viz.osteele.com

Interactive Visualizations

A small collection of interactive explainers and tools: probability and statistics, music, language models, generative geometry, and systems.

Probability & Statistics #

Foundations, estimation, and dependence. Visualizations I built while working these out for myself.

These pages aren't a complete or standard treatment of the subject. Each sits at the intersection of a topic from coursework, a topic I wanted to study more closely, and a topic I thought an interactive figure could make more accessible. All are works in progress — not externally reviewed, and in places I haven't finished checking the implementation against the math.

10 items
Measure Theory & Random Variables
Measurable spaces, probability measures, pushforward measures, densities, and importance sampling
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Notation: density form vs. measure-theoretic form
A bilingual dictionary for the two ways probability is written on this site, plus the four places where the choice of language actually matters
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Named Distributions
How Bernoulli, Poisson, Gaussian, Cauchy, chi-square, t, F, conjugate priors, and heavy-tail laws are related
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Modes of Convergence
Almost sure, in probability, in distribution, in L^p — the implication lattice, counterexamples as sample paths, Markov/Chebyshev/Chernoff bounds, and MCT/DCT/Fatou
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Calculus of Variations
First variations, Euler-Lagrange residuals, curve relaxation, and the brachistochrone race
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Sufficient Statistics
Fisher–Neyman factorization, the fiber picture, Rao–Blackwell variance collapse, and a categorical diagram showing factorization, variance, and Fisher info as one commuting square
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The Exponential Family
The canonical form, naming and the statistical-physics log-partition story, derivatives of A giving the moments of T(X), canonical links (logit, log) behind GLMs, and an interactive picker stepping through six standard members
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Fisher Information
Likelihood geometry, score functions, Fisher information, exponential families, log-partition, Jeffreys and max-entropy priors, and Bayesian updates
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Hypothesis Testing
Type-I error, type-II error, power, and decision thresholds through the classic overlapping-distributions diagram
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Distance Correlation
Distance-based dependence tests, partial distance correlation, and cases Pearson r misses
Paper notes

Random Processes #

Stochastic processes in time, frequency, and function space. Visualizations I built while working these out for myself.

Topics guided by (but not strictly following) Prof. Ercan Kuruoğlu's Fall 2025 Random Processes course at Tsinghua SIGS.

These pages aren't a complete or standard treatment of the subject. Each sits at the intersection of a topic from coursework, a topic I wanted to study more closely, and a topic I thought an interactive figure could make more accessible. All are works in progress — not externally reviewed, and in places I haven't finished checking the implementation against the math.

5 items
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Bayesian Inference #

Approximating intractable posteriors by sampling and by optimization. Visualizations I built while working these out for myself.

Material draws on Prof. Ercan Kuruoğlu's Spring 2026 Bayesian Inference and Monte Carlo Simulation course at Tsinghua SIGS.

These pages aren't a complete or standard treatment of the subject. Each sits at the intersection of a topic from coursework, a topic I wanted to study more closely, and a topic I thought an interactive figure could make more accessible. Some also take a more measure-theoretic angle than the course, or than a typical introduction. All are works in progress — not externally reviewed, and in places I haven't finished checking the implementation against the math.

12 items
Choosing a Prior
Principles of prior selection: use real prior information when you have it; otherwise group invariance, max entropy, or Jeffreys — and how the three routes disagree near boundaries
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Conjugate Priors & the Exponential Family
Why some prior–likelihood pairs update in closed form, hyperparameters as pseudo-counts, worked Beta/Normal/Gamma examples, and a table of standard pairs
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Posterior Summaries & Bayes Risk
Squared, absolute, and zero-one loss pick out the posterior mean, median, and mode — three views of the same posterior, only one of which ignores everything but the peak
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Hierarchical Bayes
Two-level Normal–Normal model, the posterior formula for borrowing strength across groups, empirical-Bayes fitting of the between-group variance, and the connection to ridge regression
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Bayesian Regression: Penalties as Priors
OLS, ridge, LASSO, and best-subset selection as MAP under four noise/prior pairs — and why the shape of the prior near zero determines whether the estimator shrinks, selects, or both
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Bayesian Graphical Models
DAG factorization, d-separation, explaining away, Dirichlet-multinomial CPT learning, and structure scoring
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Hidden Markov Models
HMM sampling, forward-backward filtering and smoothing, log-domain messages, and Viterbi versus marginal MAP paths
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Monte Carlo & MCMC
Rejection, importance sampling, Metropolis-Hastings, Gibbs, RJMCMC, simulated annealing, and when to use each method on a static target
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Kalman & Particle Filters
Sequential inference of a hidden state from noisy observations: Kalman filter for linear-Gaussian models, EKF/UKF for local linearization, particle filter for fully nonlinear non-Gaussian SSMs
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Free Energy & Variational Inference
The free-energy/ELBO identity and how it turns posterior approximation into optimization
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Variational Bayes for Gaussian Mixtures
CAVI for a 2-D Gaussian mixture with Normal–Wishart and Dirichlet priors, showing component ellipses, automatic pruning of unused components, and the ELBO trace
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Bayesian Neural Networks
Weight posteriors, predictive function ensembles, Laplace approximation, evidence, Occam's hill, and prior mismatch
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