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.
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.