Sure, I don‘t deny that. What I am saying is, that your probability model don‘t tell you which probability you have to base on a certain decision
It says which probability you have, based on what you’ve observed. If you observed that it’s Monday—you are supposed to use probability conditionally on the fact that it’s Monday, if you didn’t observe that it’s Monday you can’t lawfully use the probability conditionally on the fact that it’s Monday. Simple as that.
There is a possible confusion where people may think that they have observed “this specific thing happened” while actually they observed “any thing from some group of things happened”, which is the technicolor and rare event cases are about.
Suppose a simple experiment where the experimenter flips a fair coin and you have to guess if Tails or Heads, but you are only rewarded for the correct decision if the coin comes up Tails. Then, of course, you should still entertain unconditional probabilities P(Heads)=P(Tails)=1/2. But this uncertainty is completely irrelevant to your decision.
Here you are confusing probability and utility. The fact that P(Heads)=P(Tails)=1/2 is very much relevant to our decision making! The correct reasoning goes like this:
Which means that you shouldn’t bet on Heads at any odds
What is relevant, however, is P(Tails/Tails)=1 and P(Heads/Tails)=0, concluding you should follow the strategy always guessing Tails.
And why did you happen to decide that it’s P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of
P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are “relevant” for you decision making?
You seem to just decide the “relevance” of probabilities post hoc, after you’ve already calculated the correct answer the proper way. I don’t think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn’t need utilities as a concept.
Another way to arrive at this strategy is to calculate expected utilities setting U(Heads)=0 as you would propose. But this is not the only reasonable solution. It’s just a different route of reasoning to take into account the experimental condition that your decision counts only if the coin lands Tails.
This is not “another way”. This is the right way. It has the proper formalization and actually allows us to arrive to the correct answer even if we do not yet know it.
If the optimal betting sheme requires you to rely on P(Heads/Red or Blue)=1/2 when receiving evidence Blue, then the betting sheme demands you to ignore your total evidence.
You do not “ignore your total evidence”—you are never supposed to do that. It’s just that you didn’t actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn’t even include “Blue” only “Blue or Red”.
But I suppose it’s better to go to the comment section Another Non-Anthropic Paradox for this particular crux.
„And why did you happen to decide that it’s P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of
P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are “relevant” for you decision making?
You seem to just decide the “relevance” of probabilities post hoc, after you’ve already calculated the correct answer the proper way. I don’t think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn’t need utilities as a concept.“
No, it‘s not post hoc. The simple rule to follow is: If a certain value x of a random variable X is relevant to your decision, then base your decision on the probability of x conditional on all conditions that are known to be satisfied when your decision is actually linked to the consequences of interest. And this is P(x/Tails) and not P(x/Heads) in case of guessing X is only rewarded if X=Tails.
Of course, the rule can‘t guarantee you correct answers, since the correctness of your decision does not only depend on the proper application of the rule but also on the quality of your probability model. However, notice that this feature could be used to test a probability model. For example, David Lewis model of the original Sleeping Beauty experiment says P(Heads/Monday)=2/3 resulting in bad betting decisions in case the bet only counts on Monday and applying the rule. Thus, there must be something wrong either with the rule or with to model. Since the logic of the rule seems valid to me, it leads me to dismiss Lewis model.
„You do not “ignore your total evidence”—you are never supposed to do that. It’s just that you didn’t actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn’t even include “Blue” only “Blue or Red”.
But I suppose it’s better to go to the comment section Another Non-Anthropic Paradox for this particular crux“
I‘ve read your latest reply on this topic and I generally agree with it. As I already wrote, it is absolutely possible to create an event space that models a state of mind that is biased towards perceiving certain events (e.g. red) while neglecting others (e.g. blue). However, I find it difficult to understand how adopting such an event space that excludes an event that is relevant evidence according to your model, is not ignoring total evidence. This seems to me as if you were arguing that you don‘t ignore something because you are biased to ignore it. Or are you just saying that I was referring to the wrong mental concept, since we can only ignore what we actually do observe? Well, from my psychologist point of view, I highly doubt that simply precommitting to red is a sufficient condition to reliably prevent the human brain from classifying the perception of blue as the event „blue room“ instead of merely “a colored room (red or blue)“. I guess, most people would still subjectively experiencing themselves in a blue room.
Apart from that, is the concept of total evidence really limited to evidence that is actually observed or does it rather refer to all evidence accessible to the agent, including evidence through further investigating, reflecting, reasoning and inference beyond direct observation? Though if the evidence „blue room“ was not initially observed by the agent due to some strong, biased mindset, the evidence would be still accessible to him and could therefore be considered part of his total evidence as long the agent is able to break the mindset.
