Is Y a particular way of responding (e.g. Y = “the person dies”), or is it a variable that denotes whether the
person responds in that way (e.g. Y=1 if the person dies and 0 otherwise)? I think you meant the latter.
The latter.
How does averaging over propositional logical variables give you a random variable? I am afraid I am getting
confused by your terminology.
There is uncertainty about which unit u we are talking about (given by some p(u) we do not see). So instead of a propositional variable assignment Y(pa(y), u) = y, we have an event with a probability p{ Y(pa(y)) = y } = \sum{u : Y(pa(y),u) = y } p(u).
That sounds interesting. Do you have a link to a proof of this statement?
I am not sure I made a formal enough statement to prove. I guess:
(a) if you believe that your domain is acyclic causal, and
(b) you know what the causal structure is, and
(c) your utility is a function of the outcomes sitting in your causal system, and
(d) your actions on a variable embedded in your causal system break causal links operating from usual direct causes to the variable, and
(e) your domain isn’t “crazy” enough to demand adjustments along the lines of TDT,
then the right thing to do is to use CDT.
These preconditions hold in the HAART example. I am not sure exactly how to formalize (e) (I am not sure anyone does, this is a part of what is open).
The latter.
There is uncertainty about which unit u we are talking about (given by some p(u) we do not see). So instead of a propositional variable assignment Y(pa(y), u) = y, we have an event with a probability p{ Y(pa(y)) = y } = \sum{u : Y(pa(y),u) = y } p(u).
I am not sure I made a formal enough statement to prove. I guess:
(a) if you believe that your domain is acyclic causal, and
(b) you know what the causal structure is, and
(c) your utility is a function of the outcomes sitting in your causal system, and
(d) your actions on a variable embedded in your causal system break causal links operating from usual direct causes to the variable, and
(e) your domain isn’t “crazy” enough to demand adjustments along the lines of TDT,
then the right thing to do is to use CDT.
These preconditions hold in the HAART example. I am not sure exactly how to formalize (e) (I am not sure anyone does, this is a part of what is open).