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One thing that helped me was the realization that, at least as used in the context of information theory, entropy is a property of an individual (typically the

by glial 1y ago

One thing that helped me was the realization that, at least as used in the context of information theory, entropy is a property of an individual (typically the person receiving a message) and NOT purely of the system or message itself. > entropy quantifies uncertainty This sums it up. Uncertainty is the property of a person and not a system/message. That uncertainty is a function of both a person's model of a system/message and their prior observations. You and I may have different entropies about the content of the same message. If we're calculating the entropy of dice rolls (where the outcome is the 'message'), and I know the dice are loaded but you don't, my entropy will be lower than yours.

ninetyninenine 1y ago
Not true. The uncertainty of the dice rolls is not controlled by you. It is the property of the loaded dice itself. Here's a better way to put it. If I roll the dice infinite times. The uncertainty of the outcome of the dice will become evident in the distribution of the outcomes of the dice. Whether you or another person is certain or uncertain of this does not indicate anything. Now when you realize this you'll start to think about this thing in probability called frequentists vs. bayesian and you'll realize that all entropy is, is a consequence of probability and that the philosophical debate in probability applies to entropy as well because they are one and the same. I think the word "entropy" confuses people into thinking it's some other thing when really it's just probability at work.
- glial 1y ago
  I concede that my framing was explicitly Bayesian, but with that caveat, it absolutely is true: your uncertainty is a function of your knowledge, which is a model of the world, but is not equivalent to the world itself. Suppose I had a coin that only landed on heads. You don't know this and you flip the coin. According to your argument, for the first flip, your entropy about the outcome of the flip is zero. However, you wouldn't be able to tell me which way the coin would land, making your entropy nonzero. This is a contradiction.
  - nyrikki 1y ago
    To add to this. Both the Bayesian vs frequentist interpretations make understanding the problem challenging, as both are powerful interpretations to find the needle in the haystack, when the problem is finding the hay in the haystack. A better lens is that a recursive binary sequence (coin flips) is an algorithmically random sequence if and only if it is a Chaitin's number.[1] Chaitin's number is normal, which is probably easier understood with decimal digits meaning that with any window size, over time the distribution, the distribution of 0-9 will be the same. This is why HALT ≈ open frame ≈ system identification ≈ symbol grounding problems. Probabilities are very powerful for problems like The dining philosophers problem or the Byzantine generals problem, they are still grabbing needles every time they reach into the hay stack. Pretty much any almost all statement is a hay in the haystack problem. For example almost all real numbers are normal, but we have only found a few. We can construct them, say with .101010101 in base 2 .123123123123 in base 3 etc...but we can't access them. Given access to the true reals, you have 0 percent chance of picking a computable number, rational, etc... but a 100% chance of getting a normal number or 100% chance of getting an uncomputable number. Bayesian vs frequentist interpretations allow us to make useful predictions, but they are the map, not the territory. Bayesian iid data and Frequentist iid random variables play the exact similar roles Enthalpy, Gibbs free energy, statistical entropy, information theory entropy, Shannon Entropy etc... The difference between them is the independent variables that they depend on and the needs of the model they are serving. You can also approach the property that people often want to communicate when using the term entropy as effective measure 0 sets, null cover, martingales, kolmogorov complexity, compressibility, set shattering, etc... As a lens, null cover is most useful in my mind, as a random real number should not have any "uncommon" properties, or look more like the normal reals. This is very different from statistical methods, or any effective usable algorithm/program, which absolutely depend on "uncommon" properties. Which is exactly the hay in the problem of finding the hay haystack problem, hay is boring. [1]https://www.cs.auckland.ac.nz/~cristian/samplepapers/omegastacs.pdf https://www.cs.auckland.ac.nz/~cristian/samplepapers/omegast...
- bloppe 1y ago
  Probability is subjective though, because macrostates are subjective. The notion of probability relies on the notion of repeatability: if you repeat a coin flip infinite times, what proportion of outcomes will be heads, etc. But if you actually repeated the toss exactly the same way every time, say with a finely-tuned coin-flipping machine in a perfectly still environment, you would always get the same result. We say that a regular human flipping a coin is a single macrostate that represents infinite microstates (the distribution of trajectories and spins you could potentially impart on the coin). But who decides that? Some subjective observer. Another finely tuned machine could conceivably detect the exact trajectory and spin of the coin as it leaves your thumb and predict with perfect accuracy what the outcome will be. According to that machine, you're not repeating anything. You're doing a new thing every time.
