Why this part matters
This optional part studies how shrinkage and sparsity can improve estimation in many dimensions. The motivating contrast is the Normal mean: optimality among unbiased estimators does not prevent a biased rule from improving squared-error risk in dimension at least three.
The lectures connect James–Stein shrinkage, coordinatewise thresholding and sparse recovery. Earlier ideas return: bias–variance comparisons, Gaussian maxima, soft thresholding and convex optimization.
Optional enrichment. All three lectures in this part are optional and are not examined. Their high, mid and low labels organize a reading path within the optional material.
Learning objectives
Explain how shrinkage can improve risk despite introducing bias (Course outcome 3).
Relate Gaussian tail scales to coordinate thresholds (Course outcome 4).
Connect sparsity with thresholding and convex estimation (Course outcome 5).
Roadmap
| Lecture (all optional) | Purpose |
| 18. Stein’s phenomenon and shrinkage estimation* | Compare fixed and James–Stein shrinkage; connect posterior means to Tweedie’s formula. Student PDF. |
| 19. Denoising via shrinkage* | Study sparse representations, thresholding and orthogonal-design LASSO. Student PDF. |
| 20. Basis pursuit, compressed sensing, and LASSO* | Connect null-space geometry to sparse recovery and LASSO error bounds. Student PDF. |
What you need from earlier lectures
Where this leads
These topics connect risk comparisons with sparse modeling. Part VII: Testing and confidence sets turns to decisions and uncertainty; its lecture materials are not yet available.
Readings
Each lecture links its sources and further reading: Lecture 18 references, Lecture 19 references, and Lecture 20 references.
Optional self-study
Try these exercises before opening their complete solutions: