LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Finite-prior Bayes envelopes converge under pointwise LAN

Statement

Assume AC. In a pointwise LAN experiment Local asymptotic normality for local experiments with $K ≻ 0$ , fix $h_{1}, \dots, h_{s} \in R^{d}$ and weights $w_{i} > 0$ summing to one. Let $g : R^{d} \to [0, M]$ be Lipschitz. The Bayes values for estimating $h$ with loss $g (a - h)$ and this finite prior, allowing all measurable randomized rules, converge to the corresponding Bayes value in the Gaussian shift $Y ∣ h \sim N_{d} (h, K^{- 1})$ . No likelihood-field uniformity or measurable-selection hypothesis is required.

Facts & Assumptions

Given: AC, the pointwise LAN experiment, finite prior and bounded Lipschitz loss.

[F1]

Local asymptotic normality for local experiments gives mean-one likelihood proxies $L_{n, h}$ , pointwise total-variation approximation, and $\log L_{n, h} = h^{T} Δ_{n} - h^{T} K h / 2 + o_{P} (1)$ with $Δ_{n} \Rightarrow N_{d} (0, K)$ . Its finite log expansion also gives $P_{n, 0} (L_{n, h} = 0) \to 0$ .

[F2]

Vector Slutsky theorem and Continuous mapping theorem give joint finite-dimensional convergence after adjoining finitely many remainders that tend to zero in probability.

[F3]

A mean-one likelihood-ratio limit implies contiguity proves uniform integrability for nonnegative mean-one variables converging in law to a mean-one limit.

[F4]

Bayes risk and minimax risk defines the Bayes risk of a rule $δ$ as $r_{π} (δ) = \int R (h, δ) π (d h)$ , while Measure kernel and probability kernel describes randomized rules. In this proof the Bayes value means the infimum of $r_{π} (δ)$ over all measurable randomized rules; the infimum is local notation and is not part of the supplier's definition.

[F5]

Total variation distance for probability laws and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product imply $∣ E_{P} f - E_{Q} f ∣ \leq M ∥ P - Q ∥_{T V}$ for $0 \leq f \leq M$ , by integrating the tail probabilities of $f$ .

[F6]

Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable gives a countable dense rational grid in finite-dimensional Euclidean action space by taking finite products.

[F7]

Multivariate normal law, including singular covariance, Standard normal and normal laws, The standard normal density has total mass one, and A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not: For $Z \sim N_{d} (0, K)$ with $K ≻ 0$ , its density is proportional to $\exp (- z^{T} K^{- 1} z / 2)$ , and completing the square shows $E \exp (h^{T} Z - h^{T} K h / 2) = 1$ .

[F8]

The Lebesgue integral is linear on $L^{1} (μ)$ : Expectation commutes with the finite weighted sums defining $U_{n}$ and $U$ .

[A1]

The Axiom of Choice is retained for the LAN and Gaussian suppliers.

Proof

1.1F4F6givenconstructalgebra

For $l \in [0, \infty)^{s}$ , set $F (l) = \inf_{a} \sum_{i} w_{i} l_{i} g (a - h_{i})$ . For any $l, l^{'}$ , each action's objective changes by at most $M \sum_{i} w_{i} ∣ l_{i} - l_{i}^{'} ∣$ ; taking infima proves the same bound for $F$ . Also $0 \leq F (l) \leq M \sum_{i} w_{i} l_{i}$ . For fixed $l$ the objective is continuous in $a$ , so F6 makes its infimum equal to the infimum over a fixed enumeration $(q_{j})$ of the rational grid. For $ϵ > 0$ , choose the first $j$ whose objective is at most $F (l) + ϵ$ . This index is Borel: its defining set and the finitely many failures at smaller indices are Borel. Thus this is an explicit measurable $ϵ$ -optimal action function of $l$ .

1.2F1F2F3F7F8A1givenconstruct

Under the base law, F1 and F2 imply $(L_{n, h_{i}})_{i = 1}^{s} \Rightarrow (l_{i} (Z))_{i = 1}^{s}$ , where $Z \sim N_{d} (0, K)$ and $l_{i} (z) = \exp (h_{i}^{T} z - h_{i}^{T} K h_{i} / 2)$ . The log expansions imply that zero likelihoods have vanishing probability, so exponentiating the finite vector is legitimate. By [F7], completing the square in the Gaussian density gives $E l_{i} (Z) = 1$ . Hence $U_{n} = \sum_{i} w_{i} L_{n, h_{i}}$ converges in law to $U = \sum_{i} w_{i} l_{i} (Z)$ , and [F8] gives $E U_{n} = E U = 1$ . [F3] makes $(U_{n})$ uniformly integrable.

2.1F2F3step 1.1step 1.2algebra

For each cutoff $b$ , $F \land b$ is bounded continuous, so its expectations converge by step 1.2. The error from this cutoff is bounded by $M E [U_{n} 1_{{M U_{n} > b}}]$ , which vanishes uniformly as $b \to \infty$ ; the limit variable has the same integrable bound. Therefore $E F ((L_{n, h_{i}})_{i}) \to E F ((l_{i} (Z))_{i})$ .

3.1F1F4F5step 1.1step 2.1algebra∎

Under the proxy mixture, the predictive density is $U_{n} = \sum_{i} w_{i} L_{n, h_{i}}$ and, when $U_{n} > 0$ , the posterior weights are $w_{i} L_{n, h_{i}} / U_{n}$ . Conditional on the data, every randomized action rule has posterior risk at least $F ((L_{n, h_{i}})_{i}) / U_{n}$ , because $F$ is the infimum over deterministic actions and randomization averages those action risks. The measurable selector from step 1.1 has posterior risk at most this envelope plus $ϵ$ . Multiplying by the predictive density and integrating shows that the proxy Bayes value differs from $E_{P_{n, 0}} F ((L_{n, h_{i}})_{i})$ by at most $ϵ$ ; the set $U_{n} = 0$ has zero predictive probability. Letting $ϵ ↓ 0$ identifies the proxy Bayes value with that expectation. The same argument with $l_{i} (Z)$ identifies the limiting envelope expectation as the finite-prior Gaussian Bayes value: tilting the base Gaussian by $l_{i}$ gives $Z \sim N_{d} (K h_{i}, K)$ , and $Y = K^{- 1} Z$ has law $N_{d} (h_{i}, K^{- 1})$ . Finally [F5] bounds the actual/proxy risk difference uniformly over rules by $M \sum_{i} w_{i} ∥ P_{n, h_{i}} - L_{n, h_{i}} P_{n, 0} ∥_{T V} \to 0$ . Infima preserve this bound, proving the assertion.

Depends on

Used by

Local asymptotic minimax bound Theorem

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Sources

A. W. van der Vaart, Semiparametric Statistics, Theorem 4.14 and complete proof, pp. 46–48 (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, Theorem 3 and proof, pp. 17–18; local finite-prior proof supplied here (standard reference, not scraped)