  - canjobear 1y ago
    Probability is a bunch of numbers that add to 1. Sometimes you can use this to represent subjective beliefs. Sometimes you can use it to represent objectively existing probability distributions. For example, an LLM is a probability distribution on a following token given previous tokens. If two "observers" disagree about an LLM's probability assigned to some token, then only at most one of them can be correct. So the probability is objective.
    - bloppe 1y ago
      We're talking about 2 different things. I agree that probability is objective as long as you've already decided on the definition of the macrostate, but that definition is subjective. From an LLM's perspective, the macrostate is all the tokens in the context window and nothing more. A different observer may be able to take into account other information, such as the identity and mental state of the author, giving rise to a different distribution. Both of these models can be objectively valid even though they're different, because they rely on different definitions of the macrostate. It can be hard to wrap your head around this, but try taking it to the extreme. Let's say there's an omniscient being that knows absolutely everything there is to know about every single atom within a system. To that observer, probability does not exist, because every macrostate represents a single microstate. In order for something to be repeated (which is core to the definition of probability), it must start from the exact same microstate, and thus always have the same outcome. You might think that true randomness exists at the quantum level and that means true omniscience is impossible (and thus irrelevant), but that's not provable and, even if it were true, would not invalidate the general point that probabilities are determined by macrostate definition.
- quietbritishjim 1y ago
  You're right it reduces to Bayesian vs frequentist views of probability. But you seem to be taking an adamantly frequentist view yourself. Imagine you're not interested in whether a dice is weighted (in fact assume that it is fair in every reasonable sense), but instead you want to know the outcome of a specific roll. What if that roll has already happened, but you haven't seen it? I've cheekily covered up the dice with my hand straight after I rolled it. It's no longer random at all, in at least some philosophical points of view, because its outcome is now 100% determined. If you're only concerned about "the property of the dice itself" are you now only concerned with the property of the roll itself? It's done and dusted. So the entropy of that "random variable" (which only has one outcome, of probability 1) is 0. This is actually a valid philosophical point of view. But people that act as though the outcome is still random, allow themselves to use probability theory as if it hadn't been rolled yet, are going to win a lot more games of chance than those that refuse to. Maybe this all seems like a straw man. Have I argued against anything you actually said in your post? Yes I have: your core disagreement with OP's statement "entropy is a property of an individual". You see, when I covered up the dice with my hand, I did see it. So if you take the Bayesian view of probability and allow yourself to consider that dice roll probabilistically, then you and I really do have different views about the probability distribution of that dice roll and therefore the entropy. If I tell a third person, secretly and honestly, that the dice roll is even then they have yet another view of the entropy of the same dice roll! All at the same time and all perfectly valid.
empath75 1y ago
> If we're calculating the entropy of dice rolls (where the outcome is the 'message'), and I know the dice are loaded but you don't, my entropy will be lower than yours. That's got nothing to do with entropy being subjective. If 2 people are calculating any property and one of them is making a false assumption, they'll end up with a different (false) conclusion.
- glial 1y ago
  Entropy is based on your model of the world and every model, being a simplification and an estimate, is false.
- mitthrowaway2 1y ago
  What if I told you the dice were loaded, but I didn't tell you which face they were loaded in favor of? Then you (presumably) assign a uniform probability over one true assumption and five false assumptions. Which is the sort of situation where subjective entropy seems quite appropriate.
- canjobear 1y ago
  > If 2 people are calculating any property and one of them is making a false assumption, they'll end up with a different (false) conclusion. This implies that there is an objectively true conclusion. The true probability is objective.
  - mitthrowaway2 1y ago
    Ok. I rolled a die and the result was 5. What should the true objective probability have been for the outcome of that roll?
pharrington 1y ago
Are you basically just saying "we're not oracles"?
Geee 1y ago
It's both. The system or process has it's actual entropy, and the sequence of observations we make has a certain entropy. We can say that "this sequence of numbers has this entropy", which is slightly different from the entropy of the process which created the numbers. For example, when we make more coin tosses, our sequence of observations has an entropy which gets closer and closer to the actual entropy of the coin.

