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LIT-1847: Lecture 4: Statistical decision theory: model distance and equivalence

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StatusActive — read 2026-10-30 (NOTE-1426) in full, all 26 slides with their handwritten annotations, from the course's own copy at web.stanford.edu/class/ee378c/lecture4_annotated.pdf. Worth reading as a compact statement of Le Cam's deficiency for decision problems with [0, 1]-valued losses and of the randomization criterion: a model is ε-deficient relative to another exactly when some stochastic kernel brings it within ε of the other in total variation, uniformly in θ. Read it knowing that only the easy direction (a kernel within ε bounds every risk gap by ε) is proved line by line; the converse is sketched for a finite parameter set by one inf–sup swap, with the minimax theorem's conditions and the finiteness of the observation spaces left unstated, and the passage to an infinite parameter set is not given.
Tagsmathematical-statistics
Cited as “grounds” byCLAIM-503 — Over finitely many states and finite outcome sets, one experiment is ε-deficient relative to another in every decision problem with losses in [0, 1] exactly when a single Markov kernel brings it within ε in total variation at every state, so the deficiency computed as a linear programme is the least such ε
Cited as “paper” byNOTE-1426 — Han — Model distance and equivalence (EE378C Lecture 4)

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Han, Y. (2021). Lecture 4: Statistical decision theory: model distance and equivalence. Stanford EE378C, lecture slides with annotations, April 7, 2021.

Key takeaways

From the reading, NOTE-1426.

  • Deficiency by decisions (slide 11). For every finite Θ₀ ⊆ Θ, finite action set, loss L : Θ × A → [0, 1] and rule δ_N there is a rule δ_M with R_θ(δ_M) ≤ R_θ(δ_N) + ε for all θ ∈ Θ₀.
  • The randomization theorem (slide 13). M is ε-deficient relative to N if and only if some kernel K : 𝒳 → 𝒴 has sup_θ ‖Q_θ − K P_θ‖_TV ≤ ε.
  • Easy direction, proved (slide 14). δ_M = δ_N ∘ K, and for a loss in [0, 1] the risk gap is at most ‖Q_θ − K P_θ‖_TV, at every θ.
  • Hard direction, sketched (slide 15). For finite Θ only: write the condition as sup over losses and priors of inf over δ_M, swap the inf and the sups, read the inner term as a total variation distance, and take the action set to be 𝒴 with δ_N the identity.
  • Le Cam’s distance (slide 16) is the larger of the two directed deficiencies, equivalently the larger of two inf–sup TV programmes.

Standing in the record

Filed on 2026-10-30 for CLAIM-488. It states the criterion that claim’s first step needs in both directions and proves the direction the record already had; the converse is sketched, not proved.

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