Probabilities associated with temporal uncertainty

[stevendaryl]

This seems like a simple matter, but apparently it is controversial: Is it meaningful to talk about probabilities for temporal uncertainty?

If I find myself in a room without a clock, I might wonder what time it is. I know that I entered the room at 9:00, so it has to be later than that. I know that someone will come for me at 12:00, so it has to be earlier than that. So the current time is somewhere between 9:00 and 12:00. So it's something that I'm uncertain about. Is it possible to quantify my uncertainty via a probability distribution?

The reason why some people would say that it's undefined because "the current time" is not a random variable. On the other hand, it seems that it is perfectly meaningful to reason about the uncertainty. For example, I started working on a crossword puzzle when I first came in, and I just finished. I know from experience that there is a probability distribution on the length of time it takes me to finish a crossword puzzle, highly peaked at 1 hour. So I could use that fact to reason that it's more likely that it's 10:00 now, rather than 9:15 or 10:45.

 

[.Scott]

This seems like a simple matter, but apparently it is controversial: Is it meaningful to talk about probabilities for temporal uncertainty?

If I find myself in a room without a clock, I might wonder what time it is. I know that I entered the room at 9:00, so it has to be later than that. I know that someone will come for me at 12:00, so it has to be earlier than that. So the current time is somewhere between 9:00 and 12:00. So it's something that I'm uncertain about. Is it possible to quantify my uncertainty via a probability distribution?

The reason why some people would say that it's undefined because "the current time" is not a random variable. On the other hand, it seems that it is perfectly meaningful to reason about the uncertainty. For example, I started working on a crossword puzzle when I first came in, and I just finished. I know from experience that there is a probability distribution on the length of time it takes me to finish a crossword puzzle, highly peaked at 1 hour. So I could use that fact to reason that it's more likely that it's 10:00 now, rather than 9:15 or 10:45.

Quite aside from whether "the current time" is a random variable, certainly a measurement or estimate of the current time will have a probability distribution. If you took 10,000 people, put them in a room for 3 hours, and in each case, picked a time within that 3 hour period to quiz them on their best guess at the time, you would get an T(t) and S(t) that represented the average estimated time and the 1-standard deviation period.

I think there is a simpler, but related calculation that can be made. Let's say that you are approaching a traffic signal and you know that the light stays green for exactly 120 seconds. When the light first came into view, you were 2000 feet from the wait/go decision point. At 80 feet per second, you are now 1000 feet away and it is still green. What are your chances of making the light?

You have some information about the traffic light cycle, but not full information. That's certainly similar to having an imprecise knowledge of the current time.

 

[FactChecker]

I agree with you. It is often best to think of probability theory as the theory of guessing, given your current knowledge. If a coin has already been tossed, the outcome is determined and no longer a traditional random variable. But if you do not have a clue about the result of the toss, you can still apply probability theory to say that the odds of heads is 50/50. So you are guessing about the result of something that is no longer a random variable.
Likewise, you can apply probability theory to your problem of guessing what the current time it is, given your current knowledge, even though "current time" is not a traditional random variable. As you say, you can use information such as how long it takes to solve a puzzle to adjust your guess.

 

[Stephen Tashi]

The reason why some people would say that it's undefined because "the current time" is not a random variable. .

We have to distinguish between "not a random variable" and "not unambiguously defined as a random variable".

Suppose a fair coin is put is sealed box and has been flipped (in the past) by shaking the box. There can be a philosophical argument about whether the concept that "the coin landed heads" is a random variable. There is a distinction in mathematics between variables that have "fixed but unknown" values an those that are random variables. However if a person does choose the model the state of coin as a random variable, the definition of that random variable is unambiguous.

There are many distributions in probability theory that model picking random times. For example, we can model the time that an atom undergoes radioactive decay as a random variable. Given time t = 0 is defined as a specific clock time (in the past or future) in the laboratory, a time such as t = 2 is unambiguously defined.

However, if we consider the problem of modeling "What time is it now?" as a random variable there is the interesting problem that such a concept is self-referential The question amounts to "Here at time t, what is the probability distribution for time t?". If we accept that "here at time t" refers to a definite time, then the probability that "now" is time t is 1. Time t is "fixed but unknown, at time t".

A practical person would approach modeling "What time is it now?" by setting t = 0 to be a known time in the laboratory. Then he would specify some probability distribution for the time t at which the question "What time is it now?" is asked. The answer to "What time is it now" would be the time at which the question was asked.

The philosophical issue with that approach is that it takes the viewpoint of an observer who knows when things have taken place in the laboratory. By contrast, a person asking himself "What time is it now?" is asking an ill-posed question unless he can reference when he asks the question to some definite time in the laboratory. However, if he could do that then he would have no need to ask about the time.

I'd say the philosophical issue has no impact on people doing applied mathematics. They always take the viewpoint of observers of phenomena.
.

 

[stevendaryl]

We have to distinguish between "not a random variable" and "not unambiguously defined as a random variable".

Suppose a fair coin is put is sealed box and has been flipped (in the past) by shaking the box. There can be a philosophical argument about whether the concept that "the coin landed heads" is a random variable. There is a distinction in mathematics between variables that have "fixed but unknown" values an those that are random variables. However if a person does choose the model the state of coin as a random variable, the definition of that random variable is unambiguous.

You say there is a distinction in mathematics between "fixed but unknown" and "random variable", but I would say that the former case isn't actually modeled in mathematics at all, as far as I know. If you had a satisfactory theory of the first, then I would think that there would be no particular need for the second.

