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Probability & Statistics

Distributions, expectation, Bayes, MLE, sampling

01 ·Probability & Statistics·★ MEMBER·8 min read Probability and Statistics for AI — A Model's Output Is a Distribution Classifiers and language models do not return answers; they return probability distributions. Distributions, expectation, conditional probability and Bayes explained from the symbols up — building to the payoff: why maximum likelihood is where loss functions come from. Cross-entropy and MSE were derived, not invented. 02 ·Probability & Statistics·★ MEMBER·PAPER·10 min read Bayes' Theorem in AI — Priors, Posteriors, and Uncertainty One line of math — Bayes' theorem — turns into three workhorse tools in real AI systems: probability calibration, active learning, and Bayesian optimization. Starting from a positive medical test, we build up priors, posteriors, and uncertainty with zero background assumed. 03 ·Probability & Statistics·★ MEMBER·11 min read Thinking Bayesian — A Working Feel for Priors, Likelihoods, and Posteriors Bayesian updating is the act of feeding yesterday's posterior back in as today's prior. With a conjugate prior the whole update collapses into adding pseudo-counts, and an A/B test becomes two numbers: the probability of winning and the expected loss. From zero background to the traps in stopping rules and prior choice. 04 ·Probability & Statistics·★ MEMBER·11 min read Markov Chains from Scratch — The Process That Only Looks at Now What happens next depends only on where you are now — that single act of forgetting is a Markov chain. From transition matrices and stationary distributions to why PageRank is an eigenvector and why MCMC gets to ignore the normalizing constant, built from nothing assumed. 05 ·Probability & Statistics·FREE·13 min read A Field Guide to Probability Distributions — Where Normal, Poisson, and the Exponential Family Come From The normal and Poisson distributions aren't formulas to memorize — they're the shapes that certain situations inevitably produce. Follow three stories (adding things up, counting rare events, and refusing to assume more than you know) and the textbook zoo collapses into one river called the exponential family, with sigmoid and softmax falling out of it. 06 ·Probability & Statistics·★ MEMBER·10 min read Monte Carlo Methods from Scratch — Solving Integrals with Dice An integral you cannot solve can still be estimated: scatter random points and take the average. This piece builds up why the law of large numbers buys you only 1/√N, what importance sampling rescues, and how MCMC samples from a distribution whose normalizing constant nobody can compute. 07 ·Probability & Statistics·★ MEMBER·10 min read Hypothesis Testing and A/B Tests — How to Use a p-value, and How People Misuse It Ship the identical page to two groups and the numbers still won't match. A hypothesis test converts that gap into 'how many units of noise is this?' — that number is the p-value. From the null hypothesis up through the three failures that actually ruin experiments: power, peeking, and multiple comparisons. 08 ·Probability & Statistics·★ MEMBER·PAPER·11 min read Statistical Learning Theory — Why Does Learning Generalize? You only ever see finitely many examples, yet the model answers questions about data it has never seen. This piece builds the guarantee from scratch — Hoeffding, uniform convergence, VC dimension, PAC — then shows where it broke against deep learning, and what survives in day-to-day evaluation design. 09 ·Probability & Statistics·★ MEMBER·PAPER·10 min read Optimal Transport — The Mathematics of Moving Distributions Optimal transport measures the distance between two probability distributions as the cheapest way to haul sand from one pile into the shape of another. From the definition of the Wasserstein distance to the Sinkhorn algorithm that makes it practical, and on to WGAN, FID and Flow Matching — with numpy code and the failure modes that bite in production. 10 ·Probability & Statistics·★ MEMBER·PAPER·11 min read Kernel Methods and Gaussian Processes — The Champions Before Neural Nets Before deep learning took over, kernel SVMs held the crown in classification. This article builds the whole story from zero: the kernel trick that makes lifting into high dimensions free, Gaussian processes that put a probability distribution over functions, and Bayesian optimization that turns uncertainty into a weapon.