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Random Walks aim to explore a distribution through random perturbations. This is likely to move into low density regions, which is inefficient. Hamiltonian MC sails through a distribution much more rapidly by incorporating the distribution's structure into proposals.

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Alec Helbling

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One of the simplest algorithms for sampling from a probability distribution is Random Walk Metropolis-Hastings. It proposes new samples by taking Gaussian-distributed steps, accepting or rejecting them to maintain the target distribution. I call this pdf the "fidget spinner".
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One of the simplest algorithms for sampling from a probability distribution is Random Walk Metropolis-Hastings. It proposes new samples by taking Gaussian-distributed steps, accepting or rejecting them to maintain the target distribution. I call this pdf the "fidget spinner".

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A basic strategy for drawing from a random distribution is to first generate uniformly-distributed points, then "warp" these points so they're spread out according to the target distribution. For example, the Box-Muller transform takes uniform points to normally-distributed ones
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Langevin Monte Carlo allows you to draw samples from a probability distribution using its log gradient ∇ log p(x). By performing a sort of gradient ascent with noise you can navigate around the distribution. Langevin MC is heavily related to modern diffusion models.
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Did you know you can create the Maxwell-Boltzmann distribution with just some balls and a motor? I built this simple device to illustrate it. This is how it works: A spinning agitator pumps energy into ~400 balls, so they bounce around and collide like gas particles in a box. They escape one at a time through a tiny hole, then fall a fixed gap into bins below. Since the hole is so small, every ball leaves with no vertical speed and takes the same time to fall. Constant fall time means how far a ball flies sideways is set purely by its speed. A faster ball goes into a further bin. So each bin is really a speed. Tally all 400 and you get the Maxwell-Boltzmann distribution.
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Did you know you can create the Maxwell-Boltzmann distribution with just some balls and a motor? I built this simple device to illustrate it. This is how it works: A spinning agitator pumps energy into ~400 balls, so they bounce around and collide like gas particles in a box. They escape one at a time through a tiny hole, then fall a fixed gap into bins below. Since the hole is so small, every ball leaves with no vertical speed and takes the same time to fall. Constant fall time means how far a ball flies sideways is set purely by its speed. A faster ball goes into a further bin. So each bin is really a speed. Tally all 400 and you get the Maxwell-Boltzmann distribution.

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