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Dirichlet normal-mixture posterior contraction

1 · Prerequisites

2 · Summary

Objective

This page builds the density, finite-mixture approximation, scale-band entropy, Dirichlet simplex mass and local KL-mass estimates needed for the compact-truth Dirichlet normal-mixture contraction theorem. It also proves the posterior sieve-tail bound from the DP update and latent Pólya urn. The main rate is a Hellinger contraction rate in sampling probability; it is neither an almost-sure nor a generic minimax claim. The middle term is omitted for a compactly supported scale hyperprior. Inverse-Gamma scale tails are outside the stated upper-tail condition; this page makes no inconsistency claim about them. The interval-mass refinement removes the Gaussian-bias logarithm. A local Gaussian semigroup estimate bounds KL and log-square smoothing bias, and a coarse-interval Dirichlet neighborhood controls its discrete perturbation. The DP-dependent estimates use the current BCS-17 DP definition, construction, update and predictive-law bodies; their current Step 1 records are checked in the batch evidence. Readiness records are authoring handoff only and do not constitute independent acceptance.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The Dirichlet normal-mixture density is jointly measurable and normalized

Statement

Let $P (R)$ be the probability measures on the Borel sets of $R$ , equipped with the evaluation sigma-algebra, and let $σ > 0$ . For $F \in P (R)$ define $p_{F, σ} (x) : = \int_{R} \frac{1}{σ \sqrt{2 π}} \exp ⁣ (- \frac{(x - z)^{2}}{2 σ^{2}}) F (d z) .$ Then $(F, σ, x) \mapsto p_{F, σ} (x)$ is jointly measurable, $0 < p_{F, σ} (x) \leq (σ \sqrt{2 π})^{- 1}$ for every $(F, σ, x)$ , and $\int_{R} p_{F, σ} (x) d x = 1$ . In particular it is the Lebesgue density of the convolution of $F$ with $N (0, σ^{2})$ .

Facts & Assumptions

Given: The evaluation sigma-algebra on $P (R)$ , $F \in P (R)$ , and $σ > 0$ .

[A1]

Assume AC for the library normal location-scale density interface. The Axiom of Choice

[F1]

A positive-variance normal law has its location-scale density: $z \mapsto (σ \sqrt{2 π})^{- 1} \exp (- (x - z)^{2} / (2 σ^{2}))$ is the Lebesgue density of $N (x, σ^{2})$ and integrates to one.

[F2]

Probability measures and probability spaces and Measure kernel and probability kernel: each $F$ is a probability measure, and the evaluation map $(F, A) \mapsto F (A)$ makes $K ((F, σ, x), A) : = F (A)$ a probability kernel from $P (R) \times (0, \infty) \times R$ to $R$ .

[F3]

Measurability of integration against a kernel: integrating a nonnegative product-measurable function against a probability kernel gives a measurable function of the kernel's input.

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative measurable functions on product measure spaces may be integrated in either order.

Proof

Given: The hypotheses in the Statement.

1.1A1F2F3given

The kernel $K$ in [F2] is a probability kernel because, for fixed $(F, σ, x)$ , $A \mapsto F (A)$ is a probability measure, while for fixed Borel $A$ , $(F, σ, x) \mapsto F (A)$ is measurable by the evaluation sigma-algebra. The integrand $(σ, x, z) \mapsto (σ \sqrt{2 π})^{- 1} \exp (- (x - z)^{2} / (2 σ^{2}))$ is continuous on $(0, \infty) \times R^{2}$ and hence product-measurable. Therefore [F3] makes its integral against $K$ jointly measurable.

1.2F1givenalgebra

The integrand is strictly positive, so its integral against a probability measure is positive. It is bounded above by $(σ \sqrt{2 π})^{- 1}$ , so the integral is finite and has the stated upper bound.

1.3F1F4givenalgebra

By [F4], $\int_{R} p_{F, σ} (x) d x = \int_{R} (\int_{R} \frac{1}{σ \sqrt{2 π}} e^{- (x - z)^{2} / (2 σ^{2})} d x) F (d z) = \int_{R} 1 F (d z) = 1,$ where the inner integral is one by [F1] and $F$ has total mass one. Thus $p_{F, σ}$ is a probability density.

2.1F1F4givenalgebra∎

For every Borel set $B \subseteq R$ , Tonelli [F4] gives $\int_{B} p_{F, σ} (x) d x = \int_{R} (\int_{B} \frac{1}{σ \sqrt{2 π}} e^{- (x - z)^{2} / (2 σ^{2})} d x) F (d z)$ . The inner integral is $N (z, σ^{2}) (B)$ by [F1], so the right side is the convolution measure $(F * N (0, σ^{2})) (B)$ by its definition. Hence this density represents the convolution law in the Statement.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

A bounded likelihood ratio controls KL and log-square moments by Hellinger distance

Statement

Let $p, q$ be probability densities relative to one measure $μ$ , assume $p d μ ≪ q d μ$ , let $h^{2} : = \int (\sqrt{p} - \sqrt{q})^{2} d μ$ , and suppose $R : = {ess,sup}_{q d μ} (p / q) < \infty$ , with $p / q = 0$ on ${q = 0, p = 0}$ . Then $p ≪ q$ and for a universal constant $C$ , $K L (p d μ ∥ q d μ) \leq C h^{2} (1 + \log^{+} R), \int p {[\log (p / q)]}^{2} d μ \leq C h^{2} (1 + \log^{+} R)^{2} .$ The logarithm is assigned value zero on ${p = 0}$ ; the moment integrand is also set to zero there.

Facts & Assumptions

Given: Two probability densities $p, q$ relative to one measure, $p d μ ≪ q d μ$ , and finite $R$ as in the Statement.

[A1]

Assume AC for the library directional KL convention. The Axiom of Choice Kullback–Leibler divergence

[F1]

Root-L2 Hellinger distance between density laws: $h^{2} = \int (\sqrt{p} - \sqrt{q})^{2} d μ$ .

[F2]

Kullback–Leibler divergence: when $p d μ ≪ q d μ$ , the directional KL divergence is the integral of $\log (p / q)$ against $p d μ$ .

[F3]

The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is continuous on $(0, \infty)$ , $\log^{'} (x) = 1 / x$ , and $\log x = \int_{1}^{x} d t / t$ .

Proof

Given: The hypotheses in the Statement.

1.1A1F3givenalgebra

Put $u = p / q$ on ${q > 0}$ . The absolute-continuity premise gives $p = 0$ $μ$ -almost everywhere on ${q = 0}$ . Define $g (u) = (\sqrt{u} - 1)^{2}$ and $a (u) = (u \log u - u + 1) / g (u)$ , $b (u) = u (\log u)^{2} / g (u)$ for $u \neq 1$ , with their limiting values at $u = 1$ and $u = 0$ . At zero, set $s = 1 / u$ ; for $s \geq 1$ , [F3] gives $0 \leq \log s \leq \int_{1}^{s} t^{- 3 / 4} d t \leq 4 s^{1 / 4}$ . Hence $u \log u = - (\log s) / s \to 0$ and $u (\log u)^{2} = (\log s)^{2} / s \leq 16 s^{- 1 / 2} \to 0$ , so $a (0) = 1$ and $b (0) = 0$ . At one, $\log u / (u - 1) \to 1$ by [F3], $g (u) / (u - 1)^{2} = 1 / (\sqrt{u} + 1)^{2} \to 1 / 4$ , and $u (\log u)^{2} / (u - 1)^{2} \to 1$ . Also $u \log u - u + 1 = \int_{1}^{u} \log t d t$ , so $[u \log u - u + 1] / (u - 1)^{2} \to 1 / 2$ ; therefore $a (1) = 2$ and $b (1) = 4$ . Thus $a, b$ are continuous on $[0, 2]$ and bounded there.

2.1F3step 1.1algebra

For $u \geq 2$ , $g (u) = u (1 - u^{- 1 / 2})^{2} \geq u (1 - 2^{- 1 / 2})^{2}$ , while $0 \leq u \log u - u + 1 \leq u \log u$ and $u (\log u)^{2}$ retain their displayed forms. Hence $a (u) \leq C \log u$ and $b (u) \leq C (\log u)^{2}$ for a universal $C$ . Combining with step 1.1, for every $0 \leq u \leq R$ , $u \log u - u + 1 \leq C g (u) (1 + \log^{+} R), u (\log u)^{2} \leq C g (u) (1 + \log^{+} R)^{2} .$

3.1A1F1F2step 2.1algebra∎

Integrating the first inequality in step 2.1 against $q d μ$ gives the KL bound, since $\int q (u \log u - u + 1) d μ = \int p \log (p / q) d μ$ by $\int p = \int q = 1$ and [F2]. Integrating the second gives the log-square bound. Finally $\int q g d μ = h^{2}$ by [F1], so both right sides have the claimed factors of $h^{2}$ .

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Gaussian convolution has second-order KL and log-square error

Statement

Assume AC. Let $p \in C^{2} (R)$ be a nonnegative compactly supported probability density and suppose $J : = \int_{{p > 0}} \frac{(p^{''} (x))^{2}}{p (x)} d x < \infty .$ For $σ > 0$ , set $q_{σ} = p * ϕ_{σ}$ , where $ϕ_{σ}$ is the centered normal density with standard deviation $σ$ . Then $\int \frac{(p - q_{σ})^{2}}{q_{σ}} d x \leq \frac{4}{9} σ^{4} J, K L (p ∥ q_{σ}) \leq \frac{4}{9} σ^{4} J, \int p {[\log \frac{p}{q_{σ}}]}^{2} d x \leq \frac{4}{9} σ^{4} J .$ The logarithmic integrands are zero where $p = 0$ . No unimodality or uniform bound on $p / q_{σ}$ is assumed.

Facts & Assumptions

Given: AC and the density and integrability hypotheses of the Statement.

[F1]

The normal kernel is $ϕ_{σ} (x) = (σ \sqrt{2 π})^{- 1} e^{- x^{2} / (2 σ^{2})}$ and integrates to one. A positive-variance normal law has its location-scale density

[F2]

Cauchy–Schwarz bounds an integral of a product by the product of its two $L^{2}$ norms. Cauchy-Schwarz inequality for $L^{2}$

[F3]

Nonnegative product integrals may be taken in either order. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F4]

The directional KL divergence of two densities with absolute continuity is the integral of their log ratio under the first law. Kullback–Leibler divergence

[F5]

A parameter derivative may pass under an integral when it is measurable and dominated locally by one integrable function. Differentiation under the integral sign

[F6]

For differentiable functions with integrable derivatives on a compact interval, integration by parts holds. If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$

[F7]

If a differentiable function has an integrable derivative on a compact interval, its derivative integrates to the endpoint difference. The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$

[F8]

The natural logarithm is differentiable on $(0, \infty)$ with derivative $1 / x$ . The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

[F9]

Pointwise-convergent integrands dominated by one integrable function have convergent integrals. Dominated convergence

Proof

1.1F2givenalgebra

At every zero of the nonnegative differentiable function $p$ , $p^{'} = 0$ . At each accumulation point of its zero set, differentiation along zeros of $p^{'}$ gives $p^{''} = 0$ ; isolated zeros form a countable set. Thus $p^{''} = 0$ almost everywhere on ${p = 0}$ . Put $r = p^{''} / p$ on ${p > 0}$ and zero elsewhere. Then $p^{''} = r p$ almost everywhere, $\int r^{2} p = J$ , and [F2] gives $\int ∣ p^{''} ∣ \leq \sqrt{J}$ . Both $p$ and $p^{''}$ are continuous and compactly supported.

2.1F1F5F6F7F9step 1.1algebra

Let $P_{t} f = f * ϕ_{\sqrt{t}}$ for $t > 0$ . Direct differentiation of [F1] gives $\partial_{t} ϕ_{\sqrt{t}} = \frac{1}{2} \partial_{x}^{2} ϕ_{\sqrt{t}}$ . For fixed $x$ and $0 < ε < t$ , the differentiated integrand in $P_{s} p (x)$ is bounded on $s \in [ε, t]$ and the compact support of $p$ by $C_{x, ε, t} ∣ p (z) ∣$ ; differentiation under the integral [F5] and two integrations by parts [F6] therefore give $\partial_{s} P_{s} p = \frac{1}{2} P_{s} p^{''}$ . The boundary terms vanish because $p, p^{'}$ have compact support. Dominated convergence [F9] makes $s \mapsto P_{s} p^{''} (x)$ continuous on $[ε, t]$ , so [F7] integrates the derivative there. Gaussian approximate identities converge uniformly on continuous compactly supported functions: split their convolution integral at $∣ z ∣ \leq δ$ , where uniform continuity controls the integrand difference, and its complement, whose normal mass tends to zero by scaling and integrability of [F1]. Letting $ε ↓ 0$ consequently yields $p - P_{t} p = - \frac{1}{2} \int_{0}^{t} P_{s} p^{''} d s$ pointwise; the omitted integral over $[0, ε]$ tends to zero because $∣ P_{s} p^{''} (x) ∣ \leq ∥ p^{''} ∥_{\infty}$ .

2.2F1F2F3step 1.1algebra

Fix $t > 0$ and put $q = P_{t} p$ . It is positive everywhere because $p$ is a nonzero nonnegative density and the Gaussian kernel is positive; Tonelli [F3] and $\int ϕ_{\sqrt{t}} = 1$ show $q$ integrates to one. For $0 < s \leq t$ , the Gaussian formula gives $ϕ_{\sqrt{t}} (z) \geq \sqrt{s / t} ϕ_{\sqrt{s}} (z)$ for every $z$ , hence $q \geq \sqrt{s / t} P_{s} p$ . Applying [F2] inside the convolution and then [F3] gives $∣ P_{s} p^{''} ∣^{2} \leq (P_{s} p) P_{s} (r^{2} p)$ and therefore $\int ∣ P_{s} p^{''} ∣^{2} / q \leq \sqrt{t / s} \int P_{s} (r^{2} p) = \sqrt{t / s} J$ . These inequalities remain valid if one of the pointwise nonnegative integrals is infinite.