At the end, the experiment could be modified in a way that Beauty‘s memory about her precommittment on Sunday is erased while sleeping and brought back into her mind again by the experimenter after awoken and seeing the room. In this case, she has already observed a particular color before her Sunday mindset, which could have prevented this, is „reactivated“.
It says which probability you have, based on what you’ve observed. If you observed that it’s Monday—you are supposed to use probability conditionally on the fact that it’s Monday, if you didn’t observe that it’s Monday you can’t lawfully use the probability conditionally on the fact that it’s Monday. Simple as that.
There is a possible confusion where people may think that they have observed “this specific thing happened” while actually they observed “any thing from some group of things happened”, which is the technicolor and rare event cases are about.
Here you are confusing probability and utility. The fact that P(Heads)=P(Tails)=1/2 is very much relevant to our decision making! The correct reasoning goes like this:
P(Heads) = 1⁄2
P(Tails) = 1⁄2
U(Heads) = 0
U(Tails) = X,
E(Tails) = P(Tails)U(Tails) - P(Heads)U(Heads) = 1/2X − 0
Solving E(Tails) = 0 for X:
X = 0
Which means that you shouldn’t bet on Heads at any odds
And why did you happen to decide that it’s P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of
P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are “relevant” for you decision making?
You seem to just decide the “relevance” of probabilities post hoc, after you’ve already calculated the correct answer the proper way. I don’t think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn’t need utilities as a concept.
This is not “another way”. This is the right way. It has the proper formalization and actually allows us to arrive to the correct answer even if we do not yet know it.
You do not “ignore your total evidence”—you are never supposed to do that. It’s just that you didn’t actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn’t even include “Blue” only “Blue or Red”.
But I suppose it’s better to go to the comment section Another Non-Anthropic Paradox for this particular crux.
„And why did you happen to decide that it’s P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are “relevant” for you decision making? You seem to just decide the “relevance” of probabilities post hoc, after you’ve already calculated the correct answer the proper way. I don’t think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn’t need utilities as a concept.“
No, it‘s not post hoc. The simple rule to follow is: If a certain value x of a random variable X is relevant to your decision, then base your decision on the probability of x conditional on all conditions that are known to be satisfied when your decision is actually linked to the consequences of interest. And this is P(x/Tails) and not P(x/Heads) in case of guessing X is only rewarded if X=Tails.
Of course, the rule can‘t guarantee you correct answers, since the correctness of your decision does not only depend on the proper application of the rule but also on the quality of your probability model. However, notice that this feature could be used to test a probability model. For example, David Lewis model of the original Sleeping Beauty experiment says P(Heads/Monday)=2/3 resulting in bad betting decisions in case the bet only counts on Monday and applying the rule. Thus, there must be something wrong either with the rule or with to model. Since the logic of the rule seems valid to me, it leads me to dismiss Lewis model.
„You do not “ignore your total evidence”—you are never supposed to do that. It’s just that you didn’t actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn’t even include “Blue” only “Blue or Red”. But I suppose it’s better to go to the comment section Another Non-Anthropic Paradox for this particular crux“
I‘ve read your latest reply on this topic and I generally agree with it. As I already wrote, it is absolutely possible to create an event space that models a state of mind that is biased towards perceiving certain events (e.g. red) while neglecting others (e.g. blue). However, I find it difficult to understand how adopting such an event space that excludes an event that is relevant evidence according to your model, is not ignoring total evidence. This seems to me as if you were arguing that you don‘t ignore something because you are biased to ignore it. Or are you just saying that I was referring to the wrong mental concept, since we can only ignore what we actually do observe? Well, from my psychologist point of view, I highly doubt that simply precommitting to red is a sufficient condition to reliably prevent the human brain from classifying the perception of blue as the event „blue room“ instead of merely “a colored room (red or blue)“. I guess, most people would still subjectively experiencing themselves in a blue room.
Apart from that, is the concept of total evidence really limited to evidence that is actually observed or does it rather refer to all evidence accessible to the agent, including evidence through further investigating, reflecting, reasoning and inference beyond direct observation? Though if the evidence „blue room“ was not initially observed by the agent due to some strong, biased mindset, the evidence would be still accessible to him and could therefore be considered part of his total evidence as long the agent is able to break the mindset. At the end, the experiment could be modified in a way that Beauty‘s memory about her precommittment on Sunday is erased while sleeping and brought back into her mind again by the experimenter after awoken and seeing the room. In this case, she has already observed a particular color before her Sunday mindset, which could have prevented this, is „reactivated“.