by glial 1y ago

ninetyninenine 1y ago

Not true. The uncertainty of the dice rolls is not controlled by you. It is the property of the loaded dice itself. Here's a better way to put it. If I roll the dice infinite times. The uncertainty of the outcome of the dice will become evident in the distribution of the outcomes of the dice. Whether you or another person is certain or uncertain of this does not indicate anything. Now when you realize this you'll start to think about this thing in probability called frequentists vs. bayesian and you'll realize that all entropy is, is a consequence of probability and that the philosophical debate in probability applies to entropy as well because they are one and the same. I think the word "entropy" confuses people into thinking it's some other thing when really it's just probability at work.

glial 1y ago
I concede that my framing was explicitly Bayesian, but with that caveat, it absolutely is true: your uncertainty is a function of your knowledge, which is a model of the world, but is not equivalent to the world itself. Suppose I had a coin that only landed on heads. You don't know this and you flip the coin. According to your argument, for the first flip, your entropy about the outcome of the flip is zero. However, you wouldn't be able to tell me which way the coin would land, making your entropy nonzero. This is a contradiction.
- nyrikki 1y ago
  To add to this. Both the Bayesian vs frequentist interpretations make understanding the problem challenging, as both are powerful interpretations to find the needle in the haystack, when the problem is finding the hay in the haystack. A better lens is that a recursive binary sequence (coin flips) is an algorithmically random sequence if and only if it is a Chaitin's number.[1] Chaitin's number is normal, which is probably easier understood with decimal digits meaning that with any window size, over time the distribution, the distribution of 0-9 will be the same. This is why HALT ≈ open frame ≈ system identification ≈ symbol grounding problems. Probabilities are very powerful for problems like The dining philosophers problem or the Byzantine generals problem, they are still grabbing needles every time they reach into the hay stack. Pretty much any almost all statement is a hay in the haystack problem. For example almost all real numbers are normal, but we have only found a few. We can construct them, say with .101010101 in base 2 .123123123123 in base 3 etc...but we can't access them. Given access to the true reals, you have 0 percent chance of picking a computable number, rational, etc... but a 100% chance of getting a normal number or 100% chance of getting an uncomputable number. Bayesian vs frequentist interpretations allow us to make useful predictions, but they are the map, not the territory. Bayesian iid data and Frequentist iid random variables play the exact similar roles Enthalpy, Gibbs free energy, statistical entropy, information theory entropy, Shannon Entropy etc... The difference between them is the independent variables that they depend on and the needs of the model they are serving. You can also approach the property that people often want to communicate when using the term entropy as effective measure 0 sets, null cover, martingales, kolmogorov complexity, compressibility, set shattering, etc... As a lens, null cover is most useful in my mind, as a random real number should not have any "uncommon" properties, or look more like the normal reals. This is very different from statistical methods, or any effective usable algorithm/program, which absolutely depend on "uncommon" properties. Which is exactly the hay in the problem of finding the hay haystack problem, hay is boring. [1]https://www.cs.auckland.ac.nz/~cristian/samplepapers/omegastacs.pdf https://www.cs.auckland.ac.nz/~cristian/samplepapers/omegast...
bloppe 1y ago
Probability is subjective though, because macrostates are subjective. The notion of probability relies on the notion of repeatability: if you repeat a coin flip infinite times, what proportion of outcomes will be heads, etc. But if you actually repeated the toss exactly the same way every time, say with a finely-tuned coin-flipping machine in a perfectly still environment, you would always get the same result. We say that a regular human flipping a coin is a single macrostate that represents infinite microstates (the distribution of trajectories and spins you could potentially impart on the coin). But who decides that? Some subjective observer. Another finely tuned machine could conceivably detect the exact trajectory and spin of the coin as it leaves your thumb and predict with perfect accuracy what the outcome will be. According to that machine, you're not repeating anything. You're doing a new thing every time.
- canjobear 1y ago
  Probability is a bunch of numbers that add to 1. Sometimes you can use this to represent subjective beliefs. Sometimes you can use it to represent objectively existing probability distributions. For example, an LLM is a probability distribution on a following token given previous tokens. If two "observers" disagree about an LLM's probability assigned to some token, then only at most one of them can be correct. So the probability is objective.
  - bloppe 1y ago
    We're talking about 2 different things. I agree that probability is objective as long as you've already decided on the definition of the macrostate, but that definition is subjective. From an LLM's perspective, the macrostate is all the tokens in the context window and nothing more. A different observer may be able to take into account other information, such as the identity and mental state of the author, giving rise to a different distribution. Both of these models can be objectively valid even though they're different, because they rely on different definitions of the macrostate. It can be hard to wrap your head around this, but try taking it to the extreme. Let's say there's an omniscient being that knows absolutely everything there is to know about every single atom within a system. To that observer, probability does not exist, because every macrostate represents a single microstate. In order for something to be repeated (which is core to the definition of probability), it must start from the exact same microstate, and thus always have the same outcome. You might think that true randomness exists at the quantum level and that means true omniscience is impossible (and thus irrelevant), but that's not provable and, even if it were true, would not invalidate the general point that probabilities are determined by macrostate definition.
quietbritishjim 1y ago
You're right it reduces to Bayesian vs frequentist views of probability. But you seem to be taking an adamantly frequentist view yourself. Imagine you're not interested in whether a dice is weighted (in fact assume that it is fair in every reasonable sense), but instead you want to know the outcome of a specific roll. What if that roll has already happened, but you haven't seen it? I've cheekily covered up the dice with my hand straight after I rolled it. It's no longer random at all, in at least some philosophical points of view, because its outcome is now 100% determined. If you're only concerned about "the property of the dice itself" are you now only concerned with the property of the roll itself? It's done and dusted. So the entropy of that "random variable" (which only has one outcome, of probability 1) is 0. This is actually a valid philosophical point of view. But people that act as though the outcome is still random, allow themselves to use probability theory as if it hadn't been rolled yet, are going to win a lot more games of chance than those that refuse to. Maybe this all seems like a straw man. Have I argued against anything you actually said in your post? Yes I have: your core disagreement with OP's statement "entropy is a property of an individual". You see, when I covered up the dice with my hand, I did see it. So if you take the Bayesian view of probability and allow yourself to consider that dice roll probabilistically, then you and I really do have different views about the probability distribution of that dice roll and therefore the entropy. If I tell a third person, secretly and honestly, that the dice roll is even then they have yet another view of the entropy of the same dice roll! All at the same time and all perfectly valid.

