I am currently considering the question “Does probability have a smallest divisible unit?” and I think I’m confused.
For instance, it seems like time has a smallest divisible unit, Planck time.
Whereas the real numbers do not have a smallest divisible unit. Instead, they have a Dense order. So it seems reasonable to ask “Does probability have a smallest divisible unit?”
Then to try answering the question, if you describe a series of events which can only happen in 1 particular branch of the many worlds interpretation, and you describe something which happens in 0 branches of the many worlds interpretation, then my understanding is there is no series of events which has a probability in between those two things, which would appear to imply the concept of a smallest unit of probability is coherent and the answer is “Yes.”
However, there is an article on Infinitely divisible probability and if you can divide something infinitely, then of course, the concept of it having a smallest unit is nonsensical, and the answer would be “No.”
For instance, it seems like time has a smallest divisible unit, Planck time.
We don’t really understand what the significance of the Planck time interval is. In particular, it would be extremely surprising, given modern physics, if it were a discrete unit like the clock cycles of a computer or the steps in Conway’s game of life. It could be ‘indivisible’ in some sense, but we don’t know what sense that could be.
Then to try answering the question, if you describe a series of events which can only happen in 1 particular branch of the many worlds interpretation, and you describe something which happens in 0 branches of the many worlds interpretation, then my understanding is there is no series of events which has a probability in between those two things, which would appear to imply the concept of a smallest unit of probability is coherent and the answer is “Yes.”
Branches of the wavefunction aren’t really discrete countable things; they’re much closer to the idea of clusters of locally high density. Relatedly, even when they are approximately countable, they can come in different sizes.
Many worlds is in some ways a really bad way to understand probability. Probabilities should be based on the information available to you and should describe how justified hypotheses are given the evidence. The different possibilities don’t have to be ‘out there’ like they are in MWI, they just have to have not been ruled out by the available evidence.
How’s this? (I’m thinking here that the smallest unit would correspond to 1 possible arrangement of the Hubble volume, so the unit would be something like 1/10^70 or something. Any other state of the world is meaningless since it can’t exist.)
As usually formulated, Bayesian probability maps beliefs onto the reals between 0 and 1, and so there’s no smallest or largest probability. If you act as if there is and violate Cox’s theorem, you ought to be Dutch bookable through some set of bets that either split up extremely finely events (eg. a dice with trillions of sides) or aggregated many events. If there is a smallest physical probability, then these Dutch books would be expressible but not implementable (imagine the universe has 10^70 atoms—we can still discuss ‘what if the universe had 10^71 atoms?’).
This leads to the observed fact that an agent implementing probability with units is Dutch bookable in theory, but you will never observe you or another agent Dutch booking said agent. It’s probably also more computationally efficient.
If probability has a smallest divisible unit, it seems like there would have to be one or more least probable series of events.
If I was to anticipate that there was one or more least probable series of events, it seems like I would have to also anticipate that additional events will stop occurring in the future. If events are still taking place, a particular even more complicated series of events can continue growing more improbable than whatever I had previously thought of as a least probable event.
So it seems an alternative way of looking at this question is “Do I expect events to still be taking place in the future?” In which case I anticipate the answer is “Yes” (I have no evidence to suggest they will stop) and I think I have dissolved the more confusing question I was starting with.
Given that that makes sense to me, I think my next step is if it makes sense to other people. If I’ve come up with an explanation which makes sense only to me, that doesn’t seem likely to be helpful overall.
I don’t have an answer to the question I think you’re asking, but it’s perhaps worth noting (if only to preempt confusion) that there are different notions of probability that may provide different answers here. Probability as a mental construct that captures ones ignorance about the actual value of something in the world (e.g., what we refer to when we say a fair coin, when flipped, has a 1⁄2 probability of coming up heads) has a smallest unit that derives from the capabilities of the mind in which that construct exists, but this has nothing to do with the question of quantum measure you’re raising here.
Probability that a coin comes up heads is 0.5. Probability of N coins coming all up heads is 0.5^N. So what exactly was the original question in this context—are we asking whether there exist a smallest value of 0.5^N?
Well, if the universe has a finite time, if there is a smallest time unit, if the universe has finite number of elementary particles… this would provide some limit on the number of total coin flips in the universe. Even for infinite universes we could perhaps find some limit by specifying that the coin flips must happen in the same light cone...
