viz.osteele.com — Interactive Visualizations

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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 →

Information Theory #

Surprise, divergence, and the geometry of distributions. Visualizations I built while working these out for myself.

Unlike the coursework sections, these pages aren't tied to a class — they're topics I studied on my own. Like everything here, they're works in progress, not externally reviewed.

4 items

Entropy & Mutual Information

Average surprise, conditional entropy, shared information, binary distributions, and noisy channels

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KL Divergence

Directed distribution mismatch, support errors, and the difference between forward and reverse KL

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Optimal Transport

Wasserstein distance and the transport plan: Sinkhorn iteration, and why OT gives gradients where KL divergence does not

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Information Geometry

Probability distributions as a manifold: the probability simplex, Fisher-Rao geodesics, dual flatness, and e- vs m-projections

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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.

5 items

Poisson Processes

Rare events in time: exponential waits, Poisson counts, equivalent definitions, thinning, splitting, and process diagnostics

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Markov Chains

Discrete-time and continuous-time chains, stationary distributions, mixing, recurrence, periodicity, and CTMC rates

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LTI Systems on Random Inputs

Convolution, correlation propagation, spectra, AR(1) as a leaky integrator, and stationarity through linear filters

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Power Spectral Density

Autocorrelation, spectra, LTI shaping, and periodograms for stationary random processes

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Gaussian Processes for Regression

Priors over functions, kernels, posterior conditioning, marginal likelihood, 2-D regression, and acquisition

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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