Inferring counterfactuals about individuals requires a model that is
stronger than a Bayesian network. It requires a model of the causal
mechanism itself. We can have exactly the same conditional
probabilities and draw different inferences about causality,
depending on the (often untestable) assumptions we make about how the
(unobserved and often unobservable) casual mechanism operates.
For example, suppose we modeled John's headache using a noisy-OR,
where coffee was one cause and there is a list of other causes, each
with an independent probability of triggering a headache. We can
collapse this into a CPT on observable variables only:
John_Drank_Coffee --> John_Headache <-- Other_causes
We can also expand the model to include the unobservable triggers:
John_Drank_Coffee --> Coffee-Triggers_Headache
|
v
John_Headache
^
|
Other_Triggers_Headache <-- Other_causes
The posterior probability of Other_Triggers_Headache is true given
that John_Headache and John_Drank_Coffee are both true is the
probability that John would still have had a headache if he had not
drunk the coffee.
But that is only if we accept the causal claims of the noisy-OR. We
can define an alternative causal model with the VERY SAME joint
probability distribution on observables, that makes different
counterfactual claims. 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. 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.
This is the same kind of argument Sir Ronald Fisher made against the
claim that smoking causes cancer (very influentially for a while, I
might add!). It might be, he argued, that whatever it is that causes
people to crave tobacco also causes cancer. If they didn't smoke,
they would still get cancer, because it's this other factor that is
causing cancer. (Fisher was a pipe smoker, BTW.) The rebuttal that
persuaded people was to allow Fisher his assumption, to estimate the
model implied by his assumption, and to show that in order to get the
observed correlation between smoking and cancer, this alternate
factor that causes both a craving for tobacco and cancer would have
to be incredibly strong -- vastly stronger than typical statistical
effects in medicine and the social sciences.
Anyway, the point is that there typically are many different
counterfactual models consistent with a given distribution on
observables.
Kathy Laskey
At 5:51 AM -0700 9/23/03, Charles R. Twardy wrote:
>On Fri, 12 Sep 2003, Thomas Richardson wrote:
>}[As a side note: Structural equation models can be interpreted as relating
>}to individual causal effects - i.e. inferring causes from effects, "Would
>}John have a headache had he not drunk coffee this morning given that he
>}did drink coffee and does have a headache", but this has not been done by
>}econometricians - whereas this is not true of "causal nets".
>
>Pearl gave examples of how to do that if you have another variable P, the
>population John comes from. I believe it would go like this:
>
> Coffee --> John_Headache <-- Population
>
> * Set coffee and headache
> * Observe new distribution P' (we learned something abt John)
> * Unset coffee and headache, and make P' the new prior on P
> * Set no coffee
> * Observe probability John would have had a headache
>
>- -crt
>
>
>- --
>Charles R. Twardy, Res.Fellow, Monash University, School of CSSE
>ctwardy at alumni indiana edu +61(3) 9905 5823 (w) 5146 (fax)
>
>"in much of the rest of the world, rich people live in gated
> communities and drink bottled water." --Jared Diamond
