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by Fishysoup
2052 days ago
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Something I've had trouble finding resources for is how to apply probabilistic/Bayesian techniques and thinking to chaotic dynamical systems. People keep telling me "Look up MCMC" but I don't see the relevance to dynamical systems (further than the notion that you can maybe sample from them with MCMC somehow?) |
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If you want to condition on an event? Say you want to predict the weather on Tuesday, conditioned on the event that it rained on Sunday? Run a lot of simulations, and only keep the ones where it rained on Sunday.
Similarly: If you want to compute an expectation value? Run a lot of simulations and take the average.
If the event is unlikely? Say you want to condition on the fact that it rained on Sunday, and the high temperature was precisely 15 degrees C? Then you have a difficult problem on your hands.
(Sometimes the expectation value of the quantity you care about will depend a lot on a few rare events with outcomes many standard deviations away from the mean. Then you also have a difficult problem on your hands.)
Sometimes MCMC will work on this kind of problem, and sometimes it won't. Even if it doesn't, maybe other techniques will work.
To apply MCMC to a dynamical system, one method is write down all of the history of the system as a single object, say a single vector. You write down what your system is doing at t=1, at t=2, etc, and all that information goes into the vector. The rules governing the system determine a probability distribution over the vector space that the vector lives in. (Or more generally, the object in the object space. The vector axioms aren't important here, it's just a nice familiar example.)
Generally speaking, if you know how to describe your dynamical system, you know how to compute an non-normalized probability for any given vector. Usually you won't be able to compute a normalized probability for that vector. That's fine, since MCMC works with non-normalized probabilities.
The next step is simply to run MCMC. If you want to condition on some fact, cut out all the parts of the space where that fact doesn't hold, and then run MCMC. If you want to compute an expectation value, there are other tweaks to MCMC that are possible (i.e. importance sampling).