empath75 1y ago

> If we're calculating the entropy of dice rolls (where the outcome is the 'message'), and I know the dice are loaded but you don't, my entropy will be lower than yours. That's got nothing to do with entropy being subjective. If 2 people are calculating any property and one of them is making a false assumption, they'll end up with a different (false) conclusion.

glial 1y ago
Entropy is based on your model of the world and every model, being a simplification and an estimate, is false.
mitthrowaway2 1y ago
What if I told you the dice were loaded, but I didn't tell you which face they were loaded in favor of? Then you (presumably) assign a uniform probability over one true assumption and five false assumptions. Which is the sort of situation where subjective entropy seems quite appropriate.
canjobear 1y ago
> If 2 people are calculating any property and one of them is making a false assumption, they'll end up with a different (false) conclusion. This implies that there is an objectively true conclusion. The true probability is objective.
- mitthrowaway2 1y ago
  Ok. I rolled a die and the result was 5. What should the true objective probability have been for the outcome of that roll?

pharrington 1y ago

Are you basically just saying "we're not oracles"?

Geee 1y ago

It's both. The system or process has it's actual entropy, and the sequence of observations we make has a certain entropy. We can say that "this sequence of numbers has this entropy", which is slightly different from the entropy of the process which created the numbers. For example, when we make more coin tosses, our sequence of observations has an entropy which gets closer and closer to the actual entropy of the coin.