But is this really what the original question was about? To me it seems like the question is confused. Probability is a logical construct, not something that exist, even if it is built on things that exist.
It would be like asking “what is the smallest positive rational number” with the additional constraint that a positive number must be P/Q where P and Q are numbers of pebbles in pebble heaps that exist in this universe. If there is a limited number of particles in the universe, that puts a limit on a value of Q, so there is some minimum value of 1/Q.. but what exactly does this result mean? Even if the Q really exists, the 1/Q is just a mental construct.
I’m fairly sure the original question was trying to ask about something labelled “probability” that wasn’t (exclusively) a mental construct, which is precisely why I brought up the idea of probability as a mental construct in the first place, to pre-empt confusion. Clearly I failed at that goal, though.
I’m not exactly sure what that something-labelled-”probability” was. You may well be right that the original question was simply confused. Generally when people start incorporating events in other Everett branches into their reasoning about the world I back away and leave them to it.
The OP aside, I do expect there are value of P too small for a human brain to actually represent. Given a probability like .000000001, for example, most of us either treat the probability as zero, or stop representing it in our minds as a probability at all. That is, for most of us our representation of a probability of .000000001 is just a number, indistinguishable from our representation of a temperature-difference of .000000001 degree Celsius or a mass of .000000001 grams.
So we could like exclude computations of expressions, and consider only probabilities of “basic events”, assuming that the concept shows to be coherent. We might ask about a probability of a coin flip, but not two coins. Speaking about coins, the “quantum of probability” is simply 1⁄2, end of story.
Well, I don’t even know what could be a “basic event” at the bottom level of the universe—the more I think about it, the more I realise my ignorance of quantum physics.
I don’t see where the “basic event”/”computation of expression” distinction gets us anywhere useful. As you say, even defining it clearly is problematic, and whatever definition we use it seems that any event we actually care about is not “basic.”
It also seems pretty clear to me that my mind can represent and work with probabilities smaller than 1⁄2, so restricting ourselves to domains of discourse that don’t require smaller probabilities (e.g., perfectly fair tosses of perfectly fair coins that always land on one face or the other) seems unhelpful.
I am currently considering the question “Does probability have a smallest divisible unit?” and I think I’m confused.
For instance, it seems like time has a smallest divisible unit, Planck time. Whereas the real numbers do not have a smallest divisible unit. Instead, they have a Dense order. So it seems reasonable to ask “Does probability have a smallest divisible unit?”
Then to try answering the question, if you describe a series of events which can only happen in 1 particular branch of the many worlds interpretation, and you describe something which happens in 0 branches of the many worlds interpretation, then my understanding is there is no series of events which has a probability in between those two things, which would appear to imply the concept of a smallest unit of probability is coherent and the answer is “Yes.”
However, there is an article on Infinitely divisible probability and if you can divide something infinitely, then of course, the concept of it having a smallest unit is nonsensical, and the answer would be “No.”
How do I resolve this confusion?
We don’t really understand what the significance of the Planck time interval is. In particular, it would be extremely surprising, given modern physics, if it were a discrete unit like the clock cycles of a computer or the steps in Conway’s game of life. It could be ‘indivisible’ in some sense, but we don’t know what sense that could be.
Branches of the wavefunction aren’t really discrete countable things; they’re much closer to the idea of clusters of locally high density. Relatedly, even when they are approximately countable, they can come in different sizes.
Many worlds is in some ways a really bad way to understand probability. Probabilities should be based on the information available to you and should describe how justified hypotheses are given the evidence. The different possibilities don’t have to be ‘out there’ like they are in MWI, they just have to have not been ruled out by the available evidence.
What would you anticipate to be different if probability did/didn’t have a smallest divisible unit?
Pascal’s wager, for one thing.
How’s this? (I’m thinking here that the smallest unit would correspond to 1 possible arrangement of the Hubble volume, so the unit would be something like 1/10^70 or something. Any other state of the world is meaningless since it can’t exist.)
