Environmental and Logical Uncertainty: Reported Environmental Probabilities as Expected Environmental Probabilities under Logical Uncertainty

[Readers should be familiar with the Bayesian view of probability]

Let’s differentiate between environmental and logical uncertainty, and, consequently, environmental and logical probabilities. Environmental probabilities are the ones most of my readers will be closely familiar with. They are about the kinds of things that you can’t figure out even if you have infinite amounts of computing power until you have seen enough evidence.

Logical uncertainty is a different kind of uncertainty. For example, what is the 1,000th digit of the mathematical constant e? You know how e is defined. The definition uniquely implies what the 1,000th digit of e is and yet you’re uncertain as to the value of the 1,000th digit of e. Perhaps, you would assign logical probabilities: the probability that the 1,000th digit of e is 0 is 10%. For more detail, consider the MIRI paper Questions of Reasoning Under Logical Uncertainty by Nate Soares and Benja Fallenstein.

Now, I would like to draw attention to what happens when an agent is forced to quantify its environmental uncertainty, for example, when it needs to perform an expected value calculation. It’s good to think in terms of simplified artificial minds rather than humans, because human minds are so messy. If you think that any proper artificial intelligence would obviously know the values of its environmental probabilities, then think again: Proper ways of updating environmental probabilities based on new evidence (like Solomonoff induction) tend to be incomputable. So, an AI usually can’t quantify what exact values it should assign to certain environmental probabilities. This may remind you of the 1,000th digit of e: In both cases, there is a precise definition for something, but you can’t infer the exact numbers from that definition, because you and the AI are not intelligent enough.

Given that computing the exact probabilities is so difficult, the designers of an AI may fail with abandon and decide to implement some computable mechanism for approximating the probabilities. After all, “probabilities are subjective” anyway… Granted, an AI probably needs an efficient algorithm for quantifying its environmental uncertainty (or it needs to be able to come up with such a mechanism on its own). Sometimes you have to quickly compute the expected utility of a few actions, which requires numeric probabilities. However, any ambitious artificial intelligence should also keep in mind that there is a different, more accurate way of assigning these probabilities. Otherwise, it will forever and always be stuck with the programmers’ approximation.

The most elegant approach is to view the approximation of the correct environmental probabilities as a special case of logical induction (i.e. reasoning over logical uncertainty) possibly without even designing an algorithm for this specific task. On this view, we have logical meta-probability distributions over the correct environmental probabilities. Consider, for example, the probability P(T|E) that we assign to some physical theory T given our evidence E. There is some objectively correct subjective probability P(T|E) (assuming, for example, Solomonoff’s prior probability distribution), but the AI can’t calculate its exact value. It can, however, use logical induction to assign probabilities to statements like P(T|E) probabilities densities to statements like P(T|E)=0.368. These probabilities may be called logical meta-probabilities – they are logical probabilities about the correct environmental probabilities. With these meta-probabilities all our uncertainty is quantified again, which means we can perform  expected value calculations.

Let’s say we have to decide whether to taking action a. We know that if we take action a, one of the outcomes A, B and C will happen. The expected value of a is therefore

E[a] = P(A|a)*u(A) + P(B|a)*u(B) + P(C|a)*u(C),

where u(A), u(B) and u(C) denote the utilities an agent assigns to outcomes A, B and C, respectively. To find out the expected value of a given our lack of logical omniscience, we now calculate the “expected expected value”, where the outer expectation operator is a logical one:

E[E[a]] = E[P(A|a)*u(A) + P(B|a)*u(B) + P(C|a)*u(C)] = E[P(A|a)]*u(A) + E[P(B|a)]*u(B) + E[P(C|a)]*u(C).

The expected values E[P(A|a)], E[P(B|a)] and E[P(C|a)] are the expected environmental probabilities of the outcomes given a and can be computed using integrals. (In practice, these will have to be subject to approximation again. You can’t apply logical induction if you want to avoid an infinite regress.) These expected probabilities are the answers an agent/AI would give to questions like, “What probability do you assign to A happening?”

This view of reported environmental probabilities makes sense of a couple of intuitions that we have about environmental probabilities:

  • We don’t know which probabilities we assign to a given statement, even if we are convinced of Bayes’ theorem and a certain prior probability distribution.
  • We can update our environmental probability assignments without gathering new evidence. We can simply reconsider old evidence and compute a more accurate approximation of the proper Bayesian updating mechanism (e.g. via logical induction).
  • We can argue about probabilities with others and update. For example, people can bring new explanations of the data to our attention. This is not Bayesian evidence (ignoring that the source of such arguments may reveal its biases and beliefs through such argumentation). After all, we could have come up with these explanations ourselves. But these explanations can shift our logical meta-probabilities. (Formally: what external agents tell you can probably be viewed as (part of) a deductive process, see MIRI’s newest paper on logical induction.)

Introducing logical uncertainty into assigning environmental probabilities doesn’t solve the problem of assigning appropriate environmental probabilities. MIRI has described a logical induction algorithm, but it’s inefficient.

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