3.1F2F3step 2.1step 2.2algebra

Apply [F2] to the time integral in step 2.1 with weights $w (s) = (t / s)^{1 / 4}$ . Since $\int_{0}^{t} w (s) d s = 4 t / 3$ , it gives $∣ p - q ∣^{2} \leq \frac{1}{4} (4 t / 3) \int_{0}^{t} w (s)^{- 1} ∣ P_{s} p^{''} ∣^{2} d s$ . Divide by $q$ , integrate and use [F3] and step 2.2. The result is $\int (p - q)^{2} / q \leq \frac{1}{4} (4 t / 3) J \int_{0}^{t} (t / s)^{1 / 4} d s = (4 / 9) t^{2} J$ . In particular all these weighted integrals are finite.

4.1F4F8step 3.1algebra∎

For $u > 0$ , the function $u - 1 - \log u$ has value zero at $1$ and derivative $(u - 1) / u$ , so $\log u \leq u - 1$ and $u \log u - u + 1 \leq (u - 1)^{2}$ . For $t \geq 1$ , $t - t^{- 1} - 2 \log t$ has value zero at $1$ and derivative $(t - 1)^{2} / t^{2} \geq 0$ ; replacing $t$ by $1 / t$ gives $2 ∣ \log t ∣ \leq ∣ t - t^{- 1} ∣$ also for $0 < t < 1$ . With $u = t^{2}$ , this yields $u (\log u)^{2} \leq (u - 1)^{2}$ . At $u = 0$ , use the conventions $u \log u = u (\log u)^{2} = 0$ , for which both inequalities hold. Apply these pointwise bounds to $u = p / q$ and integrate against $q$ . Since $p, q$ integrate to one, [F4] identifies the first integral as KL; its negative part is integrable because $u \log u \geq - e^{- 1}$ . Both requested logarithmic quantities are at most the chi-square integral from step 3.1. Taking $t = σ^{2}$ proves the Statement.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Compactly supported Gaussian mixtures admit logarithmic-size finite approximations

Statement

There is a universal constant $C$ such that, for $A, σ > 0$ , $0 < η < 1 / 2$ , and every probability measure $F$ supported on $[- A, A]$ , there is a discrete probability measure $F^{'}$ on $[- A, A]$ with at most $C (A / σ \lor 1) \log (e / η)$ support points and $∥ p_{F, σ} - p_{F^{'}, σ} ∥_{1} \leq η, ∥ p_{F, σ} - p_{F^{'}, σ} ∥_{\infty} \leq η / σ .$

Facts & Assumptions

Given: $A, σ > 0$ , $0 < η < 1 / 2$ , and a probability measure $F$ on $[- A, A]$ .

[F1]

The Dirichlet normal-mixture density is jointly measurable and normalized: $p_{F, σ}$ is the mixture integral of the normal location-scale density.

[F2]

A positive-variance normal law has its location-scale density: the standard normal density is $ϕ (x) = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ and $ϕ_{σ} (x) = σ^{- 1} ϕ (x / σ)$ .

[F3]

The monic probabilists' Hermite polynomials: $H_{m}$ is monic of degree $m$ and $H_{m + 1} = x H_{m} - m H_{m - 1}$ .

[F4]

Cauchy--Schwarz bounds an $L^{1}$ integral of a product by the product of its $L^{2}$ norms. Cauchy-Schwarz inequality for $L^{2}$

[F5]

For a standard normal random variable $Z$ , $E [Z^{2}] = 1$ . Location-scale normal laws have their stated mean and variance

[F6]

Nonnegative product integrals may be interchanged. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F7]

Integration by parts holds for continuously differentiable functions on compact intervals. If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$

[F8]

A differentiable function with integrable derivative on a compact interval has endpoint difference equal to the integral of its derivative. The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$

[A1]

Assume AC for the normal-density interface. The Axiom of Choice

Proof

Given: The hypotheses in the Statement.

1.1A1F1givenconstruct

Partition $[- A, A]$ into $K \leq 2 A / σ + 1$ consecutive Borel intervals $I_{i}$ of length at most $σ$ , with left endpoints $a_{i}$ . Ignore intervals of zero $F$ -mass and let $G_{i}$ be the law of $W_{i} = (Z - a_{i}) / σ$ conditional on $Z \in I_{i}$ . Let $ℓ_{i}$ be the length of $I_{i}$ and put $b_{i} = ℓ_{i} / σ \leq 1$ , so $G_{i}$ is supported on $[0, b_{i}]$ . Approximate the moment vector $(\int w d G_{i}, \dots, \int w^{m} d G_{i})$ by finite sums obtained from partitions of $[0, b_{i}]$ ; it is therefore in the closure of the convex hull of the compact moment curve $w \mapsto (w, \dots, w^{m})$ on that interval. Every finite convex combination with more than $m + 1$ points can be reduced: affine dependence supplies coefficients $c_{j}$ , not all zero, with $\sum c_{j} = 0$ and $\sum c_{j} x_{j} = 0$ ; moving the weights along this relation until one becomes zero preserves nonnegativity and the represented point, so iteration gives at most $m + 1$ points. Thus the convex hull is the continuous image of a compact set consisting of $m + 1$ curve points and simplex weights; it is compact and closed. The moment vector belongs to it, giving a probability $G_{i}^{'}$ on at most $m + 1$ points of $[0, b_{i}]$ matching the first $m$ moments of $G_{i}$ .

1.2A1F2F3F4F5F7F8algebra

The Hermite recurrence [F3] and differentiation of $ϕ$ give $ϕ^{(r)} = (- 1)^{r} H_{r} ϕ$ for every integer $r \geq 0$ . The case $r = 0$ is immediate. If it holds at orders $r$ and $r - 1$ , differentiating $H_{r}$ 's recurrence gives $H_{r + 1}^{'} = H_{r} + x H_{r}^{'} - r H_{r - 1}^{'} = H_{r} + r (x H_{r - 1} - (r - 1) H_{r - 2}) = (r + 1) H_{r}$ ; the base cases $H_{0}^{'} = 0$ and $H_{1}^{'} = 1$ start this induction. Thus differentiating $(- 1)^{r} H_{r} ϕ$ gives $(- 1)^{r + 1} H_{r + 1} ϕ$ . Since $H_{r}$ is monic, $H_{r}^{(r)} = r!$ . Apply integration by parts [F7] repeatedly on $[- T, T]$ and let $T \to \infty$ ; the boundary terms vanish because each derivative is a polynomial times the Gaussian. This gives $\int H_{r}^{2} ϕ = (- 1)^{r} \int H_{r} ϕ^{(r)} = \int H_{r}^{(r)} ϕ = r!$ . Cauchy--Schwarz [F4] therefore gives $∥ ϕ^{(r)} ∥_{1} \leq \sqrt{r!}$ . Also $ϕ^{(r)}$ tends to zero at $- \infty$ , so [F8] on $[- T, x]$ , followed by $T \to \infty$ , gives $∣ ϕ^{(r)} (x) ∣ \leq \int_{- \infty}^{x} ∣ ϕ^{(r + 1)} (t) ∣ d t \leq \sqrt{(r + 1)!}$ and $∥ ϕ^{(r)} ∥_{\infty} \leq \sqrt{(r + 1)!}$ .

2.1F1F6F8step 1.1step 1.2algebra

Repeated application of [F8] gives the Taylor integral remainder for $w \mapsto ϕ (x - w)$ through degree $m - 1$ : $R_{m} (x, w) = \frac{(- w)^{m}}{(m - 1)!} \int_{0}^{1} (1 - t)^{m - 1} ϕ^{(m)} (x - t w) d t$ . By translation invariance and Tonelli [F6], $∥ R_{m} (\cdot, w) ∥_{1} \leq w^{m} ∥ ϕ^{(m)} ∥_{1} / m! \leq 1 / \sqrt{m!}$ ; the same formula gives $∥ R_{m} (\cdot, w) ∥_{\infty} \leq w^{m} ∥ ϕ^{(m)} ∥_{\infty} / m! \leq \sqrt{m + 1} / \sqrt{m!}$ by step 1.2. Moment matching in step 1.1 cancels the polynomial terms after integration against $G_{i} - G_{i}^{'}$ . Integrating the remainder bounds against these two probability laws gives $∥ p_{G_{i}, 1} - p_{G_{i}^{'}, 1} ∥_{1}, ∥ p_{G_{i}, 1} - p_{G_{i}^{'}, 1} ∥_{\infty} \leq 2 \sqrt{m + 1} / \sqrt{m!}$ . The bound $\log (m!) = \sum_{k = 1}^{m} \log k \geq \int_{1}^{m} \log x d x = m \log m - m + 1$ shows this error is at most $η$ when $m = ⌈ c \log (e / η) ⌉$ for a sufficiently large universal $c$ , with $η \in (0, 1 / 2)$ absorbed by increasing $c$ .

3.1F1step 1.1step 2.1algebra∎

Let $ν_{i}$ and $ν_{i}^{'}$ be the pushforwards of $G_{i}$ and $G_{i}^{'}$ , respectively, under $w \mapsto a_{i} + σ w$ , and set $F^{'} = \sum_{i} F (I_{i}) ν_{i}^{'}$ . Since $G_{i}^{'}$ is supported on $[0, b_{i}]$ , every $ν_{i}^{'}$ is supported in the closure of $I_{i}$ , so $F^{'}$ is a discrete probability measure on $[- A, A]$ with at most $K (m + 1) \leq C (A / σ \lor 1) \log (e / η)$ support points. Translation and scaling give $p_{ν_{i}, σ} (x) = σ^{- 1} p_{G_{i}, 1} ((x - a_{i}) / σ)$ and the same identity for $ν_{i}^{'}$ and $G_{i}^{'}$ . Summing these component errors with weights $F (I_{i})$ shows the $L^{1}$ error is at most $η$ and the supremum error at most $η / σ$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Countably partitioned sieves yield posterior contraction by weighted entropy

Statement

Assume AC. For each $n$ , let $Π_{n}$ be a prior on a measurable parameter space $Θ_{n}^{*}$ , and let $p_{θ}$ be a jointly measurable family of finite-valued densities against one sigma-finite measure, and let $p_{0}$ be the true density for IID data $X_{1}, \dots, X_{n}$ . Let $ϵ_{n}, δ_{n} > 0$ satisfy $δ_{n} \leq ϵ_{n}$ , $n δ_{n}^{2} \to \infty$ , and $n ϵ_{n}^{2} \to \infty$ . Suppose for a constant $C_{0} > 0$ that $Π_{n} ⁣ {θ : K L (P_{0} ∥ P_{θ}) \leq δ_{n}^{2}, P_{0} {[\log \frac{p_{0}}{p_{θ}}]}^{2} \leq δ_{n}^{2}} \geq e^{- C_{0} n δ_{n}^{2}} .$ Let $S_{n} \subseteq Θ_{n}^{*}$ be a measurable sieve with $Π_{n} (S_{n}^{c} ∣ X_{1 : n}) \to 0$ in $P_{0}^{n}$ -probability. Partition it into countably many measurable sets $S_{n, j}$ , put $π_{n, j} = Π_{n} (S_{n, j})$ , and let $N_{n, j} = N (ϵ_{n}, {p_{θ} : θ \in S_{n, j}}, h)$ be the finite covering number as defined in Covering, packing, bracketing numbers, and metric entropy, with centers in that density class, whenever $π_{n, j} > 0$ . If $e^{- n ϵ_{n}^{2}} \sum_{j : π_{n, j} > 0} \sqrt{N_{n, j} π_{n, j}} ⟶ 0,$ then for every sufficiently large fixed $M$ , the posterior mass of ${θ : h (p_{θ}, p_{0}) > M ϵ_{n}}$ converges to zero in $P_{0}^{n}$ -probability.

Facts & Assumptions

Given: A dominated IID model with a jointly measurable density family, AC, the displayed local prior mass, a measurable sieve with vanishing posterior complement, and a countable measurable partition satisfying the weighted entropy condition.

[A1]

Assume AC for the directional KL convention. The Axiom of Choice Kullback–Leibler divergence

[F1]

KL prior mass gives a posterior denominator lower bound in probability: if the local KL and log-square set has prior mass at least $e^{- C_{0} n δ_{n}^{2}}$ , then, except with probability at most $(n δ_{n}^{2})^{- 1}$ , the normalized likelihood denominator is at least $e^{- (C_{0} + 2) n δ_{n}^{2}}$ .

[F2]

Root-L2 Hellinger distance between density laws: $h^{2} (P, Q) = \int (\sqrt{p} - \sqrt{q})^{2} d μ$ and $\int \sqrt{p q} d μ = 1 - h^{2} (P, Q) / 2$ .

[F3]

Hellinger distance is a metric and controls L1 and total variation: Hellinger distance is a metric and event-supremum total variation is bounded by Hellinger distance, including for finite product laws.

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative measurable likelihood-ratio integrands may be integrated in either order.

[F5]

Dominated statistical experiments and Likelihood densities: all densities are relative to the same fixed sigma-finite measure; the supplied family is jointly measurable, so likelihood-ratio integrals are measurable.

[F6]

Identical distribution and IID families and Expectations factor over finite products of independent random variables: disjoint finite blocks of an IID sample are independent, and expectations of products of nonnegative functions of those blocks factor.

[F7]

Markov's inequality for random variables: for a nonnegative random variable $Y$ and $a > 0$ , $P (Y \geq a) \leq E [Y] / a$ .

[F8]

Probability measures and probability spaces and The measure with density $f$ relative to $μ$ : $P_{0}$ and each model law are probability measures represented by their displayed densities; integrating $p_{θ} / p_{0}$ under $P_{0}$ gives $\int_{{p_{0} > 0}} p_{θ} d μ \leq 1$ .

[F9]

Covering, packing, bracketing numbers, and metric entropy: $N (ϵ, F, h)$ is the least finite number of centers from $F$ whose closed Hellinger balls of radius $ϵ$ cover $F$ .

Proof

Given: The hypotheses in the Statement.

1.1A1F1given

Put $x_{n} = n ϵ_{n}^{2}$ , $y_{n} = n δ_{n}^{2}$ , and let $D_{n}$ be the likelihood denominator relative to $p_{0}^{n}$ . By [F1], on an event $Ω_{n}$ with $P_{0}^{n} (Ω_{n}^{c}) \leq y_{n}^{- 1}$ , $D_{n} \geq e^{- (C_{0} + 2) y_{n}}$ .

