Rich,
Thanks very much for your insightful criticism.
You are absolutely correct if the arc from Coffee_Triggers_Headache
is interpreted causally, which is the natural interpretation given
that I was talking about a causal graph. So let me apologize for
creating confusion. However, there is a hidden factor causal model
that makes the causal claims I described.
Make Hidden_Factor cause John_Drinks_Coffee with probability (almost)
100%. Also make Hidden_Factor cause Hidden_Trigger with probability
equal to the noisy-or trigger probability for coffee. Then make
Coffee_Triggers_Headache have probability 100% if John_Drinks_Coffee
and Hidden_Trigger are both true, and 0% otherwise. Then make
Other_Triggers_Headache have probability:
- 100% if Hidden_Trigger and Other are both true;
- the noisy-or trigger probability for other if Mystery_Trigger
is false and Other is true; and
- 0% otherwise.
Then set the prior probability of Mystery_Factor equal to the prior
probability that John drinks coffee.
This model is equivalent to the Noisy-Or and makes the causal claims
I described in my previous email. If we turn off John's coffee
drinking and the headache, but set Hidden_Factor to its conditional
probability given Headache and John_Drinks_Coffee, the probability of
Headache is much higher than the counterfactual probability for the
standard Noisy-or.
You are also right that the example I gave does not contradict
Charles' claim. I didn't intend to argue against his claim. My
intent was simply to note that causal claims are stronger than claims
about conditional probabilities on observed random variables.
Sometimes we can verify causal claims by experimental manipulation,
but even so, our analysis always depends on auxiliary assumptions
about the causal mechanism. We never can know what would have
happened if events had unfolded differently from how they unfolded.
Sorry for the confusion.
Kathy
>Kathy,
>
>It does not seem that you are saying what I think you intend to say.
>Consider this sentence:
>
> For example, we could draw an arc from
>>Coffee_Triggers_Headache to Other_Triggers_Headache, and change the
>>probability to 100% that Other_Triggers_Headache is true when
>>Coffee_Triggers_Headache is true, but leave everything else as it is.
>>Check it out. This doesn't change the joint distribution on (Coffee,
>>Other_causes, Headache). But it makes different counterfactual
>>claims. The claim this alternate model makes is that if John drinks
>>coffee, and there is also some other cause of a headache present, and
>>the coffee triggers a headache, then so does the other cause, so that
>>if you removed the coffee, there would still be a headache.
>
>If you removed the coffee, coffee would no longer trigger the
>headache (even if its inhibitor was not present), which means
>coffee-trigger-headache would no longer causes
>other-triggers-headache. So why would there still be a headache?
>
>
>Now in what follows you talk about a common cause of drink-coffee
>and other-trigger. This model is consistent with your claim that the
>headache would still be there if we removed coffee. However, it is
>not the model you describe above.
>
>>This
>>might occur, for example, if whatever factor causes John to
>>experience a desire to drink coffee also causes these other factors
>>to trigger a headache. Does this seem farfetched? It is not ruled
>>out by the observables! In fact, it makes EXACTLY the same
>>statistical claims about observables that the other model does.
>Can you clarify,
>
>Also, although all this in interesting, it does not seem to
>invalidate Charle's argument, which only concerns the observed
>probability distributions.
>
>
>Rich
