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".show more

Flow Matching aims to learn a "flow" that transforms... a simple source distribution (e.g. Gaussian) to an arbitrarily complex target distribution. This video shows the evolution of the marginal probability path as a source distribution is transformed to a target distribution.show more

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 onesshow more

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.show more

A discrete Markov chain is basically a random walk... on a graph, where each outgoing edge has a fixed probability. Cool fact: any initial distribution on a sufficiently nice* Markov chain converges to its stationary distribution after many steps—because it's a contraction mapping.show more

Hamiltonian Monte Carlo frames sampling from a probability distribution... as a physics problem. By endowing "particles" with momentum and simulating their energy and motion through Hamilton's equations you can efficiently explore a distribution.show more

This animation shows how a uniform grid of points... is deformed by a flow matching model. The visualization highlights how the learned flow warps the underlying space—stretching, compressing, and bending it—to transform one probability distribution into another.show more

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.show more

The π in the normal distribution has a geometric... origin. It comes from a famous mathematical result known as the Gaussian integral. ∫₋∞^∞ e^(−x²) dx = √π The interesting part is that e^(−x²) has no elementary antiderivative. Yet its integral over the entire real line can be calculated exactly. The solution involves turning a one-dimensional integration problem into a two-dimensional one. The resulting function, e^(−(x²+y²)), has circular symmetry, and polar coordinates reveal where π enters the calculation. This result is fundamental to probability theory. The familiar factor 1/(σ√(2π)) in the normal distribution comes from the same integral. It ensures that the total probability across all possible values is exactly 1. But its applications extend well beyond statistics. In engineering, Gaussian distributions are commonly used to model measurement errors and noise. With independent, zero-mean Gaussian errors of equal variance in an additive-error model, maximum-likelihood estimation is equivalent to minimizing the sum of squared residuals. Gaussian functions also appear in heat diffusion, where they describe how an initially concentrated point source spreads through an unbounded, homogeneous medium. In Fourier analysis, the transform of a Gaussian is another Gaussian, up to scaling. This property is useful in signal processing and image filtering. The same mathematical function connects probability distributions, statistical estimation, heat transfer, and frequency analysis. And the √π in its integral ultimately comes from the geometry of the plane.show more

This is a fun interactive article explaining the intuition... behind Gaussian Processes. Gaussian Processes allow you to construct a distribution over functions with a continuous domain. They are incredibly useful for applications like Bayesian regression.show more

A high-dimensional Gaussian is a soap bubble: the densest... point is the emptiest place. In high dimensions, density and mass diverge completely. For a d-dimensional standard Gaussian distribution, nearly all probability mass is concentrated in a "thin shell" or annulus with radius √d, rather than near the center. The mode has the highest density, but every sample you'll ever draw lives on a thin shell at radius √d. The typical point and the most probable point are in completely different places. Today was the n-th time this has tripped me up, so to commemorate it, I just created this visualization. 🫠show more

This simulation visualizes the Maxwell Boltzmann distribution by showing... how gas particles move at different speeds. The emerging bell curve highlights the statistical probability of finding molecules at specific velocity states.show more

Instead of pointing lookAt to a position, I tried... driving it along a line. The control over distribution is kind of nice.show more

this is my app "Kairo Alarm" i fought the... urge to keep polishing it, and shipped it in 7 days. if you're starting a new app: → build the fastest polished version possible → ship in 7–14 days max → go all-in on distribution distribution is what matters early onshow more

This is an unbelievably large mistake by the Eagles... to not go for a 4th and long 1 at the 49 down by 13 points. One of the biggest win probability mistakes I can ever remember from Siriannishow more

EVERYONE SELLING BITCOIN HERE IS SELLING THE MANIPULATION. THAT'S... THE DESIGN. The move from $125K down to $63,749 feels like the trend broke. On this map, it's the shakeout before the real move. But I won't pretend it's guaranteed. Hold $55K, and distribution follows. Lose it, and I was wrong and this was distribution all along.show more

Footage of a Ukrainian FP-2 attack drone slamming into... a Russian Wildberries distribution center in Yekaterinburg this morning. The Russian distribution center, hit by multiple Ukrainian drones, is now ablaze.show more