1.2givenalgebra

Write $T_{n} = \sum_{j} \sqrt{N_{n, j} π_{n, j}}$ and $s_{n} = e^{- x_{n}} T_{n}$ , so $s_{n} \to 0$ . Fix $K > 2 (C_{0} + 3)$ and split indices into active $J_{n}^{+} = {j : π_{n, j} \geq e^{- K x_{n}}}$ and small-mass $J_{n}^{-}$ . Since $N_{n, j} \geq 1$ , $\sum_{J_{n}^{-}} π_{n, j} \leq e^{- K x_{n} / 2} T_{n} = e^{(1 - K / 2) x_{n}} s_{n}$ . For active $j$ , $\sqrt{N_{n, j} π_{n, j}} \leq T_{n} = e^{x_{n}} s_{n}$ , hence $\log (N_{n, j} / π_{n, j}) \leq (2 + 2 K) x_{n}$ for all sufficiently large $n$ .

1.3A1F2F9givenconstruct

Fix a sufficiently large constant $M$ . If $M ϵ_{n} \geq \sqrt{2}$ , then the alternative is empty because Hellinger distance is at most $\sqrt{2}$ , and the desired posterior mass is zero for this $n$ . For the remaining $n$ , choose, by [A1], a finite $ϵ_{n}$ -cover with centers in each active stratum's density image, as allowed by [F9]. Discard centers whose closed $ϵ_{n}$ -ball meets no density in the stratum with $h (p_{θ}, p_{0}) > M ϵ_{n}$ ; at most $N_{n, j}$ centers remain, and they cover that alternative. For every retained center $q$ , the triangle inequality gives $h (q, p_{0}) > (M - 1) ϵ_{n}$ . Set $m_{n} = ⌈ \frac{2 \log 8}{(M - 1)^{2} ϵ_{n}^{2}} ⌉$ . On a block of $m_{n}$ observations define $T_{q} = 1 {\prod_{i} q (X_{i}) > \prod_{i} p_{0} (X_{i})}$ . On rejection, $p_{0}^{\otimes m_{n}} \leq \sqrt{p_{0}^{\otimes m_{n}} q^{\otimes m_{n}}}$ , so its type-I error is at most the product affinity; on acceptance, $q^{\otimes m_{n}} \leq \sqrt{p_{0}^{\otimes m_{n}} q^{\otimes m_{n}}}$ , so its type-II error under $q$ has the same bound. By [F2], this affinity is $(1 - h^{2} (q, p_{0}) / 2)^{m_{n}} \leq e^{- m_{n} h^{2} (q, p_{0}) / 2} \leq 1 / 8$ .

2.1F2F3step 1.3algebra

If $h (p_{θ}, q) \leq ϵ_{n}$ , product affinity gives $h^{2} (P_{θ}^{m_{n}}, Q^{m_{n}}) = 2 [1 - {(1 - h^{2} (p_{θ}, q) / 2)}^{m_{n}}] \leq m_{n} ϵ_{n}^{2}$ , using $1 - (1 - u)^{m} \leq m u$ for $u \in [0, 1]$ ; [F3] bounds their total variation by $\sqrt{m_{n}} ϵ_{n}$ . Since $M ϵ_{n} < \sqrt{2}$ , $ϵ_{n}^{2} < 2 / M^{2}$ and $m_{n} ϵ_{n}^{2} \leq \frac{2 \log 8}{(M - 1)^{2}} + \frac{2}{M^{2}} \leq 1 / 64$ for sufficiently large $M$ . The total variation is then at most $1 / 8$ . Therefore the block test in step 1.3 rejects with probability at least $3 / 4$ under every $p_{θ}$ in the alternative and at most $1 / 8$ under $p_{0}$ .

3.1F6F7step 2.1algebra

Divide $L_{n} = ⌊ n / m_{n} ⌋$ disjoint blocks and let $Y_{k}$ be their block-test indicators. Under each IID law the indicators are independent by [F6]; their means are at most $a = 1 / 8$ under $P_{0}$ and at least $b = 3 / 4$ under every alternative by step 2.1. For independent Bernoulli variables with means at most $a$ , exponential Markov [F7] gives $P_{0} (L_{n}^{- 1} \sum Y_{k} \geq t) \leq e^{- L_{n} D (t ∥ a)}$ ; with means at least $b$ , the negative-exponential bound gives $P_{θ} (L_{n}^{- 1} \sum Y_{k} < t) \leq e^{- L_{n} D (t ∥ b)}$ , where $D (t ∥ u) = t \log (t / u) + (1 - t) \log ((1 - t) / (1 - u))$ . These bounds follow by bounding each moment generating function by $1 - u + u e^{λ}$ (using $u = a$ for $λ > 0$ , $u = b$ for $λ < 0$ ) and optimizing over $λ$ .

4.1A1step 1.2step 2.1step 3.1algebra

The function $D (t ∥ a) - D (t ∥ b)$ is affine and strictly increasing on $[a, b]$ , is negative at $a$ , and equals $D (b ∥ a) > 0$ at $b$ . Let $t_{*}$ be its unique zero. By AC [A1], for each active $j$ choose $t_{j} \in [t_{*}, b]$ such that $L_{n} {D (t_{j} ∥ a) - D (t_{j} ∥ b)} = \log (N_{n, j} / π_{n, j})$ . This is possible after increasing $M$ : writing $A = 2 \log 8$ , $M ϵ_{n} < \sqrt{2}$ implies $A / ((M - 1)^{2} ϵ_{n}^{2}) > 2$ for sufficiently large $M$ , so the ceiling in $m_{n}$ costs at most a fixed factor; since $n ϵ_{n}^{2} \to \infty$ , the floor in $L_{n}$ also costs at most a fixed factor eventually. Hence $L_{n} \geq c M^{2} x_{n}$ for a universal $c > 0$ , while step 1.2 bounds the right side by $(2 + 2 K) x_{n}$ . Define $ϕ_{n, j}$ as the maximum of the block-majority tests over its retained centers (or zero if there are none), using threshold $t_{j}$ . Continuity on $[t_{*}, b]$ and positivity of both divergences at $t_{*}$ give a constant $c_{*} > 0$ such that $(D (t ∥ a) + D (t ∥ b)) / 2 \geq c_{*}$ there. The null union bound and step 3.1 give $P_{0}^{n} (ϕ_{n, j} = 1) \leq \sqrt{N_{n, j} π_{n, j}} e^{- c_{*} L_{n}}$ ; the same Chernoff bound for a center within $ϵ_{n}$ of each alternative gives $\sup_{θ \in S_{n, j} : h (p_{θ}, p_{0}) > M ϵ_{n}} P_{θ}^{n} (ϕ_{n, j} = 0) \leq \sqrt{N_{n, j} / π_{n, j}} e^{- c_{*} L_{n}}$ by step 2.1.

5.1F4F5step 4.1algebra

Let $ϕ_{n}$ be one on the countable union of the rejection events of $ϕ_{n, j}$ over $j \in J_{n}^{+}$ and zero otherwise; it is measurable as a countable union of measurable events. The union bound and step 4.1 give $P_{0}^{n} (ϕ_{n} = 1) \leq T_{n} e^{- c_{*} L_{n}}$ . For the tested likelihood numerator on an active stratum, Tonelli [F4], the jointly measurable likelihood interface [F5] and change of density give $E_{0} [(1 - ϕ_{n}) \int_{{θ \in S_{n, j} : h (p_{θ}, p_{0}) > M ϵ_{n}}} \prod_{i} \frac{p_{θ} (X_{i})}{p_{0} (X_{i})} Π_{n} (d θ)] \leq π_{n, j} \sup_{θ in the set} P_{θ}^{n} (1 - ϕ_{n, j}) \leq \sqrt{N_{n, j} π_{n, j}} e^{- c_{*} L_{n}} .$

6.1F7F8step 1.1step 1.2step 5.1algebra

Write $d_{n} = e^{- (C_{0} + 2) y_{n}}$ . For the active numerator $R_{n}^{+}$ , step 5.1 gives $P_{0}^{n} (ϕ_{n} = 1) \leq T_{n} e^{- c_{*} L_{n}}$ and $E_{0} [(1 - ϕ_{n}) R_{n}^{+}] \leq T_{n} e^{- c_{*} L_{n}}$ . Thus, for every fixed $ζ > 0$ , $P_{0}^{n} (R_{n}^{+} / D_{n} > ζ) \leq P_{0}^{n} (D_{n} < d_{n}) + T_{n} e^{- c_{*} L_{n}} + ζ^{- 1} d_{n}^{- 1} T_{n} e^{- c_{*} L_{n}}$ . This tends to zero for sufficiently large $M$ , since $T_{n} = e^{x_{n}} s_{n}$ , $y_{n} \leq x_{n}$ , $L_{n} \geq c M^{2} x_{n}$ , and $s_{n} \to 0$ . For the small-mass numerator $R_{n}^{-}$ , [F8] and Tonelli give $E_{0} [R_{n}^{-}] \leq \sum_{J_{n}^{-}} π_{n, j} \leq e^{- K x_{n} / 2} T_{n}$ ; hence its posterior mass tends to zero in probability by Markov and the denominator event, because $e^{(C_{0} + 2) y_{n}} e^{- K x_{n} / 2} T_{n} \leq s_{n} e^{(C_{0} + 3 - K / 2) x_{n}} \to 0$ when $K > 2 (C_{0} + 3)$ .

7.1givenstep 6.1∎

The posterior outside the $M ϵ_{n}$ -ball is the union of its active-stratum part, its small-mass-stratum part, and the sieve complement. Step 6.1 sends the first two parts to zero in probability, and the sieve-complement term tends to zero by hypothesis.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Hellinger entropy for compact-location normal mixtures in a scale band

Statement

For $A \geq 1$ , $τ > 0$ , and $0 < ϵ < 1 / 4$ , let $P_{A, τ} : = {p_{F, σ} : F ([- A, A]) = 1, τ \leq σ \leq 2 τ} .$ There is a universal constant $C$ such that $\log N (ϵ, P_{A, τ}, h) \leq C (A / τ \lor 1) {[\log ⁣ (\frac{C (A / τ \lor 1)}{ϵ})]}^{2} .$ The same bound, with a changed universal constant, holds for $P_{A, τ, ρ} : = {p_{F, σ} : F ([- A, A]^{c}) \leq ρ, τ \leq σ \leq 2 τ}$ whenever $0 \leq ρ \leq c ϵ^{2}$ for a sufficiently small universal $c > 0$ . Here $N$ is the covering number in the stated metric.

Facts & Assumptions

Given: $A \geq 1$ , $τ > 0$ , $0 < ϵ < 1 / 4$ , AC, and the mixture density of The Dirichlet normal-mixture density is jointly measurable and normalized.

[A1]

Assume AC for the normal-density and finite-support approximation interfaces. The Axiom of Choice

[F1]

Compactly supported Gaussian mixtures admit logarithmic-size finite approximations: a mixture with mixing law on $[- A, A]$ is within $L^{1}$ error $η$ of a discrete mixture with at most $C (A / σ \lor 1) \log (e / η)$ atoms.

[F2]

Hellinger distance is a metric and controls L1 and total variation: for probability densities, Hellinger distance is a metric and $h^{2} (p, q) \leq \int ∣ p - q ∣ d λ$ .

[F3]

A positive-variance normal law has its location-scale density: $ϕ_{σ} (x - z) = σ^{- 1} ϕ ((x - z) / σ)$ , with $ϕ$ the standard normal density.

[F4]

Root-L2 Hellinger distance between density laws: the Hellinger metric used here is the root-L2 distance between square-root densities.

[F5]

For a standard normal random variable $Z$ , $E [Z^{2}] = 1$ . Location-scale normal laws have their stated mean and variance

[F6]

Nonnegative product integrals may be interchanged. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F7]

Proof

Given: The hypotheses in the Statement.

1.1A1F1givenchoose

Put $K : = ⌈ C_{0} (A / τ \lor 1) \log (e / ϵ) ⌉$ . By [F1], every $p_{F, σ} \in P_{A, τ}$ is within $L^{1}$ distance $ϵ^{2} / C_{1}$ of a mixture with at most $K$ atoms, for a fixed sufficiently large $C_{1}$ . Pad shorter mixtures by zero-weight atoms.

2.1A1F2F3F4F5F6F7step 1.1givenconstruct

Differentiating the density in [F3] gives $∥ \partial_{z} ϕ_{σ} (\cdot - z) ∥_{1} = σ^{- 1} E ∣ Z ∣ \leq σ^{- 1}$ , since $2 ∣ Z ∣ \leq 1 + Z^{2}$ and $E Z^{2} = 1$ by [F5], and $∥ \partial_{σ} ϕ_{σ} (\cdot - z) ∥_{1} = σ^{- 1} E ∣ Z^{2} - 1 ∣ \leq 2 σ^{- 1}$ for $Z \sim N (0, 1)$ . Integrating these derivatives along the location and scale segments by [F7] yields $∥ ϕ_{σ} (\cdot - z) - ϕ_{σ} (\cdot - z^{'}) ∥_{1} \leq ∣ z - z^{'} ∣ / σ$ and $∥ ϕ_{σ} - ϕ_{σ^{'}} ∥_{1} \leq 2 ∣ σ - σ^{'} ∣ / (σ \land σ^{'})$ . For the first class use a scale grid in $[τ, 2 τ]$ ; for an actual band, instead grid $[τ, \min {2 τ, s_{n}}]$ , omitting $2 τ$ when that endpoint is excluded. In either case choose grid mesh at most $c_{2} τ ϵ^{2}$ , with every grid point admitted by the corresponding scale constraint. Use a location grid in $[- A, A]$ with mesh at most $c_{2} τ ϵ^{2}$ and an $L^{1}$ grid on the $K$ -simplex of mesh at most $c_{2} ϵ^{2}$ . Replacing the scale, locations and weights changes the mixture density in $L^{1}$ by at most $C c_{2} ϵ^{2}$ : integrate the componentwise location and scale bounds against the finite mixing weights, and use Tonelli [F6] for the componentwise integrals. Choose $c_{2}$ so this plus the approximation from step 1.1 is at most $ϵ^{2} / 4$ . Thus the resulting finite set, with centers in the class being covered, is an $L^{1}$ cover of radius $ϵ^{2} / 4$ , hence an $h$ -cover of radius $ϵ / 2$ by [F2].

