I announce over and over that the chronicle ordering of the post are irrelevant for beginners' favor. There are many blanks I skipped. I would fill the holes later. Variational method During my physics coursework and researches, I used this method countlessly. I even had a book of the name. It is quite simple, but also as big topic as being a book. Simply put, it is a technique to find equations and solutions (sometimes approximate solutions) by extremizing functionals which is mainly just integrals of fields, and treat the functions in the integral, as parameters.

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Yay! Finally something more directly from physics to data science. We will also have a chance to see how Metropolis-Hastings algorithm works! The Hamiltonian Monte Carlo method is a kind of Metropolis-Hastings method. One of the weak points of Monte Carlo sampling comes up with random walks. Hamiltonian Monte Carlo method (HMC) is an approach to reducing the randomizing in algorithm of the sampling. The original name was hybrid Monte Carlo method.

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In advance, I will proceed in the extension of the previous post. I will use the same target distribution function and the similar Gaussian disposal distribution. Even Python script will be better understood if you’ve already read the previous post about importance sampling. The rejection sampling could be the most familiar Monte Carlo sampling. When need to introduce Monte Carlo method to somebody, it is very intuitive and effective to give an example of computing the area of the circle (or anything) by using random samples.

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Importance sampling is the first sampling method I faced when I studied Monte Carlo method. Nevertheless, I haven’t seen many examples for the importance sampling. Maybe it is because the importance sampling is not effective for high dimensional systems. The weak point of the importance sampling is that the performance of it is determined by how well we choose the disposal distribution close to the target distribution. Here, I will present a simple example of the importance sampling.

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Exact Markov chain Monte Carlo sampling I don’t like the naming. The word exact could mislead us to understand the concept. Anyway I used the word in the title because it was the title of the chapter of the book “Information Theory, Inference, and Learning Algorithms” by David Mackay, which I studied to learn the theory. The different names of it are perfect simulation and coupling from the past. Maybe these are better names.

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