How about a more concrete example: what’s the difference between observing that I one-box and setting that I one-box?
Without a specified causal graph for Newcomb’s, this is difficult to describe. (The difference is way easier to explain in non-Newcomb’s situations, I think, like the Smoker’s Lesion, where everyone agrees on the causal graph and the joint probability table.)
Suppose we adopt the graph Prediction ← Algorithm → Box, where you choose your algorithm, which perfectly determines both Omega’s Prediction and which Boxes you take. Omega reads your algorithm, fills the box accordingly, but then before you can make your choice Professor X comes along and takes control of you, which Omega did not predict. Professor X can force you to one-box or two-box, but that won’t adjust Omega’s prediction of you (and thus which boxes are filled). Professor X might realistically expect that he could make you two-box and receive all the money, whereas you could not expect that, because you know that two-boxing means that Omega would predict that you two-boxed.
(Notice that this is different from the interpretation in which Omega can see the future, which has a causal graph like Box → Prediction, in which case you cannot surprise Omega.)
P(A|B) = P(A&B)/P(B). That is the definition of conditional probability. You appear to be doing something else.
That’s what I’m describing, but apparently not clearly enough. P(A&B) was what I meant by the ‘probability of A once we throw out all cases where it isn’t B’, renormalized by dividing by P(B).
So, do(x) refers to someone else making the decision for you? Newcomb’s problem doesn’t traditionally have a “let Professor X mind-control you” option.
(Notice that this is different from the interpretation in which Omega can see the future, which has a causal graph like Box → Prediction, in which case you cannot surprise Omega.)
In your case, you cannot surprise Omega either. Only Professor X can.
So, do(x) refers to someone else making the decision for you?
Generally, no. Newcomb’s is weird, and so examples using it will be weird.
It may be clearer to imagine a scenario where there is a default value for some node, which may depend on other variables in the system, and that you could intervene to adjust it from the default to some other value you prefer.
For example, suppose you had a button that toggles whether a fire alarm is ringing. Suppose the fire alarm is not perfectly reliable, so that sometimes it rings when there isn’t a fire, and sometimes when there’s a fire it doesn’t ring. It’s very different for you to observe that the alarm is off, and then switch the alarm on, and for you to observe that the alarm is on.
If an EDT system only has two nodes, “fire” (which is unobserved) and “alarm” (which is observed), then it doesn’t have a way to distinguish between the alarm switching on its own (when we should update our estimate of fire) and the alarm switching because we pressed the button (when we shouldn’t update our estimate of fire). We could fix that by adding in a “button” node, or by switching to a causal network where fire points to alarm but alarm doesn’t point to fire. In general, the second approach is better because it lacks degrees of freedom which it should not have (and because many graph-based techniques scale in complexity based on the number of nodes, whereas making the edges directed generally reduces the complexity, I think). It’s also agnostic to how we intervene, which allows for us to use one graph to contemplate many interventions, rather than having a clear-cut delineation between decision and nature nodes.
In your case, you cannot surprise Omega either. Only Professor X can.
Right; I meant to convey that in the Omega sees the future case, not even Professor X can surprise Omega.
Hopefully, you can tell the difference between an alarm you triggered and an alarm that you did not.
I can, and you can, but imagine that we’re trying to program a robot to make decisions in our place, and we can’t trust the robot to have our intuition.* Suppose we give it a utility function that prefers there not being a fire to there being a fire, but don’t give it control over its epistemology (so it can’t just alter its beliefs so it never believes in fires).
If we program it to choose actions which maximize P(O|a) in the two-node system, it’ll shut off the alarm in the hopes that it will make a fire less likely. If we program it to choose actions which maximize P(O|do(a)), it won’t make that mistake.
* People have built-in decision theories for simple problems, and so it often seems strange to demo decision theories on problems small enough that the answer is obvious. But a major point of mathematical decision theories is to enable algorithmic computation of the correct decision in very complicated systems. Medical diagnosis causal graphs can have hundreds, if not thousands, of nodes- and the impact on the network of adjusting some variables might be totally nonobvious. Maybe some symptoms are such that treating them has no effect on the progress of the disorder, whereas other symptoms do have an effect on the progress of the disorder, and there might be symptoms that treating them makes it slightly more likely that the disorder will be cured, but significantly less likely that we can tell if the disorder is cured, and so calculating whether or not that tradeoff is worth it is potentially very complicated.
I can, and you can, but imagine that we’re trying to program a robot to make decisions in our place, and we can’t trust the robot to have our intuition.