3.1step 2.1algebra

The scale grid has at most $C ϵ^{- 2}$ points and the location grid at most $C (1 + A / (τ ϵ^{2}))$ points. Rounding each of the first $K - 1$ weights down to a mesh proportional to $ϵ^{2} / K$ leaves the last weight nonnegative after setting it to make the sum one, gives at most $(C K / ϵ^{2})^{K}$ weight vectors, and changes their $L^{1}$ distance by at most $2 c_{2} ϵ^{2}$ . Therefore the logarithm of the cover size is at most $C \log (e / ϵ) + K [\log ⁣ (\frac{C (A / τ \lor 1)}{ϵ^{2}}) + \log ⁣ (\frac{C K}{ϵ^{2}})] \leq C (A / τ \lor 1) {[\log ⁣ (\frac{C (A / τ \lor 1)}{ϵ})]}^{2} .$ The constructed finite set is an $h$ -net of radius $ϵ / 2$ , hence also a net of radius $ϵ$ ; its cardinality therefore bounds the covering number and proves the first bound.

4.1A1F2F4F6step 2.1step 3.1givenalgebra∎

If $r : = F ([- A, A]^{c}) = 0$ , then $F$ already belongs to the compact-location class. If $0 < r \leq ρ < 1$ , let $F_{A}$ and $F_{A^{c}}$ be the normalized restrictions to $[- A, A]$ and its complement. Then $p_{F, σ} = (1 - r) p_{F_{A}, σ} + r p_{F_{A^{c}}, σ}$ , so $∥ p_{F, σ} - p_{F_{A}, σ} ∥_{1} \leq 2 r$ by integrating the mixture densities (Tonelli [F6]). By [F2], $h (p_{F, σ}, p_{F_{A}, σ}) \leq \sqrt{2 r}$ ; the triangle inequality in [F4] then transfers the cover to the tail-constrained class. The scale grid is restricted as in step 2.1, so every center has $F ([- A, A]^{c}) = 0$ and satisfies the band and cutoff constraints. For sufficiently small universal $c$ , $ρ \leq c ϵ^{2}$ makes the truncation error at most $ϵ / 2$ , so the radius- $ϵ / 2$ cover from steps 2.1 and 3.1 becomes a radius- $ϵ$ cover after the constant rescaling used in the entropy bound.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The dyadic scale bands satisfy the weighted entropy condition

Statement

Fix $A \geq 1$ independently of $n$ . The constants in the entropy estimates and the required fixed $M_{0}$ may depend on $A$ . Let $0 < a_{2} < a_{1} < 1$ , $n^{- a_{1}} ≲ σ_{n} ≲ n^{- a_{2}}$ , and $γ > 1$ . Suppose $G$ is a probability measure on $(0, \infty)$ satisfying $G ((0, s]) \leq C e^{- β s^{- γ}}$ for all sufficiently small $s > 0$ . Let $s_{n} \to 0$ with $s_{n} / σ_{n} \to \infty$ . For integers $j$ with $2^{j} σ_{n} \leq s_{n}$ , put ${\tilde{r}}_{n} : = \min {r_{n}, 1 / 4}$ and $P_{n, j} : = {p_{F, σ} : F ([- A, A]^{c}) \leq {\tilde{r}}_{n}^{2} / 8, 2^{j} σ_{n} \leq σ < 2^{j + 1} σ_{n}, σ \leq s_{n}},$ and let $π_{n, j}$ be the prior mass of the corresponding parameter band, so $π_{n, j} \leq G ([2^{j}, 2^{j + 1}))$ . Suppose $r_{n} > 0$ and, for some fixed sufficiently large $M_{0}$ (allowed to depend on $A$ ) and fixed $η > γ / [2 (γ - 1)]$ , $r_{n} \geq M_{0} \max {(n σ_{n})^{- 1 / 2} \log n, n^{- 1 / 2} σ_{n}^{- η}} .$ With $N_{n, j} = N (r_{n}, P_{n, j}, h)$ , one has $e^{- n r_{n}^{2}} \sum_{j : 2^{j} σ_{n} \leq s_{n}} \sqrt{N_{n, j} π_{n, j}} ⟶ 0.$ If instead $G$ is supported on a compact subset of $(0, \infty)$ , the same conclusion holds under the first rate term alone.

Facts & Assumptions

Given: The fixed cutoff $A \geq 1$ , AC, the polynomial bounds on $σ_{n}$ , the lower scale-tail bound on $G$ , the dyadic scale bands, and the compact-location entropy estimate.

[A1]

Assume AC for the compact-location entropy and mixture-density interfaces. The Axiom of Choice

[F1]

Hellinger entropy for compact-location normal mixtures in a scale band: for compactly supported mixing laws on $[- A, A]$ , $0 < u < 1 / 4$ , and $τ \leq σ \leq 2 τ$ , $\log N (u, {p_{F, σ}}, h) \leq C (A / τ \lor 1) [\log (C (A / τ \lor 1) / u)]^{2} .$

[F2]

Hellinger distance is a metric and controls L1 and total variation: for probability densities, Hellinger distance is a metric and $h^{2} (p, q) \leq \int ∣ p - q ∣ d λ$ .

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2givenalgebra

Write $x_{n} = n r_{n}^{2}$ and $J_{n} = \max {j : 2^{j} σ_{n} \leq s_{n}}$ . Since $s_{n} \to 0$ and $σ_{n} \geq c n^{- a_{1}}$ , $J_{n} = O (\log n)$ . Put $u_{n} : = \min {r_{n} / 2, 1 / 8}$ . If $r_{n} \leq 1 / 4$ , truncate a mixing law to $[- A, A]$ and normalize it there; the removed mass is at most $r_{n}^{2} / 8$ , so its mixture changes by $L^{1}$ at most $r_{n}^{2} / 4$ and by Hellinger at most $r_{n} / 2$ using [F2]. The compact cover at radius $u_{n} = r_{n} / 2$ then covers $P_{n, j}$ at radius $r_{n}$ . If $r_{n} > 1 / 4$ , the sieve tail is at most $1 / 128$ , so truncation changes Hellinger distance by at most $1 / 8$ ; the compact cover at radius $u_{n} = 1 / 8$ then covers the class at radius at most $1 / 4 < r_{n}$ . Thus in both cases $N_{n, j} \leq N (u_{n}, P_{A, 2^{j} σ_{n}}, h)$ . The rate lower bound implies $\log (1 / u_{n}) \leq C \log n$ and $x_{n} \geq M_{0}^{2} σ_{n}^{- 1} (\log n)^{2}, x_{n} \geq M_{0}^{2} σ_{n}^{- 2 η} .$ For $0 \leq j \leq J_{n}$ , [F1] bounds $\log N_{n, j} \leq C σ_{n}^{- 1} (\log n)^{2}$ , because the band lower scale is at least $σ_{n}$ . Since $π_{n, j} \leq 1$ , these terms contribute at most $C \log n \exp {C σ_{n}^{- 1} (\log n)^{2} / 2 - x_{n}},$ which tends to zero for sufficiently large $M_{0}$ .

1.2A1F1givenalgebra

For negative indices write $j = - k$ , $k \geq 1$ . For all sufficiently large $k$ , the scale-tail premise gives $π_{n, - k} \leq C \exp (- c 2^{γ k})$ . Also [F1] and $u_{n} \geq c n^{- 1 / 2} \log n$ imply $\log N_{n, - k} \leq C σ_{n}^{- 1} 2^{k} (\log n + k)^{2} .$ Choose $θ > 0$ and $γ_{0} < γ$ sufficiently close to $0$ and $γ$ , respectively, so $q : = 1 + θ < γ_{0}$ and $ρ : = γ_{0} / (γ_{0} - q) < 2 η$ . Since $(\log n + k)^{2} \leq C_{θ} (\log n)^{2} 2^{θ k}$ and $e^{- c 2^{γ k}} \leq e^{- c^{'} 2^{γ_{0} k}}$ , the logarithm of each summand is at most $C K_{n} 2^{q k} - c^{'} 2^{γ_{0} k}, K_{n} : = σ_{n}^{- 1} (\log n)^{2} .$ Enlarging $C$ handles the finitely many smaller $k$ .

2.1step 1.2givenalgebra

Put $p = γ_{0} / q > 1$ . As a function of $v = 2^{q k} \geq 1$ , the exponent $C K_{n} v - c^{'} v^{p}$ has maximum at most $C^{'} K_{n}^{p / (p - 1)} = C^{'} K_{n}^{ρ}$ , by differentiation. The maximizing scale is a constant multiple of $K_{n}^{1 / (p - 1)}$ , so there are $O (\log K_{n})$ indices up to any fixed multiple of that scale; their sum has logarithm at most $C^{''} K_{n}^{ρ}$ . Choose a fixed multiple $L$ large enough that for $v \geq L K_{n}^{1 / (p - 1)}$ the negative term is at least twice the positive term. Grouping the subsequent dyadic $v$ values gives a convergent geometric tail, also bounded by $\exp (C^{''} K_{n}^{ρ})$ . Therefore $\log \sum_{k \geq 1} \sqrt{N_{n, - k} π_{n, - k}} \leq C^{''} σ_{n}^{- ρ} (\log n)^{2 ρ} .$ Because $ρ < 2 η$ and $σ_{n} \leq C n^{- a_{2}}$ , this is $o (σ_{n}^{- 2 η}) = o (x_{n})$ . The negative-index contribution therefore vanishes after multiplication by $e^{- x_{n}}$ .

3.1A1F1step 1.1givenalgebra∎

If $G$ is supported on $[g_{-}, g_{+}]$ with $g_{-} > 0$ , only a fixed finite number of dyadic indices have positive mass. Their lower scales are at least $g_{-} σ_{n} / 2$ , so [F1] at radius $u_{n}$ bounds each log covering number by $C σ_{n}^{- 1} (\log n)^{2}$ . The first rate term gives $x_{n} \geq M_{0}^{2} σ_{n}^{- 1} (\log n)^{2}$ , and the finite weighted sum tends to zero for sufficiently large $M_{0}$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

A Dirichlet process assigns exponential mass to a finite-partition simplex neighborhood

Statement

Assume AC, let $α$ be a finite nonzero measure of total mass $m$ , and let $F \sim D P (α)$ . Consider a finite partition $(U_{0}, U_{1}, \dots, U_{N})$ with $α (U_{0}) = 0$ or $0 < α (U_{0}) \leq m$ . Write $α_{j} = α (U_{j})$ for $j \geq 1$ , and assume $a η^{b} \leq α_{j} \leq 1 (1 \leq j \leq N), 0 < η < 1 / 4, N η \leq 1 / 4,$ for fixed $a, b > 0$ . For every $w = (w_{1}, \dots, w_{N})$ in the probability simplex, there are constants $c, C > 0$ depending only on $a, b, m, α (U_{0})$ such that $\Pr ⁣ {\sum_{j = 1}^{N} ∣ F (U_{j}) - w_{j} ∣ + F (U_{0}) \leq 3 η, \min_{1 \leq j \leq N} F (U_{j}) \geq η^{2} / 2} \geq C \exp {- c N \log (e / η)} .$ For any compact $J \subset (0, m]$ , the constants may instead depend only on $a, b, m, J$ and are uniform over $α (U_{0}) \in J$ . When $α (U_{0}) = 0$ , the zero-base cell has $F (U_{0}) = 0$ almost surely and is omitted from the Dirichlet vector.

Facts & Assumptions

Given: Full AC, a finite nonzero base measure, the stated finite partition, and $w$ in the $N$ -simplex.

[A1]

Full AC supplies the normalized-gamma Dirichlet-process existence interface and implies Countable Choice for the finite-dimensional Dirichlet density. The Axiom of Choice

[F1]

Dirichlet process and Existence of a Dirichlet process by normalized gamma Poisson random measure: the finite-partition mass vector has the Dirichlet law with parameters $α (U_{j})$ ; zero-base cells have mass zero.

[F2]

Dirichlet distribution, including the one-category case: with positive parameters $(a_{0}, a_{1}, \dots, a_{N})$ , the simplex density is $Γ (m) \prod_{j = 0}^{N} x_{j}^{a_{j} - 1} / \prod_{j = 0}^{N} Γ (a_{j})$ , where $m = \sum_{j} a_{j}$ ; the zero-coordinate case is omitted.

[F3]

The Euler Gamma values are positive and finite for positive arguments, and $Γ (s + 1) = s Γ (s)$ for $s > 0$ . The real Gamma function by Euler's integral Euler's Gamma integral converges exactly for positive real parameters The real Gamma functional equation $Γ (s + 1) = s Γ (s)$

[F4]

The real Gamma function is continuous on $(0, \infty)$ . The real Gamma function is smooth and its derivatives are logarithmic moments

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2givenchooseconstruct

If $α (U_{0}) = 0$ and $N = 1$ , [F1]–[F2] give $F (U_{0}) = 0$ and $F (U_{1}) = w_{1} = 1$ almost surely, so the event has probability one. In all remaining cases there are at least two positive-parameter cells. By [F1]–[F2], if $α (U_{0}) > 0$ the vector $(X_{0}, X_{1}, \dots, X_{N}) : = (F (U_{0}), F (U_{1}), \dots, F (U_{N}))$ has the displayed Dirichlet density; if $α (U_{0}) = 0$ , set $X_{0} = 0$ and use the $N$ -coordinate Dirichlet density. Choose an index $i_{*}$ with $w_{i_{*}} = \max_{j} w_{j} \geq 1 / N$ . For every $j \neq i_{*}$ restrict $x_{j}$ to $[\max (w_{j} - η^{2}, η^{2} / 2), \min (w_{j} + η^{2}, 1)]$ , an interval of length at least $η^{2} / 2$ . If $α (U_{0}) > 0$ , also restrict $x_{0}$ to $[η^{2} / 2, η^{2}]$ and define $x_{i_{*}} = 1 - x_{0} - \sum_{j \neq i_{*}} x_{j}$ ; if $α (U_{0}) = 0$ , define $x_{i_{*}} = 1 - \sum_{j \neq i_{*}} x_{j}$ . The deviations of the coordinates other than $i_{*}$ sum to at most $N η^{2}$ , so $∣ x_{i_{*}} - w_{i_{*}} ∣ \leq N η^{2}$ . Consequently the total event deviation is at most $2 N η^{2} + η^{2} \leq 3 η$ ; and $x_{i_{*}} \geq 1 / N - N η^{2} \geq η^{2} / 2$ because $N η \leq 1 / 4$ . These choices lie in the simplex, satisfy the event, and their integration region has coordinate volume at least $(η^{2} / 2)^{N}$ when $α (U_{0}) > 0$ and $(η^{2} / 2)^{N - 1}$ when $α (U_{0}) = 0$ .

