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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.

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Sources

Subhashis Ghosal and Aad W. van der Vaart, Posterior convergence rates of Dirichlet mixtures at smooth densities, Theorem 1, equation (2.2), and §9 proof (standard reference, not scraped)