Two views of one object: DP(α)
— the concentration α controls how a unit of probability splits into clusters
① Stick-breaking — GEM(α): break a unit stick, βₖ ~ Beta(1, α), πₖ = βₖ·∏(1−βⱼ)
② Pólya urn — draw balls one-by-one: new color with prob ∝ α, an existing color ∝ its count
Same α, two mechanisms — the explicit construction (top) and the sequential predictive rule (bottom). Both are the Dirichlet process: a few big clusters and a long tail of small ones. Drag α and watch the number of clusters change.