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habanero comments on What's Wrong with Evidential Decision Theory? - Less Wrong
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Let's look at a slightly more complicated example:
A patient comes in to the hospital because he's sick, his vitals are taken (C), and based on C, a doctor prescribes medication A. Sometime later, the patient dies (Y). Say this happens for lots of patients, and we form an empirical distribution p(C,A,Y) from these cases. Things we may want to represent:
marginal probability of death: p(Y)
the policy the doctor followed when prescribing the medicine: p(A|C)
the probability that someone would die given that they were given the medicine: p(Y|A)
the "causal effect" of medicine on death: ???
The issue with the "causal effect" is that the doctor's policy, if it is sensible, will be more likely to prescribe A to people who are already very sick, and not prescribe A to people who are mostly healthy. Thus it may very well turn out that p(Y|A) is higher than the probability p(Y) (this happens for example with HIV drugs). But surely this doesn't mean that medicine doesn't help! What we need is the probability of death given that the doctor arbitrarily decided to give medicine, regardless of background status C. This arbitrary decision "decouples" the influence of the patient's health status and the influence of the medicine in the sense that if we average over health status for patients after such a decision was made, we would get just the effect of the medicine itself.
For this we need to distinguish an "arbitrary" decision to give medicine, divorced from the policy. Call this arbitrary decision A* . You may then claim that we can just use rules of conditional probabilities on events Y,C,A, and A* , and we do not need to involve anything else. Of course, if you try to write down sensible axioms that relate A* with Y,C,A you will notice that essentially you are writing down standard axioms of potential outcomes (one way of thinking about potential outcomes is they are random variables after a hypothetical "arbitrary decision" like A* ). For example, if were to arbitrarily decide to prescribe medicine just precisely for the patients the doctor prescribed medicine to, we would get that p(Y|A) = p(Y|A* ) (this is known as the consistency axiom). You will find that your A* functions as an intervention do(a).
There is a big literature on how intervention events differ from observation events (they have different axiomatizations, for example). You may choose to use a notation for interventions that does not look different from observations, but the underlying math will be different if you wish to be sensible. That is the point. You need new math beyond math about evidence (which is what probability theory is).
It seems to me that we often treat EDT decisions with some sort of hindsight bias. For instance, given that we know that the action A (turning on sprinklers) doesn't increase the probability of the outcome O (rain) it looks very foolish to do A. Likewise, a DT that suggests doing A may look foolish. But isn't the point here that the deciding agent doesn't know that? All he knows is, that P(E|A)>P(E) and P(O|E)>P(O). Of course A still might have no or even a negative causal effect on O, but yet we have more reason the believe otherwise. To illustrate that, consider the following scenario:
Imagine you find yourself in a white room with one red button. You have no idea why you're there and what this is all about. During the first half hour you are undecided whether you should press the button. Finally your curiosity dominates other considerations and you press the button. Immediately you feel a blissful release of happiness hormones. If you use induction it seems plausible to infer that considering certain time intervalls (f.i. of 1 minute) P(bliss|button) > P(bliss). Now the effect has ceased and you wished to be shot up again. Is it now rational to press the button a second time? I would say yes. And I don't think that this is controversial. And since we can repeat the pattern with further variables it should also work with the example above.
From that point of view it doesn't seem to be foolish at all - talking about the sprinkler again - to have a non-zero credence in A (turning sprinkler on) increasing the probability of O (rain). In situations with that few knowledge and no further counter-evidence (which f.i. might suggest that A might have no or a negative influence on O) this should lead an agent to do A.
Considering the doctor, again, I think we have to stay clear about what the doctor actually knows. Let's imagine a doctor who lost all his knowledge about medicine. Now he reads one study which shows that P(Y|A) > P(Y). It seems to me that given that piece of information (and only that!) a rational doctor shouldn't do A. However, once he reads the next study he can figure out that C (the trait "cancer") is confounding the previous assessment because most patients who are treated with A show C as well, whereas ~A don't. This update (depending on the probability values respectively) will then lead to a shift favoring the action A again.
To summarize: I think many objections against EDT fail once we really clarify what the agent knows respectively. In scenarios with few knowledge EDT seems to give the right answers. Once we add further knowledge an EDT updates his beliefs and won’t turn on the sprinkler in order to increase the probability of rain. As we know it from the hindsight bias it might be difficult to really imagine what actually would be different if we didn’t know what we do know now.
Maybe that's all streaked with flaws, so if you find some please hand me over the lottery tickets ; )