„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“.
„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“.