That isn’t how probability works. Birthdays are roughly evenly distributed throughout the year, so a July birthday is just as likely as a January birthday. Just because we decide to mark a certain day as the beginning of the year doesn’t mean that people are most likely to be born half a year from that day.
If you’re saying that the “expected birthday”, in the sense of “minimizing the average squared difference between your guess and the true value,” is in the middle of the year, then that is true—but only as long as December and January are considered to be a year apart, which in reality they are not, since each year is followed by another. (An analogy—time is like a helix projected onto a circle, and the circle is the Year, with a point on the circle considered to be the endpoint.) Dates might be a uniquely bad example for this reason: the beginning and end of the window are generally arbitrary, while (ferex) the beginning and end of a civilization are far less arbitrary. (It would be very strange to define the beginning of a civilization as halfway through its lifespan, and then say that the beginning happened immediately after the end. It would not be very strange to have the new year begin on July 1st.)
But that’s not exactly it. For one, that definition is probably neither what you meant nor an actual definition of “more likely”. Additionally, even if we’re talking about the year 1967 specifically (thus invalidating my argument from circularity), people aren’t more likely to be born in July than January (barring variation in how often sex occurs). I think that you’re misunderstanding the anthropic doomsday argument, which says that we are with 90% prior certainty in the middle 90% of humanity, so our prior should be that if X people have lived so far, the future will contain at most 9X people. It doesn’t say that we’re probably very close to the midpoint—in fact, it assumes that the chance of being any particular person is uniform!
Your birthday and galactic location examples both have a similar problem—arbitrary framing. I could just as easily argue that we’d be most likely to find ourselves at the galactic center. You can transform a random distribution into another one with a different midpoint. For example, position-in-galaxy to distance-from-galactic-center.
It’s possible that you got this idea from looking at bell curves, in which the middle is more common than the edges, even if you compare equally-sized intervals. But that’s a property of the distribution, not something inherent in middles—in fact, with a bimodal distribution, the edges are more common than the middle.
No, I don’t use bell curve distribution. By saying “middle in the year” I mean everything which is not 1 of January or 31 of December. Surely I understand that people have the same chance to be born in July and in December.
The example was needed to demonstrate real (but weak) predictive power of mediocrity reasoning. For example, I could claim that it is very unlikely that you was born in any of these dates (31 December or 1 of Janury), and most likely you was born somewhere between them; the same way I could claim that it is unlikely that it is exactly midnight on your clock.
And this is not depending on the choice of the starting point or framing. If our day change will be in 17.57, it would be still unlikely that your time now is 17.57.
I interpreted “the middle” as a point and its near surroundings, which explains some of the disagreement. (All of your specific examples in your original post were near the midpoint, which didn’t clarify which interpretation you intended.)
I think that the more fundamental rule isn’t about middles, and (as demonstrated here) that’s easily misinterpreted without including many qualifiers and specifics. “Larger intervals are more likely, to the extent that the distribution is flat” is more fundamental, but there are so many ways to define large intervals that it doesn’t seem very useful here. It all depends on what you call a very unusual point—if it’s the middle that’s most unusual to me, then my version of the doomsday argument says “we will probably either die out soon or last for a very long time, but not last exactly twice as long as we already have.” (In this case, my large interval would be time-that-civilization-exists minus (midpoint plus neighborhood of midpoint), and my small interval would be midpoint plus neighborhood of midpoint.)
That isn’t how probability works. Birthdays are roughly evenly distributed throughout the year, so a July birthday is just as likely as a January birthday. Just because we decide to mark a certain day as the beginning of the year doesn’t mean that people are most likely to be born half a year from that day.
If you’re saying that the “expected birthday”, in the sense of “minimizing the average squared difference between your guess and the true value,” is in the middle of the year, then that is true—but only as long as December and January are considered to be a year apart, which in reality they are not, since each year is followed by another. (An analogy—time is like a helix projected onto a circle, and the circle is the Year, with a point on the circle considered to be the endpoint.) Dates might be a uniquely bad example for this reason: the beginning and end of the window are generally arbitrary, while (ferex) the beginning and end of a civilization are far less arbitrary. (It would be very strange to define the beginning of a civilization as halfway through its lifespan, and then say that the beginning happened immediately after the end. It would not be very strange to have the new year begin on July 1st.)
But that’s not exactly it. For one, that definition is probably neither what you meant nor an actual definition of “more likely”. Additionally, even if we’re talking about the year 1967 specifically (thus invalidating my argument from circularity), people aren’t more likely to be born in July than January (barring variation in how often sex occurs). I think that you’re misunderstanding the anthropic doomsday argument, which says that we are with 90% prior certainty in the middle 90% of humanity, so our prior should be that if X people have lived so far, the future will contain at most 9X people. It doesn’t say that we’re probably very close to the midpoint—in fact, it assumes that the chance of being any particular person is uniform!
Your birthday and galactic location examples both have a similar problem—arbitrary framing. I could just as easily argue that we’d be most likely to find ourselves at the galactic center. You can transform a random distribution into another one with a different midpoint. For example, position-in-galaxy to distance-from-galactic-center.
It’s possible that you got this idea from looking at bell curves, in which the middle is more common than the edges, even if you compare equally-sized intervals. But that’s a property of the distribution, not something inherent in middles—in fact, with a bimodal distribution, the edges are more common than the middle.
No, I don’t use bell curve distribution. By saying “middle in the year” I mean everything which is not 1 of January or 31 of December. Surely I understand that people have the same chance to be born in July and in December.
The example was needed to demonstrate real (but weak) predictive power of mediocrity reasoning. For example, I could claim that it is very unlikely that you was born in any of these dates (31 December or 1 of Janury), and most likely you was born somewhere between them; the same way I could claim that it is unlikely that it is exactly midnight on your clock.
And this is not depending on the choice of the starting point or framing. If our day change will be in 17.57, it would be still unlikely that your time now is 17.57.
I interpreted “the middle” as a point and its near surroundings, which explains some of the disagreement. (All of your specific examples in your original post were near the midpoint, which didn’t clarify which interpretation you intended.)
I think that the more fundamental rule isn’t about middles, and (as demonstrated here) that’s easily misinterpreted without including many qualifiers and specifics. “Larger intervals are more likely, to the extent that the distribution is flat” is more fundamental, but there are so many ways to define large intervals that it doesn’t seem very useful here. It all depends on what you call a very unusual point—if it’s the middle that’s most unusual to me, then my version of the doomsday argument says “we will probably either die out soon or last for a very long time, but not last exactly twice as long as we already have.” (In this case, my large interval would be time-that-civilization-exists minus (midpoint plus neighborhood of midpoint), and my small interval would be midpoint plus neighborhood of midpoint.)