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IlyaShpitser comments on Evidential Decision Theory, Selection Bias, and Reference Classes - Less Wrong
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Ok -- the data is as I describe above. You don't get any more data. What is your EDT solution to this example?
You didn't give any data, just a problem description. Am I to assume that there is a bunch of {A0, L0, A1, Y} records are available? And you say that the policy for giving A1 is known, is the information that this decision is based on (health status) also available?
In any case, you end up with the problem of estimating a causal structure from observational data, which is a challenging problem. But I don't see what this has to do with EDT vs another DT. Wouldn't this other decision theory face exactly the same problem?
You have (let's say infinitely many to avoid dealing with stats issues) records for { A0, L0, A1, Y }. You know they come from the causal graph I specified (complete with an unobserved confounder for health status on which no records exist. You don't need to learn the graph, you just need to tell me whether HAART is killing people or not and why, using EDT.
There is no single 'right answer' in this case. The answer will depend on your prior for the confounder.
As others have noted, the question "is HAART killing people?" has nothing to do with EDT, or any other decision theory for that matter. The question that decision theories answer is "should I give HAART to person X?"
I think I disagree with both of these assertions. First, there is the "right answer," and it has nothing to do with priors or Bayesian reasoning. In fact there is no model uncertainty in the problem -- I gave you "the truth" (the precise structure of the model and enough data to parameterize it precisely so you don't have to pick or average among a set of alternatives). All you have to do is answer a question related to a single parameter of the model I gave you. The only question is which parameter of the model I am asking you about. Second, it's easy enough to rephrase my question to be a decision theory question (I do so here:
http://lesswrong.com/lw/hwq/evidential_decision_theory_selection_bias_and/9cdk).
To quote your other comment:
You put the patient on HAART if and only if
V(HAART) > V(!HAART), whereIn these formulas
HAARTmeans "(decide to) put this patient on HAART" anddeathmeans "this patient dies".For concreteness, we can assume that the utility of death is low, say 0, while the utility of !death is positive. Then the decision reduces to
So if you give me
P(!death|HAART)andP(!death,!HAART)then I can give you a decision.Ok. This is wrong. The problem is P(death|HAART) isn't telling you whether HAART is bad or not (due to unobserved confounding). I have already specified that there is confounding by health status (that is, HAART helps, but was only given to people who were very sick). What you need to compare is
for various values of A1, and A0.
Note that I defined
HAARTas "put this patient on HAART", not the probability of death when giving HAART in general (maybe I should have used a different notation).If I understand your model correctly then
with the confounding variable H1, the health at time t=L0, which influences the choice of A1. You didn't specify how L0 was determined, is it fixed or does it also depend on the patient's health? Your formula above suggests that it depends only on the choice A0.
Now a new patient comes in, and you want to know whether you should pick A0=true/false and A1=true/false. Now for the new patient x, you want to estimate
P(death[x] | A0[x],A1[x]). If it was just about A0[x], then it would be easy, since the assignment was randomized, so we know that A0 is independent of any confounders. But this is not true for A1, in fact, we have no good data with which to estimate A1[x], since we only have samples where A1 was chosen according to the health-status based policy.Yes, you should have. The notation "P(!death|HAART)" means "find every record with HAART, and calculate the percentage of them with !death." This is how EDT as an epistemic approach generates numbers to use for decisions. Why am I specifying as an epistemic approach? Because EDT and CDT ask for different sorts of information with which to make decisions, and thus have different natural epistemologies. CDT asks for "P(!death| do(HAART))", which is not the sort of information EDT asks for, and thus not the sort of information an EDT model has access to.
To go back to an earlier statement:
IlyaShpitser is asking you how you would calculate those from empirical data. The EDT answer uses the technical notation you used before, and it's the suboptimal way to do things.
Really? My impression is that the observational records are good enough to get some knowledge. (Indeed, they must be good enough; lives are on the line, and saying "I don't have enough info" will result in more deaths than "this is the best I can do with the existing info.")
EDT does not answer this question, at least, the definition of EDT I found on wikipedia makes no mention of it. Can you point me to a description of EDT that includes the estimation of probabilities?
I should have said "to estimate the effect of A1[x]".
Sure, you can do something to make an estimate. But as I understand it, estimating causal models (which is what you need to estimate A1[x]) from observational data is a hard problem. That is why clinical trials use randomization, and studies that don't try very hard to control for all possible confounders.