2.1A1F1F2F3F4step 1.1algebracases∎

For $0 < a_{j} \leq 1$ , split the Euler integral for $Γ (1 + a_{j})$ at $1$ . On $(0, 1]$ , $t^{a_{j}} \leq 1$ , and on $[1, \infty)$ , $t^{a_{j}} \leq t$ , so $Γ (1 + a_{j}) \leq 1 + \int_{1}^{\infty} t e^{- t} d t = 1 + 2 / e$ . The functional equation in [F3] gives $Γ (a_{j}) \leq (1 + 2 / e) / a_{j}$ . Put $c_{Γ} = (1 + 2 / e)^{- 1}$ . When $α (U_{0}) > 0$ , the normalizing coefficient in [F2] is at least $[Γ (m) / Γ (α (U_{0}))] c_{Γ}^{N} \prod_{j = 1}^{N} a_{j}$ ; when $α (U_{0}) = 0$ and $N \geq 2$ , it is at least $Γ (m) c_{Γ}^{N} \prod_{j = 1}^{N} a_{j}$ . These Gamma factors are positive by [F3]. On the region from step 1.1, each interval coordinate satisfies $x_{j}^{a_{j} - 1} \geq 1$ . If $α (U_{0}) > 1$ , its factor is at least $(η^{2} / 2)^{m}$ ; if $0 < α (U_{0}) \leq 1$ , it is at least one. For $α (U_{0}) > 0$ , integrating the density over that region is therefore at least $[Γ (m) / Γ (α (U_{0}))] (c_{Γ} a / 2)^{N} η^{(b + 2) N}$ , multiplied by $(η^{2} / 2)^{m}$ when $α (U_{0}) > 1$ . For $α (U_{0}) = 0$ and $N \geq 2$ , the coordinate volume $(η^{2} / 2)^{N - 1} \geq (η^{2} / 2)^{N}$ gives a lower bound $Γ (m) (c_{Γ} a / 2)^{N} η^{(b + 2) N}$ . If $α (U_{0})$ ranges in a compact $J \subset (0, m]$ , [F4] bounds $Γ (α (U_{0}))$ above uniformly, so $Γ (m) / Γ (α (U_{0}))$ is uniformly bounded below; all other Gamma factors here are fixed and positive. Since $a, b$ are fixed, $(c_{Γ} a / 2)^{N} η^{(b + 2) N} \geq \exp {- c_{1} N \log (e / η)}$ for some $c_{1} > 0$ . When present, the residual factor $(η^{2} / 2)^{m}$ is at least $\exp {- c_{2} N \log (e / η)}$ because $m$ is fixed and $N \geq 1$ . Combining these bounds gives $C \exp {- c N \log (e / η)}$ with constants depending only on $a, b, m, α (U_{0})$ , uniformly when $α (U_{0}) \in J$ . The probability-one case in step 1.1 also satisfies the claimed bound after taking $C \leq 1$ . The resulting probability is the one in the Statement by [F1].

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The mixture prior places local KL and log-square mass near a compact smooth truth

Statement

Assume AC. Let $p_{0}$ be a bounded probability density supported on a compact set $K \subset R$ . Let $α$ be a finite nonzero measure with a continuous density bounded below by a positive constant on a neighborhood of $K$ . Assume $h (p_{0}, p_{0} * ϕ_{σ}) \leq C σ^{2}$ for all sufficiently small $σ > 0$ , where $ϕ_{σ}$ is the centered normal density with standard deviation $σ$ . Let $F \sim D P (α)$ and let $S \sim G$ be independent, where $G$ is any probability measure on $(0, \infty)$ . Suppose $n^{- a_{1}} ≲ σ_{n} ≲ n^{- a_{2}}$ for $0 < a_{2} < a_{1} < 1$ . Set $σ = σ_{n} S$ and $δ_{n} : = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2} \log n} .$ Then, for some constant $C_{1} > 0$ and all sufficiently large $n$ , $Π_{n} ⁣ {(F, σ) : K L (P_{0} ∥ P_{F, σ}) \leq C_{1} δ_{n}^{2}, P_{0} {[\log \frac{p_{0}}{p_{F, σ}}]}^{2} \leq C_{1} δ_{n}^{2}} \geq e^{- C_{1} n δ_{n}^{2}} .$ Here $p_{F, σ}$ is the density of The Dirichlet normal-mixture density is jointly measurable and normalized, and all logarithmic integrands are set to zero on ${p_{0} = 0}$ .

Facts & Assumptions

Given: AC; the compactly supported bounded density $p_{0}$ ; the stated Gaussian-convolution Hellinger bound; the finite base measure $α$ with a positive continuous density near $K$ ; the independent priors $F \sim D P (α)$ and $S \sim G$ ; and the polynomial bounds on $σ_{n}$ .

[A1]

AC supplies $D P (α)$ and the directional KL convention. The Axiom of Choice Existence of a Dirichlet process by normalized gamma Poisson random measure Kullback–Leibler divergence

[F1]

The Dirichlet normal-mixture density is jointly measurable and normalized: $p_{F, σ} (x) = \int ϕ_{σ} (x - z) F (d z)$ is a positive finite-valued jointly measurable density.

[F2]

Compactly supported Gaussian mixtures admit logarithmic-size finite approximations: a probability mixing law on a compact interval has a discrete approximation with logarithmically many atoms and controlled $L^{1}$ and supremum mixture errors.

[F3]

A Dirichlet process assigns exponential mass to a finite-partition simplex neighborhood: for a finite partition whose interval-cell base masses are bounded below by $c η^{2}$ , a DP vector has probability at least $e^{- c^{'} N \log (e / η)}$ of being within $O (η)$ of any target weight vector while every interval-cell mass is at least $η^{2} / 2$ . The residual-cell mass may be zero (then that coordinate is omitted); for fixed positive residual mass the constants may depend on its value, uniformly on compact subsets of $(0, m]$ .

[F4]

A bounded likelihood ratio controls KL and log-square moments by Hellinger distance: if $h^{2} (p, q)$ is small and $∥ p / q ∥_{\infty} \leq R < \infty$ , then $K L (p ∥ q) \leq C h^{2} (1 + \log^{+} R)$ and $P [(\log (p / q))^{2}] \leq C h^{2} (1 + \log^{+} R)^{2}$ .

[F5]

Dirichlet process and Existence of a Dirichlet process by normalized gamma Poisson random measure: the finite evaluation vector of $F$ on a measurable partition has the Dirichlet law with parameters given by $α$ .

[F6]

Root-L2 Hellinger distance between density laws and Hellinger distance is a metric and controls L1 and total variation: $h$ is the root-L2 distance between square-root densities, satisfies the triangle inequality, and $h^{2} (p, q) \leq \int ∣ p - q ∣ d λ$ .

[F7]

A positive-variance normal law has its location-scale density: $ϕ_{σ} (x - z) = σ^{- 1} ϕ ((x - z) / σ)$ for the standard normal density $ϕ$ .

[F8]

For a standard normal random variable $Z$ , $E [Z^{2}] = 1$ . Location-scale normal laws have their stated mean and variance

[F9]

Nonnegative product integrals may be interchanged. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Proof

Given: The hypotheses in the Statement.

1.1F2givenchoose

Since $G$ is a probability on $(0, \infty) = ⋃_{m \geq 1} [1 / m, m]$ , choose $0 < s_{-} < s_{+} < \infty$ with $g : = G ([s_{-}, s_{+}]) > 0$ . Put $σ_{-} = s_{-} σ_{n}$ . Apply the moment-matching proof of [F2] to the law $P_{0}$ at scale $σ_{-}$ , restricting each compact moment curve to $K$ intersected with the closure of its partition interval. The same convex-hull reduction and remainder bounds then give a finite discrete probability $F_{n}$ supported on $K$ with at most $C σ_{n}^{- 1} \log n$ atoms and $∥ p_{F_{n}, σ_{-}} - p_{P_{0}, σ_{-}} ∥_{1} \leq η_{n}$ , where $η_{n} : = σ_{n}^{8}$ .

1.2A1F3F5F7F8givenconstruct

Choose an open neighborhood $V$ of $K$ on which the density of $α$ is bounded above and below by positive constants. By compactness, choose a fixed finite union $D$ of disjoint compact intervals with $K \subset int D \subset D \subset V$ . Move each atom of $F_{n}$ to a nearest point of a lattice of mesh $η_{n} σ_{-}$ , merge coincident atoms, and call the resulting law ${\tilde{F}}_{n}$ . Each atom moves by at most $η_{n} σ_{-} / 2$ ; for every $σ \geq σ_{-}$ this changes the mixture in $L^{1}$ by at most $η_{n} / 2$ , since $∥ \partial_{z} ϕ_{σ} (\cdot - z) ∥_{1} = E ∣ Z ∣ / σ \leq 1 / σ \leq 1 / σ_{-}$ , using $2 ∣ Z ∣ \leq 1 + Z^{2}$ and [F8]. The relocated locations are separated by at least $η_{n} σ_{-}$ . Around each take a centered cell of length $ℓ_{n} = η_{n} σ_{-} / 2$ . In each component of $D$ , split every intervening or end gap of length $g \geq ℓ_{n}$ into $q = ⌈ 2 g / σ_{-} ⌉$ equal subintervals. For large $n$ , the end gaps have length at least $ℓ_{n}$ because $K \subset int D$ , and gaps between atom-cells have length at least $ℓ_{n}$ by lattice separation. If $q = 1$ , the subinterval length lies in $[ℓ_{n}, σ_{-} / 2]$ ; if $q \geq 2$ , then $g > σ_{-} / 2$ and $q \leq 4 g / σ_{-}$ , so each subinterval has length in $[σ_{-} / 4, σ_{-} / 2]$ . The resulting Borel partition $U_{1}, \dots, U_{N}$ of $D$ has every cell length between $c η_{n} σ_{-}$ and $σ_{-} / 2$ , every atom-cell diameter at most $2 η_{n} σ_{-}$ , and $N \leq C (m_{n} + σ_{-}^{- 1}) \leq C σ_{n}^{- 1} \log n$ , where $m_{n}$ is the number of atoms. Let $w_{j} = {\tilde{F}}_{n} (U_{j})$ , so $\sum_{j} w_{j} = 1$ . For all large $n$ , $c η_{n} σ_{n} \leq α (U_{j}) \leq 1$ and $c η_{n} σ_{n} \geq c^{'} η_{n}^{2}$ . Also $N η_{n} \leq C σ_{n}^{7} \log n \to 0$ . The residual cell is $U_{0} = R ∖ D$ , so $α (U_{0})$ is fixed, possibly zero. Apply [F3] with $a = c^{'}$ , $b = 2$ , and $η = η_{n}$ ; its constant may depend on this fixed residual mass. We obtain an event $E_{n}$ with $Π_{D P} (E_{n}) \geq e^{- C N \log (e / η_{n})} \geq e^{- C σ_{n}^{- 1} (\log n)^{2}}$ , on which $F (U_{0}) \leq 3 η_{n}$ , $\sum_{j} ∣ F (U_{j}) - w_{j} ∣ \leq 3 η_{n}$ , and $F (U_{j}) \geq η_{n}^{2} / 2$ for every $j$ .

2.1F1F2F6F7F9givenstep 1.1step 1.2algebra

For $F$ in $E_{n}$ , compare $p_{F, σ}$ with $p_{{\tilde{F}}_{n}, σ}$ , uniformly for $s_{-} σ_{n} \leq σ \leq s_{+} σ_{n}$ . On atom cells, replacing $F ∣_{U_{j}}$ by its mass at the center costs at most $C η_{n}$ in $L^{1}$ by the [F7] location derivative bound and cell diameter. The event bounds make the total mass on gap cells and $U_{0}$ at most $C η_{n}$ and the sum of weight discrepancies at most $C η_{n}$ . Since each normal density has $L^{1}$ norm one, $∥ p_{F, σ} - p_{{\tilde{F}}_{n}, σ} ∥_{1} \leq C η_{n}$ . For every such $σ$ , the relocation estimate in step 1.2 also gives $∥ p_{{\tilde{F}}_{n}, σ} - p_{F_{n}, σ} ∥_{1} \leq C η_{n}$ . If $σ > σ_{-}$ , completing the square gives $ϕ_{σ_{-}} * ϕ_{v} = ϕ_{σ}$ for $v = \sqrt{σ^{2} - σ_{-}^{2}}$ ; at $σ = σ_{-}$ the identity is immediate. Convolution by a probability density contracts $L^{1}$ by Tonelli [F9], so $∥ p_{F_{n}, σ} - p_{P_{0}, σ} ∥_{1} \leq ∥ p_{F_{n}, σ_{-}} - p_{P_{0}, σ_{-}} ∥_{1} \leq η_{n}$ . Thus [F6] gives $h (p_{F, σ}, p_{P_{0}, σ}) \leq C \sqrt{η_{n}}$ , and the assumed bias bound gives $h (p_{0}, p_{F, σ}) \leq C (σ_{n}^{2} + \sqrt{η_{n}}) \leq C σ_{n}^{2}$ .

3.1F1F4F6F7givenstep 1.2step 2.1algebra

For every $x \in K$ , the interval $[x - σ_{-}, x + σ_{-}]$ contains a full cell $U_{j}$ because $D$ contains a fixed neighborhood of $K$ and all cells have length at most $σ_{-} / 2$ . On this interval $ϕ_{σ} (x - z) \geq c / σ$ for $s_{-} σ_{n} \leq σ \leq s_{+} σ_{n}$ . Thus [F1] and the cell-mass lower bound in step 1.2 give $p_{F, σ} (x) \geq c η_{n}^{2} / σ_{n}$ . Since $p_{0}$ is bounded and vanishes outside $K$ , $R_{n} : = ∥ p_{0} / p_{F, σ} ∥_{\infty} \leq C σ_{n} η_{n}^{- 2} = C σ_{n}^{- 15}$ . Step 2.1 gives $h (p_{0}, p_{F, σ}) \leq C σ_{n}^{2}$ , and the polynomial lower bound on $σ_{n}$ gives $1 + \log^{+} R_{n} \leq C \log n$ . Apply [F4] to $p = p_{0}$ and $q = p_{F, σ}$ to obtain both local KL quantities at most $C σ_{n}^{4} (\log n)^{2} \leq C δ_{n}^{2}$ .

