I’m not sure I see the relevance of the bus example to anthropic reasoning. Below I explain why (maybe I spent too long on this; ended up hyperfocusing). Note that all uses of ‘average’ and ‘expectation’ are in the technical, mathematical sense.
By a similar argument (your random arrival time has the mean and median in the middle between two buses), the time you should expect to wait for the bus to arrive is the same as the time you have already been waiting
If the ‘random arrival time has the mean in the middle between two busses,’ one should expect to wait time equal to the remaining wait time in that average situation.
One could respond that we don’t know the interval between busses, and thus don’t know the remaining wait time; but this does not seem to be a reason to expect the bus to arrive after the time you’ve been waiting doubles (from the view of neither anthropic nor non-anthropic reasoners).
The bus example has some implicit assumptions about the interval between events (i.e., one assumes that busses operate on the timescales of human schedules). In the odd scenario where one was clueless when an event would happen, and only knew (a) it would eventually happen, (b) it happened at least once before, and (c) it’s happening on an interval, then that one would have no choice but to reason that the event could have happened most recently at any point in the past, and thus could happen next at any point in the future.
Or, more precisely, given the event could not have first happened earlier than the starting point of the universe, an anthropic reasoner could reason that the average observer will exist at the mid-point between (a) the average expected time of the initial event: half-way between when the universe began and now, and (b) when the event will next happen. Their ‘expectation’ would still shift forwards as time passes (correctly from their perspective; it would seem like they were updating on further evidence about which times the event has not happened at), but it would not start out equal to the time since they first started waiting.
This still wouldn’t seem quite analogous to real-world anthropic reasoning to me, though, because in reality an anthropic reasoner doesn’t expect the amount of observers to be evenly distributed over their range. There are a few possible distributions of the amount of observers over time which we consider plausible (some of which are mentioned in the original post), and in none of these distributions is the average in the center as it is with busses.
I’m not sure I see the relevance of the bus example to anthropic reasoning. Below I explain why (maybe I spent too long on this; ended up hyperfocusing). Note that all uses of ‘average’ and ‘expectation’ are in the technical, mathematical sense.
If the ‘random arrival time has the mean in the middle between two busses,’ one should expect to wait time equal to the remaining wait time in that average situation.
One could respond that we don’t know the interval between busses, and thus don’t know the remaining wait time; but this does not seem to be a reason to expect the bus to arrive after the time you’ve been waiting doubles (from the view of neither anthropic nor non-anthropic reasoners).
The bus example has some implicit assumptions about the interval between events (i.e., one assumes that busses operate on the timescales of human schedules). In the odd scenario where one was clueless when an event would happen, and only knew (a) it would eventually happen, (b) it happened at least once before, and (c) it’s happening on an interval, then that one would have no choice but to reason that the event could have happened most recently at any point in the past, and thus could happen next at any point in the future.
Or, more precisely, given the event could not have first happened earlier than the starting point of the universe, an anthropic reasoner could reason that the average observer will exist at the mid-point between (a) the average expected time of the initial event: half-way between when the universe began and now, and (b) when the event will next happen. Their ‘expectation’ would still shift forwards as time passes (correctly from their perspective; it would seem like they were updating on further evidence about which times the event has not happened at), but it would not start out equal to the time since they first started waiting.
This still wouldn’t seem quite analogous to real-world anthropic reasoning to me, though, because in reality an anthropic reasoner doesn’t expect the amount of observers to be evenly distributed over their range. There are a few possible distributions of the amount of observers over time which we consider plausible (some of which are mentioned in the original post), and in none of these distributions is the average in the center as it is with busses.