That "coming clean" bit really highlights the difference between theory and reality; to operationalize Metropolis-Hastings requires a lot of extra work.

I really respect the way the 'like' and 'sub' requests weren't explicit. It was made clear they were also important and prioritized them by making their consideration a natural part of the exposition. Bravo.

This is brilliant, as a physicist turned pseudo-statistician, the physics-motivated approach of MCMC makes more sense to me than the statistical approach.
Fun story: despite bearing his name, Nicholas Metropolis didn't contribute at all to the algorithm. The concept was originally developed by physicist Marshall Rosenbluth at Los Alamos for studying thermonuclear explosions. His wife Arianna Rosenbluth (also a physicist) turned the concept into code. The Rosenbluths are the true pioneers. They struggled with a consistent way to pick an appropriate acceptance function. One day having dinner with Edward Teller and his wife Augusta Teller, they described the problem, the implementation, and the challenge. The Tellers being a couple of physicist+mathematician suggested the obvious choice for a physicist: simply use the Boltzmann distribution. And the first MCMC algorithm was born. What did Metropolis do? He had the keys of the computer lab, and the alphabetical order used for naming authors made his last name the name of the algorithm. I personally call it the Rosenbluth algorithm.

I have been searching for a clear explanation for the acceptance rate in the Metropolis algorithm without success. This is an amazing and brilliant explanation. How flipping between the transition distribution and the stationary distribution makes the idea crestal clear. Thank you very much for sharing such a brilliant explanation.

I've never seen the acceptance rate expression explained/derived so intuitively - great job!

By the way, to get your RWM to work, judging by the traceplots it seems that you would have needed to add some sort of pre-conditioning to change the variance of the proposal distribution for each dimension of your target separately, since alpha and beta seem to be on a different "scale" to the p variables, which is why finding a common "h" that allowed the chain to mix fast enough in all directions without getting stuck would be borderline impossible. RWM (even with a Gaussian proposal rather than uniform) tends to struggle with hierarchical models and high-dimensional models more generally because of this idea, but thankfully we have things like STAN using HMC. Would love to see an explainer on NUTS/HMC in your explanatory style if you ever feel like it!

That was the best explanation of MCMC I have heard. I'm going to listen to it ten more times and see I start to understand. This stuff is really hard for people with no stats or higher math background, but it's fascinating and fun to learn, however slowly

I never would have thought Metropolis was actually the guy’s name

Happy to see you covering this topic. MCMC is what made Bayesian statistics viable beyond conjugate priors and simple distributions. There's a lot more to dig into including Hamiltonian Monte Carlo (what NUTS is based on) and Gibbs sampling, as well as convergence metrics. Bayesian Data Analysis 3rd ed. (Gelman, Carlin, Stern, Dunson, Vehtari, & Rubin) is the go to book for this.

That's actually a pretty cool idea for a city trip activity

Thank you for a stimulating video on an important statistical method. The hand-written equation appearing on-screen from 0:10 to 0:22 and briefly captioned “Despite how it looks, this is just an average,” is a fundamental building block of thermodynamics. It was at the core of my doctoral dissertation project in polymer physical chemistry in the 1960s. A generation earlier the average geometric properties of a polymer such as its end-to-end distance or radius of gyration (average distance of component atoms from center of mass) were simulated with pencil and paper as a Brownian motion type random walk with a fixed step length representing the distance between atoms along the connected chain, often constrained to a 2- or 3-dimensional spatial lattice. Their mathematical properties were delineated by Einstein and others. However, a Brownian particle can visit the same lattice point indefinitely, while two real atoms cannot occupy the same point in space. This gave rise to the concept of the self-avoiding random walk, a much more difficult mathematical problem. The 1950s saw the proliferation of electronic computers like the Illiac, that made it possible to simulate more realistic average polymer properties with self-avoiding walks, but it was quickly found that the proportion of walks without self-intersections decreased rapidly as the chain length (no. of steps) increased. A few clever biasing techniques involving re-starting a self-intersecting walk improved things slightly but these were only minor tweaks.

In the 1960s Moti Lal, a chemist at Unilever Labs in the UK, became aware of Metropolis’s paper and applied it to the polymer problem. However, his available computing power (IBM 360/50) confined his polymers to 30 monomer units on a 2-dimensional spatial lattice. As a graduate student at NYU in 1968 I had access to a Control Data Corp. CDC 6600 Cray Supercomputer at the Courant Institute of Mathematical Sciences (they let us chemists use their facilities). I used the Metropolis-Lal method to generate polymers with free rotation about their backbone bonds, i.e., not restricted to an artificial lattice. I used your hand-written equation with Boltzmann weighting exp(-E/kT) to generate average geometric and thermodynamic properties from thousands of Monte Carlo samples. The Metropolis importance sampling algorithm generated samples from more “important” regions of polymer conformation space so it took fewer samples and far less computer time to get stable averages. I could even generate numerical distributions of end-to-end distances of polymers with sufficient accuracy to discriminate among several distribution functions proposed by theoreticians.

Pretty cool explanation of the acceptance function! I remember seeing this method in my computational statistics class.

Next gibbs sampling, exponential families

There's a cool application of this underlying principle with markov chains in that you can coordinate uncoordinated agents without having to track or communicate with them in real time.

Taking your Portland Example, you would get your 100 people stationed at each activity in roughly the correct proportions by giving them a list of the other activities and the probability that they should stay put or go to each other place.   They'll all shuffle around each hour, but they'll adhere closely to the target.

This has applications for things like,  say,  a swarm of Drones spelling out a word in the sky.  Sure, you could calculate a specific 3d coordinate for each specific drone and transmit that info to each of them 1 by 1 every time you want to change the word.   Or you could just broadcast the 'pixel' locations you want the drones to occupy, and how densely filled each pixel is, and let them individually, randomly pick which space to occupy.

Especially if you let them constantly cycle their positions,  it'll do a good job with the task.

The audio quality on your videos is progressively better. Keep up the good work!

I heard Portland is pretty nice this time of year.  I even think they did a show about Portland called Portlandia. Also based on what I've heard the "drinking craft brew" percentage should be increased by roughly 50 points.

Not a statistician, but a machine learning person -- one of the more fascinating things about this algorithm to me is you can use it to explore a solution space by turning it on its head and inventing a probability distribution that makes better solutions more likely. It's not super widely used but it has applications. Important work that deserves better visibility!

Finally the MCMC video! I like your explanation of the acceptance function and the comind clean section

havent watched the entire thing but it reminds me of that thing where you can survey everyone a random thing (eg: how many jelly beans in a jar) and you'll get very close to the answer

The cancellation of the constant K was beautiful and really highlighted the power of the algorithm. Great explanation!

At 6:34: your f(X) must be non negative. Also, care must be taken about how f(X) behaves at infinity (see heavy tail distributions).