I’m going to start with the concrete question about the sun rising and then segue into related stuff that I suspect has more value. My concise reply is “whatever shminux means/meant by the sun not rising”, because my example is just a derivative of your example, from the opening post:
The odds of the sun rising tomorrow, without any additional data, is about 1-1/average number of consecutive sunrises.
I agree that my example is ambiguous. I think your questions exaggerate the ambiguity. According to how I hear people communicate, and thus how I think you are communicating:
The “sun rising” does not refer to it getting bright in the morning, because it starts to get bright before the sun rises, and because the sun does not rise less after a full moon, or in areas of high light pollution, or on a cloudy day. However, morning brightness can be evidence for the sun rising.
The “sun rising” does not refer to my individual perception of the sun rising, because if I am indoors when the sun rises, and do not see the sun rising, I do not say that the sun did not rise that day. However, my perception of the sun rising can be evidence for the sun rising.
An eclipsed sun is still the sun, we say “don’t look at the sun during an eclipse”. The sun continues rising during an eclipse if it eclipsed while it is rising.
When the sun is obscured by clouds, mist, supervolcano eruption, asteroid impact, invading alien space craft, etc we say that we cannot see the sun rise. We don’t say that the sun did not rise.
But I agree that it’s an ambiguous example. In particular, close to the poles there are days in the year when the sun does not rise, and people say things like “the sun rises every morning at the equator but only once a year at the north pole”. So the statement “the sun is rising” is true in some places and times and false in others, and the example is ambiguous because it doesn’t specify the place. There is also some ambiguity around disputing definitions that I don’t think is very illuminating and I raise mostly so that nobody else has to.
So the ambiguities don’t update my assigned probabilities and the additional thought we’re putting into it doesn’t seem to be very decision-relevant, so I don’t think it’s paying off for us. I feel like we’ve roughly gone from “epsilon” to “it depends, but epsilon”. Do you think it’s paying off? If so, how? If not, is that evidence against your thesis?
I don’t think there is anything special about “epsilon” here. Consider another probability question: “Conditional on me flipping this unbiased coin, will it land heads?”. My probability is 50%, I’m not an ideal Bayesian. But an ideal Bayesian would give a number very slightly less than 50% due to “edge” and “does not land” and other more exotic possibilities. If “epsilon” is a cop-out then is “50%” a cop-out?
My conclusion so far: in the rare case where the pay-off of an outcome is “huge” and its probability is “epsilon” then both “huge” and “epsilon” are unhelpful and more thought is useful. In other scenarios, “huge” and “epsilon” are sufficient.
I agree that the sunrise example may not be the best one, I just wanted to show that definitions tend to become ambiguous when there are many disjunctive possibilities to get to roughly the same outcome.
Consider another probability question: “Conditional on me flipping this unbiased coin, will it land heads?”. My probability is 50%, I’m not an ideal Bayesian. But an ideal Bayesian would give a number very slightly less than 50% due to “edge” and “does not land” and other more exotic possibilities. If “epsilon” is a cop-out then is “50%” a cop-out?
It’s kind of the same example, what are the odds of the coin not landing either heads or tails? There are many possibilities, including those you listed, and it’s work if you care about this edge case (no pun intended). If you care about slight deviations from 50%, then do the work, otherwise just say “50% for my purpose”, no epsilon required.
I’m going to start with the concrete question about the sun rising and then segue into related stuff that I suspect has more value. My concise reply is “whatever shminux means/meant by the sun not rising”, because my example is just a derivative of your example, from the opening post:
I agree that my example is ambiguous. I think your questions exaggerate the ambiguity. According to how I hear people communicate, and thus how I think you are communicating:
The “sun rising” does not refer to it getting bright in the morning, because it starts to get bright before the sun rises, and because the sun does not rise less after a full moon, or in areas of high light pollution, or on a cloudy day. However, morning brightness can be evidence for the sun rising.
The “sun rising” does not refer to my individual perception of the sun rising, because if I am indoors when the sun rises, and do not see the sun rising, I do not say that the sun did not rise that day. However, my perception of the sun rising can be evidence for the sun rising.
An eclipsed sun is still the sun, we say “don’t look at the sun during an eclipse”. The sun continues rising during an eclipse if it eclipsed while it is rising.
When the sun is obscured by clouds, mist, supervolcano eruption, asteroid impact, invading alien space craft, etc we say that we cannot see the sun rise. We don’t say that the sun did not rise.
But I agree that it’s an ambiguous example. In particular, close to the poles there are days in the year when the sun does not rise, and people say things like “the sun rises every morning at the equator but only once a year at the north pole”. So the statement “the sun is rising” is true in some places and times and false in others, and the example is ambiguous because it doesn’t specify the place. There is also some ambiguity around disputing definitions that I don’t think is very illuminating and I raise mostly so that nobody else has to.
So the ambiguities don’t update my assigned probabilities and the additional thought we’re putting into it doesn’t seem to be very decision-relevant, so I don’t think it’s paying off for us. I feel like we’ve roughly gone from “epsilon” to “it depends, but epsilon”. Do you think it’s paying off? If so, how? If not, is that evidence against your thesis?
I don’t think there is anything special about “epsilon” here. Consider another probability question: “Conditional on me flipping this unbiased coin, will it land heads?”. My probability is 50%, I’m not an ideal Bayesian. But an ideal Bayesian would give a number very slightly less than 50% due to “edge” and “does not land” and other more exotic possibilities. If “epsilon” is a cop-out then is “50%” a cop-out?
My conclusion so far: in the rare case where the pay-off of an outcome is “huge” and its probability is “epsilon” then both “huge” and “epsilon” are unhelpful and more thought is useful. In other scenarios, “huge” and “epsilon” are sufficient.
I agree that the sunrise example may not be the best one, I just wanted to show that definitions tend to become ambiguous when there are many disjunctive possibilities to get to roughly the same outcome.
It’s kind of the same example, what are the odds of the coin not landing either heads or tails? There are many possibilities, including those you listed, and it’s work if you care about this edge case (no pun intended). If you care about slight deviations from 50%, then do the work, otherwise just say “50% for my purpose”, no epsilon required.