A robot would always be able to tell if it’s an alarm it triggered. Humans are the ones that are bad at it. Did you actually decide to smoke because EDT is broken, or are you just justifying it like that and you’re actually doing it because you have smoking lesions?
If we program it to choose actions which maximize P(O|a) in the two-node system, it’ll shut off the alarm in the hopes that it will make a fire less likely.
Once it knows its sensor readings, knowing whether or not it triggers the alarm is no further evidence for or against a fire.
Without a specified causal graph for Newcomb’s, this is difficult to describe. (The difference is way easier to explain in non-Newcomb’s situations, I think, like the Smoker’s Lesion, where everyone agrees on the causal graph and the joint probability table.)
Suppose we adopt the graph Prediction ← Algorithm → Box, where you choose your algorithm, which perfectly determines both Omega’s Prediction and which Boxes you take. Omega reads your algorithm, fills the box accordingly, but then before you can make your choice Professor X comes along and takes control of you, which Omega did not predict. Professor X can force you to one-box or two-box, but that won’t adjust Omega’s prediction of you (and thus which boxes are filled). Professor X might realistically expect that he could make you two-box and receive all the money, whereas you could not expect that, because you know that two-boxing means that Omega would predict that you two-boxed.
(Notice that this is different from the interpretation in which Omega can see the future, which has a causal graph like Box → Prediction, in which case you cannot surprise Omega.)
That’s what I’m describing, but apparently not clearly enough. P(A&B) was what I meant by the ‘probability of A once we throw out all cases where it isn’t B’, renormalized by dividing by P(B).
So, do(x) refers to someone else making the decision for you? Newcomb’s problem doesn’t traditionally have a “let Professor X mind-control you” option.
In your case, you cannot surprise Omega either. Only Professor X can.
Generally, no. Newcomb’s is weird, and so examples using it will be weird.
It may be clearer to imagine a scenario where there is a default value for some node, which may depend on other variables in the system, and that you could intervene to adjust it from the default to some other value you prefer.
For example, suppose you had a button that toggles whether a fire alarm is ringing. Suppose the fire alarm is not perfectly reliable, so that sometimes it rings when there isn’t a fire, and sometimes when there’s a fire it doesn’t ring. It’s very different for you to observe that the alarm is off, and then switch the alarm on, and for you to observe that the alarm is on.
If an EDT system only has two nodes, “fire” (which is unobserved) and “alarm” (which is observed), then it doesn’t have a way to distinguish between the alarm switching on its own (when we should update our estimate of fire) and the alarm switching because we pressed the button (when we shouldn’t update our estimate of fire). We could fix that by adding in a “button” node, or by switching to a causal network where fire points to alarm but alarm doesn’t point to fire. In general, the second approach is better because it lacks degrees of freedom which it should not have (and because many graph-based techniques scale in complexity based on the number of nodes, whereas making the edges directed generally reduces the complexity, I think). It’s also agnostic to how we intervene, which allows for us to use one graph to contemplate many interventions, rather than having a clear-cut delineation between decision and nature nodes.
Right; I meant to convey that in the Omega sees the future case, not even Professor X can surprise Omega.
Hopefully, you can tell the difference between an alarm you triggered and an alarm that you did not.
I can, and you can, but imagine that we’re trying to program a robot to make decisions in our place, and we can’t trust the robot to have our intuition.* Suppose we give it a utility function that prefers there not being a fire to there being a fire, but don’t give it control over its epistemology (so it can’t just alter its beliefs so it never believes in fires).
If we program it to choose actions which maximize P(O|a) in the two-node system, it’ll shut off the alarm in the hopes that it will make a fire less likely. If we program it to choose actions which maximize P(O|do(a)), it won’t make that mistake.
* People have built-in decision theories for simple problems, and so it often seems strange to demo decision theories on problems small enough that the answer is obvious. But a major point of mathematical decision theories is to enable algorithmic computation of the correct decision in very complicated systems. Medical diagnosis causal graphs can have hundreds, if not thousands, of nodes- and the impact on the network of adjusting some variables might be totally nonobvious. Maybe some symptoms are such that treating them has no effect on the progress of the disorder, whereas other symptoms do have an effect on the progress of the disorder, and there might be symptoms that treating them makes it slightly more likely that the disorder will be cured, but significantly less likely that we can tell if the disorder is cured, and so calculating whether or not that tradeoff is worth it is potentially very complicated.
A robot would always be able to tell if it’s an alarm it triggered. Humans are the ones that are bad at it. Did you actually decide to smoke because EDT is broken, or are you just justifying it like that and you’re actually doing it because you have smoking lesions?
Once it knows its sensor readings, knowing whether or not it triggers the alarm is no further evidence for or against a fire.