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What Is Entropy?

jasonfantl.com

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Re: What Is Entropy?

#81

Post by kgwgk »

kgwgk wrote: post #79
canjobear wrote: post #77

Earlier quoted context omitted.

An LLM distribution exists in the physical world, just as much as this comment does. It didn’t exist before the model was trained. It has relation to the physical world: it assigns probabilities to subword units of text. It has commercial value that it wouldn’t have if its objective probability values were different.

> It has relation to the physical world: it assigns probabilities to subword units of text. How is that probability assignment linked to the physical world exactly? In the physical world the computer will produce a token. You rejected before that it was about predicting the token that would be produced.

Or maybe you mean that the probability assignments are not about the output of a particular LLM implementation in the real world but about subword units of text in the wild.

In that case how could two different LLMs do different assigments to the same physical world without being wrong? Would they be “objective” but unrelated to the “object”?

Re: What Is Entropy?

#82

Post by bloppe »

canjobear wrote: post #55
bloppe wrote: post #51

Earlier quoted context omitted.

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 ris…

Suppose you're training a language model by minimizing cross entropy, and the omniscient being is watching. In each step, your model instantiates some probability distribution, whose gradients are computed. That distribution exists, and is not deterministic to the omniscient entity.

An LLM is given a definition of the macrostate which creates the probability distribution, but a different definition of the macrostate (such as would be known to the omniscient being) would create a different distribution. According to the omniscient entity, the vast majority of long combinations of tokens would have zero probability because nobody will ever write them down in that order. The infinite monkey theorem is misleading in this regard. The odds of producing Shakespeare's works completely randomly before the heat death of the universe are practically zero, even if all the computing power in the world were dedicated to the cause.

Re: What Is Entropy?

#83

Post by asdf_snar »

I throw these quotes by Y. Oono into the mix because they provide viewpoints which are in some tension with those who take -\sum_x p(x) log p(x) definition of entropy as fundamental.

> Boltzmann’s argument summarized in Exercise of 2.4.11 just derives Shannon’s formula and uses it. A major lesson is that before we use the Shannon formula important physics is over.

> There are folklores in statistical mechanics. For example, in many textbooks ergodic theory and the mechanical foundation of statistical mechanics are discussed even though detailed mathematical explanations may be missing. We must clearly recognize such topics are almost irrelevant to statistical mechanics. We are also brainwashed that statistical mechanics furnishes the foundation of thermodynamics, but we must clearly recognize that without thermodynamics statistical mechanics cannot be formulated. It is a naive idea that microscopic theories are always more fundamental than macroscopic phenomenology.

sources: http://www.yoono.org/download/inst.pdf http://www.yoono.org/download/smhypers12.pdf

Re: What Is Entropy?

#84

Post by kgwgk »

canjobear wrote: post #55
bloppe wrote: post #51

Earlier quoted context omitted.

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 ris…

Suppose you're training a language model by minimizing cross entropy, and the omniscient being is watching. In each step, your model instantiates some probability distribution, whose gradients are computed. That distribution exists, and is not deterministic to the omniscient entity.

What’s non deterministic there?

That “probability distribution” is just a mathematical function assigning numbers to tokens, defined using a model that the person creating the model and the omniscent entity know, applying a set of deterministic mathematical functions to a sequence of observed inputs that the person creating the model and the omniscent entity also know.

Re: What Is Entropy?

#85

Post by dist-epoch »

frank20022 wrote: post #78
infogulch wrote: post #54

Earlier quoted context omitted.

Half a year after that talk Wolfram appeared on a popular podcast [1] to discuss his book on the Second Law of Thermodynamics [2]. That discussion contained the best one-sentence description of entropy I've ever heard: > Entropy is the logarithm of the number of states that are consistent with what you know about a system. [1]: Mystery of Entropy FINALLY Solved After 50 Years? (Stephen Wolfram) - Machine Learning Str…

By that definition, the entropy of a game of chess decreases with time because as the game moves on there are less possible legal states. Did I get that right?

Is about subjective knowledge, not objective.

So entropy is not related to the number of remaining legal states.

If I know the seed of a PRNG, the entropy of the numbers it generates is zero for me. If I don't know the seed, it has very high entropy.

https://www.quantamagazine.org/what-is-entropy-a-measure-of-...

Re: What Is Entropy?

#86

Post by namaria »

3abiton wrote: post #80
xavivives wrote: post #68

Over the last few months, I've been developing an unorthodox perspective on entropy [1] . It defines the phenomenon in much more detail, allowing for a unification of all forms of entropy. It also defines probability through the same lens. I define both concepts fundamentally in relation to priors and possibilities: - Entropy is the relationship between priors and ANY possibility, relative to the entire space of poss…

I am curious why the word "entropy" encompasses so many concepts? Wouldn't it have made sense to just give each concept a different word?

Yes. There are different concepts called 'entropy', sometimes merely because their mathematical formulation looks very similar.

It means different things in different contexts and an abstract discussion of the term is essentially meaningless.

Even discussions within the context of the second law of thermodynamics are often misleading because people ignore much of the context in which the statistical framing of the law was formulated. Formal systems and all that... These are not general descriptions of how nature works, but formal systems definitions that allow for some calculations.

I find the study of symmetries by Noether much more illuminating in general than trying to generalize conservation laws as observed within certain formal models.

Re: What Is Entropy?

#87

Post by tsimionescu »

This goes through all definitions of entropy, except the very first one, which is also the one that is in fact measurable and objective: the variation in entropy is the amount of heat energy that the system exchanges with the environment at a given temperature during a reversible process. While tedious, this can be measured, and it doesn't depend on any subjective knowledge about the system. Any two observers will agree on this value, even if one knows all of the details of every single microstate.

Re: What Is Entropy?

#88

Post by prof-dr-ir »

3abiton wrote: post #80
xavivives wrote: post #68

Over the last few months, I've been developing an unorthodox perspective on entropy [1] . It defines the phenomenon in much more detail, allowing for a unification of all forms of entropy. It also defines probability through the same lens. I define both concepts fundamentally in relation to priors and possibilities: - Entropy is the relationship between priors and ANY possibility, relative to the entire space of poss…

I am curious why the word "entropy" encompasses so many concepts? Wouldn't it have made sense to just give each concept a different word?

Whenever there is an entropy, it can be defined as

S = - sum_n p_n log( p_n )

where the p_n is a probability distribution: for n = 1...W, p_n >= 0 and sum_n p_n = 1. This is always the underlying equation, the only thing that changes is the probability distribution.

Re: What Is Entropy?

#89

Post by FilosofumRex »

Boltzmann and Gibbs turn in their graves, every time some information theorist mutilates their beloved entropy. Shanon & Von Neumann were hacking a new theory of communication, not doing real physics and never meant to equate thermodynamic concepts to encoding techniques - but alas now dissertations are written on it.

Entropy can't be a measure of uncertainty, because all the uncertainty is in the probability distribution p(x) - multiplying it with its own logarithm and summing doesn't tell us anything new. If it did, it'd violate quantum physics principles including the Bell inequality and Heisenberg uncertainty.

The article never mentions the simplest and most basic definition of entropy, ie its units (KJ/Kelvin), nor the 3rd law of thermodynamics which is the basis for its measurement.

“Every physicist knows what entropy is. Not one can write it down in words.” Clifford Truesdell

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