4.1givenstep 1.2step 3.1algebra∎

The scale event $S \in [s_{-}, s_{+}]$ has probability $g > 0$ and is independent of $F$ . Thus the subset $E_{n} \times {S \in [s_{-}, s_{+}]}$ is contained in the displayed local KL/log-square set and has prior probability at least $g e^{- C σ_{n}^{- 1} (\log n)^{2}}$ . By the definition of $δ_{n}$ , $n δ_{n}^{2} \geq σ_{n}^{- 1} (\log n)^{2}$ , so increasing $C_{1}$ absorbs $g$ and proves the required lower bound.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Interval masses give a log-free local KL neighborhood for the mixture prior

Statement

Assume AC. Let $p_{0} \in C^{2} (R)$ be a probability density with compact support $K$ , a finite union of intervals. Suppose $J = \int_{{p_{0} > 0}} (p_{0}^{''})^{2} / p_{0} < \infty$ and there are $a, c > 0$ such that $P_{0} (I) \geq c λ (I)^{a}$ for every interval $I \subseteq K$ . Let $F \sim D P (α)$ , where $α$ is a finite nonzero measure with continuous strictly positive density on a neighborhood of $K$ , and independently let $S \sim G$ for any probability measure $G$ on $(0, \infty)$ . Suppose $n^{- a_{1}} ≲ σ_{n} ≲ n^{- a_{2}}$ for $0 < a_{2} < a_{1} < 1$ , and put $σ = σ_{n} S$ . Then for constants $C_{1}, C_{2} > 0$ and all sufficiently large $n$ , $Π_{n} {K L (P_{0} ∥ P_{F, σ}) \leq C_{1} σ_{n}^{4}, P_{0} {[\log \frac{p_{0}}{p_{F, σ}}]}^{2} \leq C_{1} σ_{n}^{4}} \geq \exp {- C_{2} σ_{n}^{- 1} (\log n)^{2}} .$ Consequently, for $d_{n} = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2}}$ , the same two local quantities are at most $C_{1} d_{n}^{2}$ on a set of prior mass at least $e^{- C_{3} n d_{n}^{2}}$ . Logarithmic integrands vanish where $p_{0} = 0$ .

Facts & Assumptions

Given: AC and all hypotheses of the Statement.

[F1]

Gaussian convolution of this truth has KL and log-square error at most $(4 / 9) σ^{4} J$ . Gaussian convolution has second-order KL and log-square error

[F2]

A compactly supported mixing law in an interval of length at most a fixed multiple of $τ$ has a discrete normal-mixture approximation at scale $τ$ with $O (\log (e / η))$ atoms, $L^{1}$ error at most $η$ and supremum error at most $η / τ$ . Compactly supported Gaussian mixtures admit logarithmic-size finite approximations

[F3]

For a finite partition with interval-cell base masses in $[c η^{2}, 1]$ and $N η \leq 1 / 4$ , a DP finite vector has probability at least $C \exp {- c^{'} N \log (e / η)}$ of having total target-weight discrepancy plus residual mass at most $3 η$ ; the constants are uniform when the residual-cell base mass ranges in a fixed compact subset of $(0, m]$ . A Dirichlet process assigns exponential mass to a finite-partition simplex neighborhood

[F4]

The mixture density is jointly measurable, positive and finite. The Dirichlet normal-mixture density is jointly measurable and normalized

[F5]

The Gaussian kernel has formula $ϕ_{σ} (x) = (σ \sqrt{2 π})^{- 1} e^{- x^{2} / (2 σ^{2})}$ . A positive-variance normal law has its location-scale density

[F6]

The natural logarithm is differentiable on $(0, \infty)$ with derivative $1 / x$ . The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

Proof

1.1F2givenchooseconstruct

Choose $0 < s_{-} < s_{+} < \infty$ with $g : = G ([s_{-}, s_{+}]) > 0$ , possible because these compact intervals exhaust $(0, \infty)$ . Set $b_{n} = σ_{n}$ , $τ = s_{-} b_{n}$ , $B = a + 6$ and $η = b_{n}^{B}$ . For all large $n$ , partition each nondegenerate component of $K$ into equal consecutive Borel intervals of lengths in $[τ / 8, τ / 4]$ : use $⌈ 4 ℓ / τ ⌉$ pieces for a component of length $ℓ$ . Endpoints can be assigned arbitrarily because $P_{0}$ has a density. The support of a continuous density has no isolated points, so there are no isolated support components to treat. There are $O (b_{n}^{- 1})$ coarse cells $I_{k}$ , each of mass at least $c^{'} b_{n}^{a}$ . Translate the conditional probability $P_{0} (\cdot ∣ I_{k})$ to an interval centered at zero, apply [F2] at scale $τ$ , and translate back. Multiplying by $P_{0} (I_{k})$ and summing gives a finite probability $F_{n}$ with at most $C b_{n}^{- 1} \log (e / η)$ atoms in the closures of these cells, each carrying its original coarse-cell label, and $∥ p_{F_{n}, τ} - p_{P_{0}, τ} ∥_{\infty} \leq η / τ$ . The total weight of atoms with label $k$ is exactly $P_{0} (I_{k})$ .

2.1givenstep 1.1constructalgebra

Put $δ = η τ$ . Move each atom to a nearest point of the global lattice of mesh $δ$ and merge atoms at the same location. Preserve its fractional label weights when merging. The resulting distinct locations $z_{j}$ are separated by at least $δ$ , with weights $w_{j}$ summing to one, and still number $N \leq C b_{n}^{- 1} \log (e / η)$ . Let $U_{j} = [z_{j} - δ / 4, z_{j} + δ / 4)$ and $U_{0} = R ∖ ⋃_{j = 1}^{N} U_{j}$ . These are disjoint partition cells. The base density is bounded above and below on a fixed compact neighborhood of $K$ , so $c_{0} η τ \leq α (U_{j}) \leq C_{0} η τ \leq 1$ eventually. Since $B > 1$ , $η τ \geq c_{1} η^{2}$ and $N η \to 0$ . Also $\sum_{j} α (U_{j}) \leq C N δ \to 0$ , so, writing $m = α (R) > 0$ , we have $α (U_{0}) \in [m / 2, m]$ eventually.

3.1F3step 2.1algebra

Apply [F3] with tolerance $η$ to get an event $E_{n}$ on which $\sum_{j} ∣ F (U_{j}) - w_{j} ∣ + F (U_{0}) \leq 3 η$ and $Π_{D P} (E_{n}) \geq \exp {- C b_{n}^{- 1} (\log n)^{2}}$ . Its constants can be taken uniform in $n$ : in the proof of [F3] the only residual-parameter factors are $Γ (m) / Γ (α (U_{0}))$ and a lower bound $(η^{2} / 2)^{m}$ for the residual power. The gamma ratio is bounded below on the fixed compact interval $[m / 2, m]$ , and the latter power already has fixed exponent $m$ . The remaining constants use only the fixed bound $c_{1} η^{2} \leq α (U_{j}) \leq 1$ . Finally $\log (e / η) \leq C \log n$ by the polynomial lower bandwidth bound.

4.1F4F5step 1.1step 2.1step 3.1algebra

For $τ \leq σ \leq s_{+} b_{n}$ , the original discrete approximation still satisfies $∥ p_{F_{n}, σ} - p_{P_{0}, σ} ∥_{\infty} \leq C η / b_{n}$ . Indeed the Gaussian convolution identity $ϕ_{u} * ϕ_{v} = ϕ_{\sqrt{u^{2} + v^{2}}}$ follows by completing the square and integrating; convolution with a probability kernel contracts the supremum norm by the triangle inequality. Use $u = τ$ , $v = \sqrt{σ^{2} - τ^{2}}$ , treating equality by the original approximation. Differentiating [F5] gives $∥ ϕ_{σ}^{'} ∥_{\infty} \leq C σ^{- 2}$ . Atom relocation by at most $δ / 2$ therefore changes the mixture by at most $C δ / σ^{2} \leq C η / b_{n}$ . For $F \in E_{n}$ , replacing each $F ∣_{U_{j}}$ by its mass at $z_{j}$ costs at most the same derivative bound times $δ / 4$ ; the weight discrepancies and residual mass cost at most $3 η ∥ ϕ_{σ} ∥_{\infty}$ . Consequently, uniformly over $F \in E_{n}$ and $τ \leq σ \leq s_{+} b_{n}$ , one has $∥ p_{F, σ} - p_{P_{0}, σ} ∥_{\infty} \leq C η / b_{n}$ .

4.2F4F5step 1.1step 2.1step 3.1algebra

For any $x \in K$ choose a coarse cell whose closure contains $x$ . Its length is at most $τ / 4$ , so every original atom with that label, and every associated relocated atom-cell, lies in $[x - τ, x + τ]$ eventually. Let $J_{x}$ be the set of relocated atom indices carrying a positive fraction of this label's weight. Merging preserves the full label weight, so $\sum_{j \in J_{x}} w_{j} \geq P_{0} (I_{k}) \geq c^{'} b_{n}^{a}$ . The vector discrepancy on $E_{n}$ implies $F ([x - τ, x + τ]) \geq \sum_{j \in J_{x}} F (U_{j}) \geq c^{'} b_{n}^{a} - 3 η \geq c^{'} b_{n}^{a} / 2$ . On this neighborhood, [F5] is at least $c^{''} / b_{n}$ for every $τ \leq σ \leq s_{+} b_{n}$ . Thus $p_{F, σ} (x) \geq c^{'''} b_{n}^{a - 1}$ . The same argument directly with the coarse-cell mass under $P_{0}$ gives $p_{P_{0}, σ} (x) \geq c^{'''} b_{n}^{a - 1}$ , after decreasing the constant.

5.1F1F6step 4.1step 4.2algebra

Put $q = p_{P_{0}, σ}$ and $r = p_{F, σ}$ . The mean-value theorem for the logarithm, whose derivative is given by [F6], and steps 4.1, 4.2 give $\sup_{x \in K} ∣ \log (q (x) / r (x)) ∣ \leq C η b_{n}^{- a} = C b_{n}^{6}$ . By [F1], $K L (P_{0} ∥ Q) \leq C b_{n}^{4}$ and $P_{0} [\log (p_{0} / q)]^{2} \leq C b_{n}^{4}$ uniformly in these scales. Adding the bounded log perturbation proves $K L (P_{0} ∥ P_{F, σ}) \leq C b_{n}^{4} + C b_{n}^{6} \leq C^{'} b_{n}^{4}$ . The inequality $(u + v)^{2} \leq 2 u^{2} + 2 v^{2}$ similarly gives $P_{0} [\log (p_{0} / r)]^{2} \leq C^{'} b_{n}^{4}$ . All integrals exist by the displayed bounds, and integrands vanish off ${p_{0} > 0}$ .

6.1givenstep 3.1step 5.1algebra∎

The scale event $S \in [s_{-}, s_{+}]$ has fixed probability $g > 0$ and is independent of $F$ . Therefore $E_{n}$ times this event is contained in the displayed KL/log-square set and has probability at least $g \exp {- C b_{n}^{- 1} (\log n)^{2}}$ . Increase $C_{2}$ to absorb $g$ . Since $n d_{n}^{2} \geq b_{n}^{- 1} (\log n)^{2}$ and $d_{n}^{2} \geq b_{n}^{4}$ , the consequent local-radius form follows as well. This proves all assertions without a unimodality hypothesis.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The posterior excludes large scales and mixing mass far from compact data

Statement

Assume AC. Let $p_{0}$ be supported on a compact interval $I = [L, U]$ with $L < U$ , let $F \sim D P (α)$ for a finite nonzero base measure whose density is bounded below by a positive constant on a neighborhood of $I$ , and let $S \sim G$ be independent with $σ = σ_{n} S$ . Put $R = (U - L) / 2$ and, for $d > 0$ , $I_{d} = [L - d, U + d]$ . Observe IID data from $p_{0}$ under the location-mixture density $p_{F, σ}$ . Let $ϵ_{n}, δ_{n} > 0$ with $n ϵ_{n}^{2} \to \infty$ and $n δ_{n}^{2} \to \infty$ . Suppose the local KL and log-square prior-mass condition gives a denominator event $P_{0}^{n} {D_{n} \geq e^{- C_{0} n δ_{n}^{2}}} ⟶ 1,$ where $D_{n} = \int \prod_{i} p_{F, σ} (X_{i}) / p_{0} (X_{i}) Π_{n} (d F, d σ)$ . Choose $s_{n} \to 0$ and put $q_{n} = 2 Φ (- R / s_{n})$ , where $Φ$ is the standard normal distribution function. If $n q_{n} - (C_{0} + 2) n δ_{n}^{2} \to \infty, n ϵ_{n}^{- 2} s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})} \to 0,$ then the posterior complement of $S_{n} : = {(F, σ) : σ \leq s_{n}, F (I_{d}^{c}) \leq ϵ_{n}^{2}}$ converges to zero in $P_{0}^{n}$ -probability.

Facts & Assumptions

Given: Full AC; compactly supported truth; base density bounded below by a positive constant on a neighborhood of $I$ ; the DP normal-mixture model and independent scale prior; the denominator event; and the two displayed asymptotic conditions.

[A1]

Full AC supplies the DP construction and the directional KL convention. The Axiom of Choice Existence of a Dirichlet process by normalized gamma Poisson random measure Kullback–Leibler divergence

[F1]

Dirichlet-process posterior update: conditionally on latent IID locations $θ_{1 : n} ∣ F \sim F^{\otimes n}$ , the posterior law of $F$ is $D P (α + \sum_{i} δ_{θ_{i}})$ .

[F2]

Dirichlet-process Pólya-urn predictive law: conditionally on all latent locations except $θ_{i}$ , the prior predictive law of $θ_{i}$ is $(α + \sum_{k \neq i} δ_{θ_{k}}) / (α (R) + n - 1)$ , by exchangeability.

[F3]

Dirichlet-process posterior update and Dirichlet-process Pólya-urn predictive law: under the posterior $D P (α + \sum_{i} δ_{θ_{i}})$ , the next latent location has predictive probability $(α (A) + \sum_{i} 1_{A} (θ_{i})) / (α (R) + n)$ ; because its conditional law given $F$ is $F$ , this is the posterior mean of $F (A)$ .

[F4]

KL prior mass gives a posterior denominator lower bound in probability: the local KL/log-square prior-mass condition gives the displayed lower bound for $D_{n}$ with probability tending to one.

