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Probabilistic Machine Learning · Chapter 4

Explore a Gaussian

Move the controls. Predict what will change, then inspect the distribution.

Change the covariance

Correlation ρ 0.50

SD of X₁ 1.00

SD of X₂ 1.00

Projection angle 45°

Observed X₂ 2.00

Sample size n 1

Prior variance multiplier 1.00

Data mean, coordinate 1 2.00

Data mean, coordinate 2 2.00

Show illustrative samples

Predict first.

Precision adds in each direction

S_n⁻¹ = S₀⁻¹ + nK⁻¹

μ_n = S_n(S₀⁻¹μ₀ + nK⁻¹x̄)

Prior mean μ₀ = (0, 0). Individual observations have known covariance K = diag(4, 1).

The data mean has covariance K/n. Increasing n makes that evidence more precise while the displayed data mean stays fixed.

PriorData mean / likelihoodPosterior

Illustrative models and synthetic samples, not measured height data. Contours mark Mahalanobis distances Δ = 1 and 2; their enclosed probability depends on dimension. All calculations run locally. Based on Chapter 4, Multivariate Random Variables.