As usually formulated, Bayesian probability maps beliefs onto the reals between 0 and 1, and so there’s no smallest or largest probability. If you act as if there is and violate Cox’s theorem, you ought to be Dutch bookable through some set of bets that either split up extremely finely events (eg. a dice with trillions of sides) or aggregated many events. If there is a smallest physical probability, then these Dutch books would be expressible but not implementable (imagine the universe has 10^70 atoms—we can still discuss ‘what if the universe had 10^71 atoms?’).
This leads to the observed fact that an agent implementing probability with units is Dutch bookable in theory, but you will never observe you or another agent Dutch booking said agent. It’s probably also more computationally efficient.
Good answer to help me focus.
If probability has a smallest divisible unit, it seems like there would have to be one or more least probable series of events.
If I was to anticipate that there was one or more least probable series of events, it seems like I would have to also anticipate that additional events will stop occurring in the future. If events are still taking place, a particular even more complicated series of events can continue growing more improbable than whatever I had previously thought of as a least probable event.
So it seems an alternative way of looking at this question is “Do I expect events to still be taking place in the future?” In which case I anticipate the answer is “Yes” (I have no evidence to suggest they will stop) and I think I have dissolved the more confusing question I was starting with.
Given that that makes sense to me, I think my next step is if it makes sense to other people. If I’ve come up with an explanation which makes sense only to me, that doesn’t seem likely to be helpful overall.
Makes sense to me.
I don’t have an answer to the question I think you’re asking, but it’s perhaps worth noting (if only to preempt confusion) that there are different notions of probability that may provide different answers here. Probability as a mental construct that captures ones ignorance about the actual value of something in the world (e.g., what we refer to when we say a fair coin, when flipped, has a 1⁄2 probability of coming up heads) has a smallest unit that derives from the capabilities of the mind in which that construct exists, but this has nothing to do with the question of quantum measure you’re raising here.
Probability that a coin comes up heads is 0.5. Probability of N coins coming all up heads is 0.5^N. So what exactly was the original question in this context—are we asking whether there exist a smallest value of 0.5^N?
Well, if the universe has a finite time, if there is a smallest time unit, if the universe has finite number of elementary particles… this would provide some limit on the number of total coin flips in the universe. Even for infinite universes we could perhaps find some limit by specifying that the coin flips must happen in the same light cone...
But is this really what the original question was about? To me it seems like the question is confused. Probability is a logical construct, not something that exist, even if it is built on things that exist.
It would be like asking “what is the smallest positive rational number” with the additional constraint that a positive number must be P/Q where P and Q are numbers of pebbles in pebble heaps that exist in this universe. If there is a limited number of particles in the universe, that puts a limit on a value of Q, so there is some minimum value of 1/Q.. but what exactly does this result mean? Even if the Q really exists, the 1/Q is just a mental construct.
I’m fairly sure the original question was trying to ask about something labelled “probability” that wasn’t (exclusively) a mental construct, which is precisely why I brought up the idea of probability as a mental construct in the first place, to pre-empt confusion. Clearly I failed at that goal, though.
I’m not exactly sure what that something-labelled-”probability” was. You may well be right that the original question was simply confused. Generally when people start incorporating events in other Everett branches into their reasoning about the world I back away and leave them to it.
The OP aside, I do expect there are value of P too small for a human brain to actually represent. Given a probability like .000000001, for example, most of us either treat the probability as zero, or stop representing it in our minds as a probability at all. That is, for most of us our representation of a probability of .000000001 is just a number, indistinguishable from our representation of a temperature-difference of .000000001 degree Celsius or a mass of .000000001 grams.
So we could like exclude computations of expressions, and consider only probabilities of “basic events”, assuming that the concept shows to be coherent. We might ask about a probability of a coin flip, but not two coins. Speaking about coins, the “quantum of probability” is simply 1⁄2, end of story.
Well, I don’t even know what could be a “basic event” at the bottom level of the universe—the more I think about it, the more I realise my ignorance of quantum physics.
I don’t see where the “basic event”/”computation of expression” distinction gets us anywhere useful. As you say, even defining it clearly is problematic, and whatever definition we use it seems that any event we actually care about is not “basic.”
It also seems pretty clear to me that my mind can represent and work with probabilities smaller than 1⁄2, so restricting ourselves to domains of discourse that don’t require smaller probabilities (e.g., perfectly fair tosses of perfectly fair coins that always land on one face or the other) seems unhelpful.