Why Ising model : 3 reasons for relevance Studying Ising model can be useful to understand phase transition of various systems. Hopfield network or Boltzmann machine to the neural network is just a generalized form of Ising model. Ising model is also useful as a statistical model in its own right. Ising model $\boldsymbol{x}$ is the state of an Ising model with $N$ spins be a vector in which each component $\boldsymbol x_n$ takes values $-1$ or $+1$.

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Slice sampling algorithm A single transition $(x,u) \rightarrow (x',u')$ of a one-dimensional slice sampling algorithm has the following steps. (1). evaluate $P^* (x)$ (2). draw a vertical coordinate $u' \sim$ Uniform$(0,P^* (x))$ (3). create a horizontal interval $(x_l, x_r)$ enclosing $x$  3a. draw $r \sim$ Uniform$(0,1)$  3b. $x_l := x-rw$  3c. $x_r := x+(1-r)w$  3d. while $(P^* (x_l) > u')$ ${x_l := x-rw}$  3e. while $(P^* (x_r) > u')$ ${x_r:= x+w}$

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Efficient Monte Carlo sampling This post is on the extension of the post about Hamiltonian Monte Carlo method. Therefore, I assume the readers already read the post. Overrelaxation also reduces the random property of the Monte Carlo sampling, and speeds up the convergence of the Markov chain. Gibbs sampling In advance of studying over relaxation, we study Gibbs sampling. In the general case of a system with K variables, a single iteration involves sampling one parameter at a time.

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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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