work with, eg to compute normalizing constants for; furthermore I don’t think the analogous updating process will share the desirable property that \(\Pr(\varphi_i\varphi_1)=\Pr(\varphi_i\wedge\varphi_1)/\Pr(\varphi_1)\). However, the fact that these distributions allow updating by relaxed conditioning, keep (fully conditioned) truthvalues between 0 and 1, and have reasonable (at least, possiblyincreasing) behavior for conditional variances, makes them seem potentially appealing.