[F5]

A positive-variance normal law has its location-scale density: the latent observation likelihood is $ϕ_{σ} (x - θ) = (σ \sqrt{2 π})^{- 1} e^{- (x - θ)^{2} / (2 σ^{2})}$ .

[F6]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative likelihood numerators may be integrated in either order.

[F7]

Markov's inequality for random variables: for a nonnegative random variable $Y$ and $a > 0$ , $P (Y \geq a) \leq E [Y] / a$ .

[F8]

Dirichlet process and Existence of a Dirichlet process by normalized gamma Poisson random measure: $D P (α)$ is an evaluation-measurable random probability measure with the stated Dirichlet finite-partition laws.

[F9]

The integral function of a continuous integrand is differentiable with derivative equal to the integrand. The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive

Proof

Given: The hypotheses in the Statement.

1.1A1F5F9givenconstruct

The latent representation is $θ_{i} ∣ F \overset{i i d}{\sim} F$ and $X_{i} ∣ θ_{i}, σ \sim N (θ_{i}, σ^{2})$ independently. Use the test $φ_{n} = 1 {some X_{i} \notin I}$ . Its type-I error is zero. Write $Φ (t) = \int_{- \infty}^{t} ϕ (z) d z$ ; [F9] gives $Φ^{'} (t) = ϕ (t)$ . The normal formula [F5] is even and satisfies $ϕ^{'} (t) = - t ϕ (t)$ , so it decreases on $[0, \infty)$ . For fixed $θ$ and $σ$ , set $g (θ) = P (θ + σ Z \in I) = Φ ((U - θ) / σ) - Φ ((L - θ) / σ)$ . Then $g^{'} (θ) = σ^{- 1} [ϕ ((L - θ) / σ) - ϕ ((U - θ) / σ)]$ . For $θ \leq (L + U) / 2$ , $∣ L - θ ∣ \leq ∣ U - θ ∣$ , so $g^{'} (θ) \geq 0$ ; for $θ \geq (L + U) / 2$ the inequality reverses and $g^{'} (θ) \leq 0$ . Thus $g$ is maximized at the midpoint, where the component leakage is $2 Φ (- R / σ)$ . Every component with $σ > s_{n}$ therefore puts probability at least $q_{n} = 2 Φ (- R / s_{n})$ outside $I$ , and $P_{F, σ}^{n} (φ_{n} = 0) \leq (1 - q_{n})^{n} \leq e^{- n q_{n}}$ .

1.2F1F3F7F8givenalgebra

By [F8] the latent DP model is a measurable prior law. Introduce the latent locations $θ_{1 : n}$ . Given these locations, $X_{1 : n}$ and $σ$ , the observation likelihood is independent of $F$ , so [F1] gives the conditional law $D P (α + \sum_{i} δ_{θ_{i}})$ . Given this updated $F$ , the next latent location has conditional law $F$ ; applying [F3] to that predictive draw gives $E [F (A) ∣ θ_{1 : n}, X_{1 : n}, σ] = \frac{α (A) + \sum_{i} 1_{A} (θ_{i})}{α (R) + n}$ for every Borel set $A$ . With $A = I_{d}^{c}$ , conditional Markov inequality yields $Π_{n} (F (A) > ϵ_{n}^{2}, σ \leq s_{n} ∣ X_{1 : n}) \leq \frac{α (A) + \sum_{i} P (θ_{i} \in A, σ \leq s_{n} ∣ X_{1 : n})}{ϵ_{n}^{2} (α (R) + n)} .$

1.3A1F2F5givenalgebra

Fix an index $i$ and condition on $θ_{- i}, X_{1 : n}, σ$ . The latent vector is exchangeable because its conditional law given $F$ is IID. Reordering $θ_{i}$ last and applying [F2] shows its prior predictive law given $θ_{- i}$ is $H_{i} = (α + \sum_{k \neq i} δ_{θ_{k}}) / (α (R) + n - 1)$ . Bayes' formula multiplies this measure by the likelihood $ϕ_{σ} (X_{i} - t)$ . Almost surely $X_{i} \in I$ . On $A = I_{d}^{c}$ , $∣ X_{i} - t ∣ \geq d$ , so the likelihood numerator is at most $C σ^{- 1} e^{- d^{2} / (2 σ^{2})}$ . For $σ \leq s_{n}$ , the denominator is at least the base-measure contribution $\frac{1}{α (R) + n - 1} \int_{X_{i} - σ}^{X_{i} + σ} ϕ_{σ} (X_{i} - t) α (d t) \geq \frac{c}{α (R) + n - 1},$ because the base density is bounded below near $I$ and $[X_{i} - σ, X_{i} + σ]$ lies there for all sufficiently large $n$ . Consequently $P (θ_{i} \in A ∣ θ_{- i}, X_{1 : n}, σ) \leq C n σ^{- 1} e^{- d^{2} / (2 σ^{2})} \leq C n s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})},$ where $s^{- 1} e^{- d^{2} / (2 s^{2})}$ is increasing on $0 < s < d$ .

2.1F4F6F7step 1.1algebra

Let $N_{n}^{H}$ be the likelihood-ratio numerator integrated over ${σ > s_{n}}$ . By change of density and [F6], $E_{0} [(1 - φ_{n}) N_{n}^{H}] \leq e^{- n q_{n}}$ . On the denominator event, [F7] therefore gives, for every $ζ > 0$ , $P_{0}^{n} {Π_{n} (σ > s_{n} ∣ X_{1 : n}) > ζ} \leq o (1) + ζ^{- 1} e^{(C_{0} + 2) n δ_{n}^{2} - n q_{n}} ⟶ 0.$

3.1F4F7step 2.1step 1.2step 1.3givenalgebra∎

Substitute the step 1.3 bound into step 1.2 and use $α (A) \leq α (R)$ . The resulting upper bound is $\frac{α (R)}{ϵ_{n}^{2} (α (R) + n)} + C n ϵ_{n}^{- 2} s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})} + Π_{n} (σ > s_{n} ∣ X_{1 : n}),$ which tends to zero by $n ϵ_{n}^{2} \to \infty$ , the second premise, and step 2.1. Thus the posterior probability of $F (I_{d}^{c}) > ϵ_{n}^{2}$ tends to zero. Combining with step 2.1 proves the sieve-complement claim.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Dirichlet normal-mixture posterior contraction

Statement

Assume AC. Let $p_{0}$ be a finite-valued probability density on $R$ with compact support. Assume $p_{0} \in C^{2} (R)$ and, with the integrands set to zero on ${p_{0} = 0}$ , $\int_{{p_{0} > 0}} [{(\frac{p_{0}^{''}}{p_{0}})}^{2} + {(\frac{p_{0}^{'}}{p_{0}})}^{4}] p_{0} d x < \infty,$ and assume there is $C < \infty$ such that $h (p_{0}, p_{0} * ϕ_{σ}) \leq C σ^{2}$ for all sufficiently small $σ > 0$ . Here $h$ is the root-L2 Hellinger distance of Root-L2 Hellinger distance between density laws, and $ϕ_{σ}$ is the centered normal density with standard deviation $σ$ . Let $F \sim D P (α)$ , where $α$ is a finite nonzero measure with a continuous, strictly positive Lebesgue density on an open interval containing $supp (p_{0})$ , and independently let $S \sim G$ on $(0, \infty)$ . For $σ = σ_{n} S$ , define $p_{F, σ} (x) = \int ϕ_{σ} (x - z) F (d z),$ and observe $X_{1}, \dots, X_{n}$ IID from $p_{0}$ . Suppose $n^{- a_{1}} ≲ σ_{n} ≲ n^{- a_{2}}, 0 < a_{2} < a_{1} < 1, a_{2} \geq (4 + γ)^{- 1},$ and, for some $β > 0$ and $γ > 1$ , $G ((0, s]) ≲ e^{- β s^{- γ}} (s ↓ 0), G ((s, \infty)) ≲ e^{- β s^{γ}} (s \to \infty) .$ Here $u_{n} ≲ v_{n}$ means $u_{n} \leq C v_{n}$ eventually for a fixed constant $C$ . For every fixed $η > γ / [2 (γ - 1)]$ , put $ϵ_{n} = \max {(n σ_{n})^{- 1 / 2} \log n, n^{- 1 / 2} σ_{n}^{- η}, σ_{n}^{2} \log n} .$ Then for every sufficiently large fixed $M$ , $Π_{n} ⁣ (h (p_{F, σ}, p_{0}) > M ϵ_{n} ∣ X_{1 : n}) ⟶ 0$ in $P_{0}^{n}$ -probability. If $G$ is compactly supported in $(0, \infty)$ , the middle term $n^{- 1 / 2} σ_{n}^{- η}$ may be omitted. The conclusion is only in probability; it supplies no almost-sure contraction or generic minimax-optimality assertion.

Facts & Assumptions

Given: Full AC; the compactly supported smooth truth and displayed smoothness/convolution conditions; the finite DP base measure with its density condition; independent scale law; polynomial bandwidth and scale-tail hypotheses; and an IID sample from $p_{0}$ .

[A1]

Full AC supplies the normalized-gamma DP construction and the directional KL convention. The Axiom of Choice Existence of a Dirichlet process by normalized gamma Poisson random measure Kullback–Leibler divergence

[F1]

Dirichlet process and Existence of a Dirichlet process by normalized gamma Poisson random measure: $D P (α)$ is a measurable random probability measure with the stated Dirichlet finite-partition laws.

[F2]

The Dirichlet normal-mixture density is jointly measurable and normalized: $(F, σ, x) \mapsto p_{F, σ} (x)$ is jointly measurable, positive, finite, and integrates to one.

[F3]

The mixture prior places local KL and log-square mass near a compact smooth truth: for $d_{n} = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2} \log n}$ , the local KL and log-square neighborhood has mass at least $e^{- C n d_{n}^{2}}$ , with both quantities at most $C d_{n}^{2}$ .

[F4]

KL prior mass gives a posterior denominator lower bound in probability: local KL/log-square mass at least $e^{- n {\overset{ˉ}{d}}_{n}^{2}}$ gives, with probability at least $1 - (n {\overset{ˉ}{d}}_{n}^{2})^{- 1}$ , a likelihood denominator at least $e^{- 3 n {\overset{ˉ}{d}}_{n}^{2}}$ .

[F5]

The posterior excludes large scales and mixing mass far from compact data: given that denominator bound, for any compact interval $I = [L, U]$ containing the truth, put $R = (U - L) / 2$ and $I_{d} = [L - d, U + d]$ . If $s_{n} \to 0$ , $n q_{n} - 5 n {\overset{ˉ}{d}}_{n}^{2} \to \infty$ for $q_{n} = 2 Φ (- R / s_{n})$ , where $Φ$ is the standard normal distribution function, and $n ξ_{n}^{- 2} s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})} \to 0$ , then the posterior complement of ${F (I_{d}^{c}) \leq ξ_{n}^{2}, σ \leq s_{n}}$ vanishes, provided $n ξ_{n}^{2} \to \infty$ and the DP base density is bounded below near $I$ .

[F6]

The dyadic scale bands satisfy the weighted entropy condition: for a fixed $A \geq 1$ and dyadic scale bands whose sieve satisfies $F ([- A, A]^{c}) \leq \min {r_{n}, 1 / 4}^{2} / 8$ , the weighted entropy condition holds when $r_{n}$ dominates the displayed first and middle rate terms; for compact scale support it holds with only the first term.

[F7]

Countably partitioned sieves yield posterior contraction by weighted entropy: local KL/log-square mass, a sieve with vanishing posterior complement, and the weighted countable-band entropy condition imply contraction in the same Hellinger metric at every sufficiently large fixed multiple of the target radius.

[F8]

Dominated statistical experiments and Likelihood densities: the sampling experiment is dominated by Lebesgue measure, and the supplied jointly measurable mixture densities define its likelihoods.

[F9]

Root-L2 Hellinger distance between density laws: $h$ in the conclusion is the root-L2 Hellinger distance between square-root densities.

Proof

Given: The hypotheses in the Statement.

1.1A1F3F4givenalgebra

Put $d_{n} = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2} \log n}$ . By [F3], choose $C_{1} \geq 1$ so the prior mass is at least $e^{- C_{1} n d_{n}^{2}}$ on a set where both local quantities are at most $C_{1} d_{n}^{2}$ . Set ${\overset{ˉ}{d}}_{n} = \sqrt{C_{1}} d_{n}$ . Then the mass is at least $e^{- n {\overset{ˉ}{d}}_{n}^{2}}$ and both quantities are at most ${\overset{ˉ}{d}}_{n}^{2}$ . By [F4], except on an event of probability at most $(n {\overset{ˉ}{d}}_{n}^{2})^{- 1}$ , the likelihood denominator is at least $e^{- 3 n {\overset{ˉ}{d}}_{n}^{2}}$ . Also $n {\overset{ˉ}{d}}_{n}^{2} \geq C_{1} σ_{n}^{- 1} (\log n)^{2} \to \infty$ .

1.2F5givenconstructalgebra

Choose a compact interval $I = [L, U]$ containing $supp (p_{0})$ in its interior and contained in the open interval where the density of $α$ is continuous and strictly positive. Such an interval exists because the support is compact; its base density is bounded below on $I$ . Put $R = (U - L) / 2 > 0$ and $κ = \max {a_{1}, 1 - 4 a_{2}, 0} < 1$ . The polynomial bandwidth bounds give $n {\overset{ˉ}{d}}_{n}^{2} \leq C n^{κ} (\log n)^{2}$ . Choose $s_{n} = c_{0} (\log n)^{- 1 / 2}$ with $c_{0}$ large enough that for some $θ < 1 - κ$ , $q_{n} : = 2 Φ (- R / s_{n}) \geq n^{- θ}$ eventually: integrating the standard normal density on $[R / s_{n}, R / s_{n} + 1]$ gives the claimed lower bound once $R^{2} / (2 c_{0}^{2}) < θ$ . Hence $n q_{n} - 5 n {\overset{ˉ}{d}}_{n}^{2} \to \infty$ . Set $r_{n} = M_{0} ϵ_{n}$ for a fixed sufficiently large $M_{0}$ , and $ξ_{n} = \min {r_{n}, 1 / 4} / \sqrt{8}$ . We have $n ξ_{n}^{2} \to \infty$ and $ξ_{n}^{- 2} \leq C n$ : if $r_{n} < 1 / 4$ , use $ϵ_{n} \geq n^{- 1 / 2} \log n$ for all large $n$ , and otherwise $ξ_{n}$ is fixed. Choose $d > 0$ with $d^{2} / (2 c_{0}^{2}) > 4$ and $A \geq 1$ large enough that $I_{d} = [L - d, U + d] \subseteq [- A, A]$ . Then $n ξ_{n}^{- 2} s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})} \to 0$ . Applying [F5] gives posterior concentration on ${F (I_{d}^{c}) \leq ξ_{n}^{2}, σ \leq s_{n}}$ . Since $[- A, A]^{c} \subseteq I_{d}^{c}$ , the larger sieve $S_{n} = {F ([- A, A]^{c}) \leq ξ_{n}^{2}, σ \leq s_{n}}$ also has $Π_{n} (S_{n}^{c} ∣ X_{1 : n}) \to 0$ .

