I can’t think of any interesting connection between conservation of probability and Liouville’s theorem. Can you elaborate on what you’re thinking about? Conservation of probability just tells us that the total probability must always add up to 1, so if an update increases the probability of some hypothesis, there must be a corresponding decrease in probability of some other hypotheses. Since phase space volume is a finite measure, it must of course satisfy this sort of conservation. If the volume of some region of phase space increases, there must be a corresponding decrease in the volume of some other region.
But Liouville’s theorem tells us something different. It tells us that if we start with a region of phase space A and follow it along as it evolves under a Hamiltonian flow into another region B, then the volumes of A and B will be identical. This feature isn’t forced on us by the mere fact that volume is a measure. The only constraint placed by conservation of probability (in this case, conservation of measure) is that the total volume of phase space must remain constant under the flow. But this is compatible with the volume of some subset of phase space changing as it evolves under the flow, as long as this change is couteracted by another change elsewhere.
The conservation of total phase space volume (or what you might call the conservation of probability) simply follows from the fact that phase space volume is a finite measure. This places a constraint on possible phase space flows. There is no flow that can increase the total measure of phase space. Liouville’s theorem doesn’t just follow from the nature of the volume measure; it follows from the geometric structure of phase space. In a Hamiltonian system, phase space is a symplectic manifold. The flow must preserve the symplectic structure, and a consequence is that it must preserve the natural volume measure. So here it’s not just the general mathematics of measures that’s relevant, it’s the particular symmetries of Hamiltonian phase space. The symplectic structure of Hamiltonian mechanics places a further constraint on the flow, and this time it’s a continuity constraint.
Additionally, if you look at the probability distributions that are actually used in statistical mechanics, these distributions don’t obey the Liouville kind of conservation of probability, but they obviously obey the Bayesian kind of conservation of probability (since they are probability distributions). If the distributions obeyed Liouville’s theorem, there would not be an increase in entropy. Increasing entropy requires the probability distribution to spread out, but Liouville’s theorem forbids this. We get increase of entropy by working with coarse-grained probability distributions that aren’t governed by Liouville’s theorem
OK, I just read the link in your post, and I realized that you’re referring to something different when you talk about conservation of probability in Bayesian epistemology. I still don’t think it has all that much to do with Liouville’s theorem, but some of the stuff I wrote above is a little bit irrelevant. Stupid pragmatist! That’ll teach me to mouth off without first looking at the links.
Still, my main point stands. The Bayesian version of conservation of probability just follows from the mathematics of probability (plus Bayesian updating). The Liouvillean version follows from the geometric structure of the space over which the probability distributions are defined.
I can’t think of any interesting connection between conservation of probability and Liouville’s theorem. Can you elaborate on what you’re thinking about? Conservation of probability just tells us that the total probability must always add up to 1, so if an update increases the probability of some hypothesis, there must be a corresponding decrease in probability of some other hypotheses. Since phase space volume is a finite measure, it must of course satisfy this sort of conservation. If the volume of some region of phase space increases, there must be a corresponding decrease in the volume of some other region.
But Liouville’s theorem tells us something different. It tells us that if we start with a region of phase space A and follow it along as it evolves under a Hamiltonian flow into another region B, then the volumes of A and B will be identical. This feature isn’t forced on us by the mere fact that volume is a measure. The only constraint placed by conservation of probability (in this case, conservation of measure) is that the total volume of phase space must remain constant under the flow. But this is compatible with the volume of some subset of phase space changing as it evolves under the flow, as long as this change is couteracted by another change elsewhere.
The conservation of total phase space volume (or what you might call the conservation of probability) simply follows from the fact that phase space volume is a finite measure. This places a constraint on possible phase space flows. There is no flow that can increase the total measure of phase space. Liouville’s theorem doesn’t just follow from the nature of the volume measure; it follows from the geometric structure of phase space. In a Hamiltonian system, phase space is a symplectic manifold. The flow must preserve the symplectic structure, and a consequence is that it must preserve the natural volume measure. So here it’s not just the general mathematics of measures that’s relevant, it’s the particular symmetries of Hamiltonian phase space. The symplectic structure of Hamiltonian mechanics places a further constraint on the flow, and this time it’s a continuity constraint.
Additionally, if you look at the probability distributions that are actually used in statistical mechanics, these distributions don’t obey the Liouville kind of conservation of probability, but they obviously obey the Bayesian kind of conservation of probability (since they are probability distributions). If the distributions obeyed Liouville’s theorem, there would not be an increase in entropy. Increasing entropy requires the probability distribution to spread out, but Liouville’s theorem forbids this. We get increase of entropy by working with coarse-grained probability distributions that aren’t governed by Liouville’s theorem
OK, I just read the link in your post, and I realized that you’re referring to something different when you talk about conservation of probability in Bayesian epistemology. I still don’t think it has all that much to do with Liouville’s theorem, but some of the stuff I wrote above is a little bit irrelevant. Stupid pragmatist! That’ll teach me to mouth off without first looking at the links.
Still, my main point stands. The Bayesian version of conservation of probability just follows from the mathematics of probability (plus Bayesian updating). The Liouvillean version follows from the geometric structure of the space over which the probability distributions are defined.