You might argue that by not specifying how the conditional probabilities P(O=oj|a) are calculated, I have taken out the interesting part of the decision theory.
Basically, yes. What separates EDT and CDT is whether they condition on the joint probability distribution or use the do operator on the causal graph; there’s no other difference. This is a productive difference, and so obscuring it is counterproductive.
What you call “EEDT” I would call “expected value calculation,” and then I would use “decision theory” to describe the different ways of estimating conditional probabilities (as you put it). It is right that expected value calculation is potentially nonobvious, and so saying “we use expected values” is meaningful and important, but I think that the language to convey the concepts you want to convey already exists, and you should use the language that already exists.
I still think you can mess around with the notion of random variables, as described in this post, to get a better understanding of what you are actually prediction. But I suppose that this can be confusing to others.
What you call “EEDT” I would call “expected value calculation,” and then I would use “decision theory” to describe the different ways of estimating conditional probabilities (as you put it).
I think there are three steps in the reduction:
A “decision problem”: “given my knowledge, what action should I take”, which is answered by a “decision theory/procedure”
Expected values: “given my knowledge, what is the expected value of taking action A”
Conditional probability estimation: “given my knowledge, what is my best guess for the probability P(X|Y)”
The reduction from 1 to 2 is fairly obvious, just take the action that maximizes expected value. I think this is common to all decision theories.
The reduction of 2 to 3 is what is done by EEDT. To me this step was not so obvious, but perhaps there is a better name for it.
Basically, yes. What separates EDT and CDT is whether they condition on the joint probability distribution or use the do operator on the causal graph; there’s no other difference.
So the difference is in how to solve step 3, I agree. I wasn’t trying to obscure anything, of course. Rather, I was trying to advocate that we should focus on problem 3 directly, instead of problem 1.
It is right that expected value calculation is potentially nonobvious, and so saying “we use expected values” is meaningful and important, but I think that the language to convey the concepts you want to convey already exists, and you should use the language that already exists.
Do you have some more standard terms I should use?
The reduction from 1 to 2 is fairly obvious, just take the action that maximizes expected value. I think this is common to all decision theories.
This doesn’t hold in prospect theory, in which probabilities are scaled before they are used. (Prospect theory is a descriptive theory of buggy human decision-making, not a prescriptive decision theory.) [Edit] Actually, I think this is a disagreement in how you go from 2 to 3. In order to get a disagreement on how you go from 1 to 2, you’d have to look at the normal criticisms of VNM.
It’s also not universal in decision theories involving intelligent adversaries. Especially in zero-sum games, you see lots of minimax solutions (minimize my opponent’s maximum gain). This difference is somewhat superficial, because minimax does reduce to expected value calculation with the assumption that the opponent will always choose the best option available to him by our estimation of his utility function. If you relax that assumption, then you’re back in EV territory, but now you need to elicit probabilities on the actions of an intelligent adversary, which is a difficult problem.
There’s a subtle point here about whether the explicit mechanisms matter. Expected value calculation is flexible enough that any decision procedure can be expressed in terms of expected value calculation, but many implicit assumptions in EV calculation can be violated by those procedures. For example, an implicit assumption of EV calculation is consequentialism in the sense that the preferences are over outcomes, but one could imagine a decision procedure where the preferences are over actions instead of over outcomes, or more realistically action-outcome pairs. We can redefine the outcomes to include the actions, and thus rescue the procedure, but it seems worthwhile to have a separation between consequentialist and deontological positions. If you relax the implicit assumption that probabilities should be treated linearly, then prospect theory can be reformulated as an EV calculation, where outcomes are outcome-probability pairs. But again it seems worthwhile to distinguish between decision algorithms which treat probabilities linearly and nonlinearly.
So the difference is in how to solve step 3, I agree. I wasn’t trying to obscure anything, of course. Rather, I was trying to advocate that we should focus on problem 3 directly, instead of problem 1. … Do you have some more standard terms I should use?
I’m not sure how much we disagree at this point, if you see the difference between EDT and CDT as a disagreement about problem 3.
Basically, yes. What separates EDT and CDT is whether they condition on the joint probability distribution or use the do operator on the causal graph; there’s no other difference. This is a productive difference, and so obscuring it is counterproductive.
What you call “EEDT” I would call “expected value calculation,” and then I would use “decision theory” to describe the different ways of estimating conditional probabilities (as you put it). It is right that expected value calculation is potentially nonobvious, and so saying “we use expected values” is meaningful and important, but I think that the language to convey the concepts you want to convey already exists, and you should use the language that already exists.
An update: your comment (among others) prompted me to do some more reading. In particular, the Stanford Encyclopedia of Philosophy article on Causal Decision Theory was very helpful in making the distinction between CDT and EDT clear to me.
I still think you can mess around with the notion of random variables, as described in this post, to get a better understanding of what you are actually prediction. But I suppose that this can be confusing to others.
I’m glad it helped!
I think there are three steps in the reduction:
A “decision problem”: “given my knowledge, what action should I take”, which is answered by a “decision theory/procedure”
Expected values: “given my knowledge, what is the expected value of taking action A”
Conditional probability estimation: “given my knowledge, what is my best guess for the probability P(X|Y)”
The reduction from 1 to 2 is fairly obvious, just take the action that maximizes expected value. I think this is common to all decision theories.
The reduction of 2 to 3 is what is done by EEDT. To me this step was not so obvious, but perhaps there is a better name for it.
So the difference is in how to solve step 3, I agree. I wasn’t trying to obscure anything, of course. Rather, I was trying to advocate that we should focus on problem 3 directly, instead of problem 1.
Do you have some more standard terms I should use?
This doesn’t hold in prospect theory, in which probabilities are scaled before they are used. (Prospect theory is a descriptive theory of buggy human decision-making, not a prescriptive decision theory.) [Edit] Actually, I think this is a disagreement in how you go from 2 to 3. In order to get a disagreement on how you go from 1 to 2, you’d have to look at the normal criticisms of VNM.
It’s also not universal in decision theories involving intelligent adversaries. Especially in zero-sum games, you see lots of minimax solutions (minimize my opponent’s maximum gain). This difference is somewhat superficial, because minimax does reduce to expected value calculation with the assumption that the opponent will always choose the best option available to him by our estimation of his utility function. If you relax that assumption, then you’re back in EV territory, but now you need to elicit probabilities on the actions of an intelligent adversary, which is a difficult problem.
There’s a subtle point here about whether the explicit mechanisms matter. Expected value calculation is flexible enough that any decision procedure can be expressed in terms of expected value calculation, but many implicit assumptions in EV calculation can be violated by those procedures. For example, an implicit assumption of EV calculation is consequentialism in the sense that the preferences are over outcomes, but one could imagine a decision procedure where the preferences are over actions instead of over outcomes, or more realistically action-outcome pairs. We can redefine the outcomes to include the actions, and thus rescue the procedure, but it seems worthwhile to have a separation between consequentialist and deontological positions. If you relax the implicit assumption that probabilities should be treated linearly, then prospect theory can be reformulated as an EV calculation, where outcomes are outcome-probability pairs. But again it seems worthwhile to distinguish between decision algorithms which treat probabilities linearly and nonlinearly.
I’m not sure how much we disagree at this point, if you see the difference between EDT and CDT as a disagreement about problem 3.