1.3F2F6givenalgebra

Partition $S_{n}$ into dyadic scale bands $S_{n, j} = S_{n} \cap {2^{j} σ_{n} \leq σ < 2^{j + 1} σ_{n}}$ for $2^{j} σ_{n} \leq s_{n}$ . Their prior masses are bounded by $G ([2^{j}, 2^{j + 1}))$ . Since $ξ_{n}^{2} = \min {r_{n}, 1 / 4}^{2} / 8$ , [F6] applies at radius $r_{n}$ and gives $e^{- n r_{n}^{2}} \sum_{j} \sqrt{N (r_{n}, {p_{F, σ} : (F, σ) \in S_{n, j}}, h) Π_{n} (S_{n, j})} ⟶ 0.$ For compactly supported $G$ , only finitely many bands have positive prior mass and the first rate term suffices.

2.1F5F6givenstep 1.1step 1.2algebra

If $G$ is compactly supported, put $ϵ_{n}^{c} = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2} \log n}$ , $r_{n}^{c} = M_{0} ϵ_{n}^{c}$ and $ξ_{n}^{c} = \min {r_{n}^{c}, 1 / 4} / \sqrt{8}$ . Since $σ_{n} \to 0$ , $n (ϵ_{n}^{c})^{2} \geq σ_{n}^{- 1} (\log n)^{2} \to \infty$ and $(ϵ_{n}^{c})^{- 2} \leq C n$ . Thus $n (ξ_{n}^{c})^{2} \to \infty$ and $(ξ_{n}^{c})^{- 2} \leq C n$ . Also ${\overset{ˉ}{d}}_{n} \leq \sqrt{C_{1}} ϵ_{n}^{c}$ . Retain $I, s_{n}, d, A$ from step 1.2: its denominator, leakage bound, and exponential factor do not depend on the target radius, so the hypotheses of [F5] hold with $ϵ_{n}^{c}$ and $ξ_{n}^{c}$ in place of $ϵ_{n}$ and $ξ_{n}$ . It yields posterior sieve complement convergence for $S_{n}^{c} = {F ([- A, A]^{c}) \leq (ξ_{n}^{c})^{2}, σ \leq s_{n}}$ . The compact branch of [F6] supplies the weighted entropy condition at radius $r_{n}^{c}$ using only the first rate term.

3.1A1F1F2F3F6F7F8F9step 1.1step 1.2step 1.3step 2.1algebra∎

The prior is the measurable DP law from [F1], and [F2], [F8] give its jointly measurable density likelihoods. We have ${\overset{ˉ}{d}}_{n} \leq \sqrt{C_{1}} ϵ_{n}$ , so increasing $M_{0}$ if necessary ensures ${\overset{ˉ}{d}}_{n} \leq r_{n}$ ; also $n {\overset{ˉ}{d}}_{n}^{2}, n r_{n}^{2} \to \infty$ . Thus [F3] supplies the local prior-mass condition for [F7]. For general $G$ , steps 1.2 and 1.3 provide a sieve with vanishing posterior complement and its weighted entropy condition. If $G$ is compactly supported, step 2.1 instead provides these two conditions at radius $r_{n}^{c}$ , and its bound ${\overset{ˉ}{d}}_{n} \leq \sqrt{C_{1}} ϵ_{n}^{c}$ allows $M_{0}$ to be chosen so ${\overset{ˉ}{d}}_{n} \leq r_{n}^{c}$ . Apply [F7] with local radius ${\overset{ˉ}{d}}_{n}$ and target radius $r_{n}$ in the general case, or $r_{n}^{c}$ in the compact case. It follows that for every sufficiently large fixed $M_{1}$ , the posterior outside $M_{1} M_{0} ϵ_{n}$ (respectively $M_{1} M_{0} ϵ_{n}^{c}$ ) tends to zero in $P_{0}^{n}$ -probability. Since $M_{1} M_{0}$ can be any sufficiently large fixed constant, this proves the stated rates. The measurability and dominated-likelihood requirements are supplied by [F1], [F2], and [F8], and [F9] fixes the conclusion's metric convention.

CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

An interval-mass condition removes the Gaussian-bias logarithm

Statement

Under the hypotheses of Dirichlet normal-mixture posterior contraction, assume additionally that $supp (p_{0})$ is a finite union of intervals and there are $a, c > 0$ such that $P_{0} (I) \geq c λ (I)^{a}$ for every interval $I$ contained in the support. Then the source's sharpened Hellinger contraction rate is $\max {(n σ_{n})^{- 1 / 2} \log n, n^{- 1 / 2} σ_{n}^{- η}, σ_{n}^{2}}, η > \frac{γ}{2 (γ - 1)} .$ If $G$ is compactly supported in $(0, \infty)$ , the middle term is omitted.

Facts & Assumptions

Given: AC and the exact hypotheses of Dirichlet normal-mixture posterior contraction, together with the finite-union interval-mass condition in the Statement.

[F1]

The main theorem fixes the smooth compact truth, the finite DP base measure, the independent scale law, the polynomial bandwidth bounds and both scale-tail assumptions. Dirichlet normal-mixture posterior contraction

[F2]

With $d_{n} = \max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2}}$ , the local KL and log-square set at radius a constant multiple of $d_{n}$ has prior mass at least $e^{- C n d_{n}^{2}}$ . Interval masses give a log-free local KL neighborhood for the mixture prior

[F3]

A local set with both quantities at most ${\overset{ˉ}{d}}_{n}^{2}$ and mass at least $e^{- n {\overset{ˉ}{d}}_{n}^{2}}$ gives a denominator at least $e^{- 3 n {\overset{ˉ}{d}}_{n}^{2}}$ except on an event of probability at most $(n {\overset{ˉ}{d}}_{n}^{2})^{- 1}$ . KL prior mass gives a posterior denominator lower bound in probability

[F4]

The posterior sieve-complement bound applies when its denominator, Gaussian leakage and distant-location conditions hold. The posterior excludes large scales and mixing mass far from compact data

[F5]

For a fixed $A \geq 1$ , the dyadic scale-band weighted entropy condition holds at a sufficiently large multiple of the first and middle terms of the stated rate, or of the first term alone for compactly supported $G$ . The dyadic scale bands satisfy the weighted entropy condition

[F6]

Local KL/log-square mass, a vanishing posterior sieve complement and the weighted countable-band entropy condition imply posterior contraction at every sufficiently large fixed multiple of the target radius. Countably partitioned sieves yield posterior contraction by weighted entropy

[F7]

The mixture densities are jointly measurable, positive, finite and normalized. The Dirichlet normal-mixture density is jointly measurable and normalized

Proof

1.1F1F2F3givenchoosealgebra

The $C^{2}$ compact truth in [F1] satisfies the $J$ hypothesis of [F2] by the stated derivative integrability condition. All other hypotheses of [F2] are either in [F1] or the interval-mass premise. Choose a fixed $C \geq 1$ large enough that, with ${\overset{ˉ}{d}}_{n} = \sqrt{C} d_{n}$ , the local set has both quantities at most ${\overset{ˉ}{d}}_{n}^{2}$ and mass at least $e^{- n {\overset{ˉ}{d}}_{n}^{2}}$ . By [F3], its denominator is at least $e^{- 3 n {\overset{ˉ}{d}}_{n}^{2}}$ with probability tending to one, since $n {\overset{ˉ}{d}}_{n}^{2} \geq C σ_{n}^{- 1} (\log n)^{2} \to \infty$ .

2.1F1step 1.1choosealgebra

Let $ϵ_{n}$ denote the displayed improved rate, omitting its middle term if $G$ is compactly supported. Put $r_{n} = M_{0} ϵ_{n}$ for a sufficiently large fixed $M_{0}$ and $ξ_{n} = \min {r_{n}, 1 / 4} / \sqrt{8}$ . Then ${\overset{ˉ}{d}}_{n} \leq r_{n}$ after increasing $M_{0}$ , $n ξ_{n}^{2} \to \infty$ and $ξ_{n}^{- 2} \leq C^{'} n$ : use $ϵ_{n} \geq (n σ_{n})^{- 1 / 2} \log n \geq n^{- 1 / 2} \log n$ eventually when $r_{n} < 1 / 4$ , while otherwise $ξ_{n}$ is fixed. Choose a compact interval $I = [L, U]$ containing the truth support in its interior, within the interval on which the base density is positive. Put $R = (U - L) / 2 > 0$ and $κ = \max {a_{1}, 1 - 4 a_{2}, 0} < 1$ . The bandwidth bounds give $n {\overset{ˉ}{d}}_{n}^{2} \leq C^{'} n^{κ} (\log n)^{2}$ . Choose $s_{n} = c_{0} (\log n)^{- 1 / 2}$ with $c_{0}$ large enough that $q_{n} = 2 Φ (- R / s_{n}) \geq n^{- θ}$ eventually for some $0 < θ < 1 - κ$ . This follows by integrating the standard normal density over $[R / s_{n}, R / s_{n} + 1]$ once $R^{2} / (2 c_{0}^{2}) < θ$ . Thus $n q_{n} - 5 n {\overset{ˉ}{d}}_{n}^{2} \to \infty$ .

3.1F4step 1.1step 2.1choosealgebra

Choose $d > 0$ with $d^{2} / (2 c_{0}^{2}) > 4$ and a fixed $A \geq 1$ large enough that $I_{d} = [L - d, U + d] \subseteq [- A, A]$ . Then $n ξ_{n}^{- 2} s_{n}^{- 1} e^{- d^{2} / (2 s_{n}^{2})} \to 0$ , by the inverse-radius bound in step 2.1. With denominator coefficient $3$ , all hypotheses of [F4] hold at local radius ${\overset{ˉ}{d}}_{n}$ and location-tail tolerance $ξ_{n}$ . It follows that the posterior complement of $S_{n} = {σ \leq s_{n}, F ([- A, A]^{c}) \leq ξ_{n}^{2}}$ vanishes in sampling probability: [F4] first gives the smaller location sieve using $I_{d}$ , and $[- A, A]^{c} \subseteq I_{d}^{c}$ .

4.1F1F5step 2.1step 3.1algebra

Partition $S_{n}$ into the measurable dyadic scale bands $S_{n, j} = S_{n} \cap {2^{j} σ_{n} \leq σ < 2^{j + 1} σ_{n}}$ . Their prior masses are bounded by $G ([2^{j}, 2^{j + 1}))$ . We have $s_{n} / σ_{n} \to \infty$ , and $ξ_{n}^{2} = \min {r_{n}, 1 / 4}^{2} / 8$ matches the tail tolerance in [F5]. Thus [F5] supplies $e^{- n r_{n}^{2}} \sum_{j} \sqrt{N (r_{n}, {p_{F, σ} : (F, σ) \in S_{n, j}}, h) Π_{n} (S_{n, j})} \to 0$ . For compactly supported $G$ , its compact branch applies with the first term alone. The preceding steps used only $d_{n}$ and the retained first term, so they also apply unchanged to this improved compact-scale rate.

5.1F2F6F7step 1.1step 2.1step 3.1step 4.1algebra∎

By [F7] the model has the jointly measurable density likelihood required in [F6]. Step 1.1 gives the local mass condition, step 3.1 gives the sieve complement, and step 4.1 gives the weighted entropy condition at target radius $r_{n}$ . Both $n {\overset{ˉ}{d}}_{n}^{2}$ and $n r_{n}^{2}$ diverge, and ${\overset{ˉ}{d}}_{n} \leq r_{n}$ . Apply [F6] to obtain posterior contraction at every sufficiently large fixed multiple of $r_{n} = M_{0} ϵ_{n}$ , in $P_{0}^{n}$ -probability. Absorbing $M_{0}$ into the final constant proves the Statement, including its compact-scale branch.

Remarks

The source proof invokes a uniform convolution likelihood-ratio bound through its unimodality lemma. Here the local Gaussian semigroup argument bounds KL and log-square bias directly from the original second-derivative integrability hypothesis. The interval-mass condition then controls only the discretization perturbation. This proves the stated refinement without adding a unimodality assumption.

5 · Examples, counterexamples and false statements

None yet.

Dirichlet normal-mixture posterior contraction

1 · Prerequisites

2 · Summary

Objective

3 · Logical flowchart

4 · Definitions, theorems and proofs

The Dirichlet normal-mixture density is jointly measurable and normalized

Statement

Facts & Assumptions

Proof

A bounded likelihood ratio controls KL and log-square moments by Hellinger distance

Statement

Facts & Assumptions

Proof

Gaussian convolution has second-order KL and log-square error

Statement

Facts & Assumptions

Proof

Compactly supported Gaussian mixtures admit logarithmic-size finite approximations

Statement

Facts & Assumptions

Proof

Countably partitioned sieves yield posterior contraction by weighted entropy

Statement

Facts & Assumptions

Proof

Hellinger entropy for compact-location normal mixtures in a scale band

Statement

Facts & Assumptions

Proof

The dyadic scale bands satisfy the weighted entropy condition

Statement

Facts & Assumptions

Proof

A Dirichlet process assigns exponential mass to a finite-partition simplex neighborhood

Statement

Facts & Assumptions

Proof

The mixture prior places local KL and log-square mass near a compact smooth truth

Statement

Facts & Assumptions

Proof

Interval masses give a log-free local KL neighborhood for the mixture prior

Statement

Facts & Assumptions

Proof

The posterior excludes large scales and mixing mass far from compact data

Statement

Facts & Assumptions

Proof

Dirichlet normal-mixture posterior contraction

Statement

Facts & Assumptions

Proof

An interval-mass condition removes the Gaussian-bias logarithm

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, counterexamples and false statements

Sources