However, if we consider the problem of modeling "What time is it now?" as a random variable there is the interesting problem that such a concept is self-referential The question amounts to "Here at time t, what is the probability distribution for time t?". If we accept that "here at time t" refers to a definite time, then the probability that "now" is time t is 1. Time t is "fixed but unknown, at time t".

In the example I gave, the question can be translated into "How many minutes has it been since I entered this room?"

I'd say the philosophical issue has no impact on people doing applied mathematics. They always take the view of observers of phenomena.

The question is: "What is the probability that the current time is 9:30 [itex]\pm[/itex] 1 minute?" (Given whatever information is available). If someone says that the question is meaningless because the current time is not a random variable, then I would say that is an example of someone letting a philosophical issue impact mathematics.

 

[Demystifier]

This seems like a simple matter, but apparently it is controversial: Is it meaningful to talk about probabilities for temporal uncertainty?

If I find myself in a room without a clock, I might wonder what time it is. I know that I entered the room at 9:00, so it has to be later than that. I know that someone will come for me at 12:00, so it has to be earlier than that. So the current time is somewhere between 9:00 and 12:00. So it's something that I'm uncertain about. Is it possible to quantify my uncertainty via a probability distribution?

The reason why some people would say that it's undefined because "the current time" is not a random variable. On the other hand, it seems that it is perfectly meaningful to reason about the uncertainty. For example, I started working on a crossword puzzle when I first came in, and I just finished. I know from experience that there is a probability distribution on the length of time it takes me to finish a crossword puzzle, highly peaked at 1 hour. So I could use that fact to reason that it's more likely that it's 10:00 now, rather than 9:15 or 10:45.

Whether time is a random variable or not is a physical (not a mathematical) question. From that perspective you might be interested in my paper
https://arxiv.org/abs/1309.0400
especially Sec. 2.2.

 

[FactChecker]

Just to clarify something -- time itself is not a random variable. In this case you have an experiment with unknown, random results that are measured in time. The resulting time given by the experiment is a random variable. Similarly, the positions on a wall are not a random variable but if one throws a dart at a wall, then the position of the dart is a random variable.

 

[Stephen Tashi]

You say there is a distinction in mathematics between "fixed but unknown" and "random variable", but I would say that the former case isn't actually modeled in mathematics at all, as far as I know.

In the calculation of confidence intervals in frequentist statistics, the parameter being estimated is assumed to have a fixed but unknown value. (The layman's interpretation of a confidence interval is usually that it says something about the probable value of the unknown parameter, but this is not correct unless one is using Bayesian statistics and computing "credible intervals".).

The question is: "What is the probability that the current time is 9:30 [itex]\pm[/itex] 1 minute?" (Given whatever information is available). If someone says that the question is meaningless because the current time is not a random variable, then I would say that is an example of someone letting a philosophical issue impact mathematics.

I agree that people who apply mathematics would take the view that the task is model when a question is asked by someone observing the questioner.

Your example assumes the person has definite knowledge "when" some time happened previously to the question. For example, in the OP, it assumes he knows when 9:00 was. If a person in his current state of knowledge at time = now says he knows when 9:00 was then why does he not know when t = now is relative to 9:00? How do we substantiate the claim "I know (right now) when 9:00 was". In your example, the claim could be substantiated by some existing physical record that gave the information "Experimental subject entered the room at 9:00 and was informed of the time". To substantiate the claim "The time now is 9:30" at time 9:35, there needs to be a physical record in existence at 9:35 such as "Experimental subject asked 'What time is it? at 9:30". This interpretation of the question "What time is it now" makes "now" a time in the past. I agree that practcial people are not troubled by such issues of semantics.

 

[stevendaryl]

Your example assumes the person has definite knowledge "when" some time happened previously to the question. For example, in the OP, it assumes he knows when 9:00 was. If a person in his current state of knowledge at time = now says he knows when 9:00 was then why does he not know when t = now is relative to 9:00?

If that were a correct argument, then there would never be a need to look at a clock. I'm assuming that people do actually want to know what time it is, and don't know. That seems to sometimes be the case in the real world (which was the motivation for inventing clocks.)

 

[Stephen Tashi]

If that were a correct argument, then there would never be a need to look at a clock.

The conventional view is that your knowledge of "when 9:00 was" is implemented by some sort of physical record that exists in your brain that associates 9:00 with some event, such as "I entered the room" . My point, is that your knowledge of "when 9:00 was" is different that the knowledge "I know when 9:00 was relative to the current time".

 

[stoomart]

This seems like a simple matter, but apparently it is controversial: Is it meaningful to talk about probabilities for temporal uncertainty?

If I find myself in a room without a clock, I might wonder what time it is. I know that I entered the room at 9:00, so it has to be later than that. I know that someone will come for me at 12:00, so it has to be earlier than that. So the current time is somewhere between 9:00 and 12:00. So it's something that I'm uncertain about. Is it possible to quantify my uncertainty via a probability distribution?

The reason why some people would say that it's undefined because "the current time" is not a random variable. On the other hand, it seems that it is perfectly meaningful to reason about the uncertainty. For example, I started working on a crossword puzzle when I first came in, and I just finished. I know from experience that there is a probability distribution on the length of time it takes me to finish a crossword puzzle, highly peaked at 1 hour. So I could use that fact to reason that it's more likely that it's 10:00 now, rather than 9:15 or 10:45.

I think it would be meaningful if you wanted to wager what time it was with a fellow person in the room with you. When reading through this thread, I imagined you entered the room at 9, quickly fell asleep, and after waking up, wondered how long you needed to wait before being picked up. In this case you could create a probability distribution from previous nap data.