Radford M. Neal
2012 · arXiv
Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious "momentum" variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be used to define complex mappings without the need to account for a hard-to-compute Jacobian factor - a property that can be exactly maintained even when the dynamics is approximated by discretizing time. In this review, I discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.
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Markov chains in essence are simple. Instead of diverging and reading all the theory, I'd recommend do it on a need basis. Learn as you go. So pick up a problem and move ahead. I don't think it is fruitful to just learn everything about Markov Chains just for the sake of it. Markov Chain Monte Carlo to sample from probability distributions is a good start - https://arxiv.org/abs/1206.1901 if you are into sampling.
Probabilistic programming uses computer science techniques to do automated statistical modeling. For example, imagine I have a coin, and I want to discover if it is biased, i.e. if it lands on heads more often than tails. In a probabilistic programming framework, I can express my model as a simple Bernoulli model, `x ~ Bernoulli(p)`, and then automatically estimate the bias parameter `p` given some data (do "inference"). You can easily do this calculation by hand or in Python, but this does not generalize to more complex real-world scenarios. For complex probabilistic models, we must rely on numerical approximations. MCMC is just one algorithm for doing this approximate inference. Another popular technique is called variational inference [2]. Another commenter mentioned HMC [3], which is just a specific instance of MCMC. [1] https://mc-stan.org/ [2] https://arxiv.org/abs/1601.00670 [3] https://arxiv.org/abs/1206.1901
Radford Neal's expository paper [1] on HMC is a beautiful overview of the area, and honestly one of the most enlightening papers I've ever read. Conceptually, HMC really isn't that difficult. The broad idea is as follows: Borrowing from statistical physics, it's possible to link energy and probability as E(x) = exp(-P(x)). So you can interpret your MCMC target as an "energy landscape". Highpoints in the landscape correspond to low probabilities in the target distribution, and valleys to high-probability regions. Now add a source of kinetic energy (an auxiliary probability distribution), and think of your MCMC sampler as a hockey puck sliding frictionlessly through the…
energy landscape. Since kinetic + potential energy is conserved, it follows that you move through contours of your joint (target, auxiliary) distribution. That means you never have to reject a sample, regardless of how far the hockey puck travels. This interpretation also shows why you need to use gradient information. To know where the hockey puck is going to move next, you need to know how steep the walls of your energy landscape are: that is, you need the gradient! [1] https://arxiv.org/pdf/1206.1901.pdf
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