(epistemic status: physicist, do simulations for a living)
Our long-term thermodynamic model Pn is less accurate than a simulation
I think it would be fair to say that the Boltzmann distribution and your instantiation of the system contain not more/less but _different kinds of_ information.
Your simulation (assume infinite precision for simplicity) is just one instantiation of a trajectory of your system. There's nothing stochastic about it, it's merely an internally-consistent static set of configurations, connected to each other by deterministic equations of motion.
The Boltzmann distribution is [the mathematical limit of] the distribution that you will be sampling from if you evolve your system, under a certain set of conditions (which are generally very good approximations to a very wide variety of physical systems). Boltzmann tells you how likely you would be to encounter a specific configuration in a run that satisfies those conditions.
I suppose you could say that the Boltzmann distribution is less *precise* in the sense that it doesn't give you a definite Boolean answer whether a certain configuration will be visited in a given run. On the other hand a finite number of runs is necessarily less *accurate* viewed as a sampling of the system's configurational space.
we can't run simulations for a long time, so we have to make do with the Boltzmann distribution
...and on the third hand, usually even for a simple system like a few-atom molecule the dimensionality of the configurational space is so enormous anyway that you have to resort to some form of sampling (propagation of equations of motion is one option) in order to calculate your partition function (the normalizing factor in the Boltzmann distribution). Yes that's right, the Boltzmann distribution is actually *terribly expensive* to compute for even relatively simple systems!
Hope these clarifications of your metaphor also help refine the chess part of your dichotomy! :)
Thanks, I didn't know that about the partition function.
In the post I was thinking about a situation where we know the microstate to some precision, so the simulation is accurate. I realize this isn't realistic.
I'm having trouble following the part about the operators. Could you spell it out in words? What do the two equations represent? Why is one a multiplication and the other a sum?
Sure: If we can predict the next move in the chess game, we can predict the next move, then the next, then the next. By iterating, we can predict the whole game. If we have a probability for each next move, we multiply them to get the probability of the game.
Conversely, if we have a probability for an entire game, then we can get a probability for just the next move by adding up all the probabilities of all games that can follow from that move.
The issue being, of course, that when we think of predicting the outcome of the chess game based on Elo score, we're not making any sort of prediction about the very next move (a feat possible only through logical microscience). A similar thing happens with the gas, where the Boltzmann distribution is not a distribution over histories. I don't think this is a coincidence.
To me it seems like for the considerations you bring up in this post, the difference between the ideal gas and the chess game is that we have a near-exact short-timescale model for the ideal gas, but we don't have a near-exact short-timescale model for the chess game.
If we knew the source code for both of the players in the chess game, we could simulate the game until someone wins, and get an accurate prediction of the outcome, that would be better than just using the ELO ratings.
Running the argument through, we would conclude that intelligence is characterized by us not having a reductionist model of it. Which seems not as ridiculous as it first sounds -- we ascribed intelligent design to eg. rain and evolution before we understood how they worked. Also, if we could near-exactly simulate chess players (in our brains, not using computers), I doubt we would see them as very intelligent.
Another disanalogy is that for an ideal gas you want to predict the microstate (which the long-timescale model doesn't get) but for the chess game you want to predict the macrostate (who wins).
Yeah, I think the fact that Elo only models the macrostate makes this an imperfect analogy. I think a better analogy would involve a hybrid model, which assigns a probability to a chess game based on whether each move is plausible (using a policy network), and whether the higher-rated player won.
I don't think the distinction between near-exact and nonexact models is essential here. I bet we could introduce extra entropy into the short-term gas model and the rollout would still be superior for predicting the microstate than the Boltzmann distribution.
The notation for the sum operator is unclear. I'd advise writing the sum as and using an subscript inside the sum so it's clearer what is being substituted where.
The sum isn't over , though, it's over all possible tuples of length . Any ideas for how to make that more clear?
Why is a chess game the opposite of an ideal gas? On short timescales an ideal gas is described by elastic collisions. And a single move in chess can be modeled by a policy network.
The difference is in long timescales: If we simulated elastic collisions for a long time, we'd end up with a complicated distribution over the microstates of the gas. But we can't run simulations for a long time, so we have to make do with the Boltzmann distribution, which is a lot less accurate.
Similarly, if we rolled out our policy network to get a distribution over chess game outcomes (win/loss/draw), we'd get the distribution of outcomes under self-play. But if we're observing a game between two players who are better players than us, we have access to a more accurate model based on their Elo ratings.
Can we formalize this? Suppose we're observing a chess game. Our beliefs about the next move are conditional probabilities of the form P1(xk+1|x0⋯xk), and our beliefs about the next n moves are conditional probabilities of the form Pn(xk+1⋯xk+n|x0⋯xk). We can transform beliefs of one type into the other using the operators
(ΠnP1)(xk+1⋯xk+n|x0⋯xk):=n−1∏i=0P1(xk+i+1|x0⋯xk+i)
(ΣnPn)(xk+1|x0⋯xk):=∑xk+2⋯∑xk+nPn(xk+1⋯xk+n|x0⋯xk)
If we're logically omniscient, we'll have ΠnP1=Pn and ΣnPn=P1. But in general we will not. A chess game is short enough that Πn is easy to compute, but Σn is too hard because it has exponentially many terms. So we can have a long-term model Pn that is more accurate than the rollout ΠnP1, and a short-term model P1 that is less accurate than ΣnPn. This is a sign that we're dealing with an intelligence: We can predict outcomes better than actions.
If instead of a chess game we're predicting an ideal gas, the relevant timescales are so long that we can't compute Πn or Σn. Our long-term thermodynamic model Pn is less accurate than a simulation ΠnP1. This is often a feature of reductionism: Complicated things can be reduced to simple things that can be modeled more accurately, although more slowly.
In general, we can have several models at different timescales, and Π and Σ operators connecting all the levels. For example, we might have a short-term model describing the physics of fundmental particles; a medium-term model describing a person's motor actions; and a long-term model describing what that person accomplishes over the course of a year. The medium-term model will be less accurate than a rollout of the short-term model, and the long-term model may be more accurate than a rollout of the medium-term model if the person is smarter than us.