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A sieve can miss the entropy bound while the posterior contracts exactly

Example

Let $Θ = [0, 1]$ , let $X_{i} ∣ θ$ be IID Bernoulli $(θ)$ , and put the full Bernoulli family $F_{n} = {p_{θ} : θ \in [0, 1]}$ in every sieve $Θ_{n} = Θ$ . Take the true value $θ_{0} = 1 / 2$ and prior $Π = δ_{1 / 2}$ . In the full density $L^{1}$ metric, the entropy premise of IID posterior contraction by entropy and prior mass fails for $ϵ_{n} = (\log n)^{1 / 4} / \sqrt{n}$ at all sufficiently large $n$ , although the other displayed conditions hold and the posterior mass outside every positive-radius ball is exactly zero for every data vector.

Facts & Assumptions

Given: The Bernoulli model, its jointly measurable mass function $p_{θ} (1) = θ$ , $p_{θ} (0) = 1 - θ$ , the point prior at $θ_{0} = 1 / 2$ , and $n \geq 3$ .

[F1]

IID posterior contraction by entropy and prior mass defines the data-fiber posterior by likelihood-weighted normalization on finite positive evidence, and its contraction conclusion is in $P_{0}^{n}$ -probability under the stated entropy, sieve-tail, and local KL/log-square conditions.

[F2]

Sieve metric entropy produces uniform exponential tests defines $D (u, F_{n}, d_{1})$ using finite sets whose pairwise full density $L^{1}$ distances exceed $u$ .

[F3]

Kullback–Leibler divergence uses the direction $K L (P_{0} ∥ P_{θ}) = P_{0} \log (p_{0} / p_{θ})$ .

[A1]

Assume the Axiom of Choice The Axiom of Choice for the imported directional KL convention; all displayed Bernoulli densities and the point-prior update are explicit.

[F4]

Probability measures and probability spaces: the IID Bernoulli sampling law $P_{0}^{n}$ is a probability measure on the finite data vectors.

[F5]

The natural logarithm as the inverse of the exponential function and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm give the natural-log product and quotient identities; The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t gives $\log x = \int_{1}^{x} d t / t$ for $x > 0$ .

[F6]

Identical distribution and IID families defines the IID sampling regime.

Verification

Given: The model, sieve, prior, truth, and rate sequence in the Example.

1.1F1F5givenalgebra

Put $t_{n} : = \log n$ . For $n \geq 3$ , $t_{n} > 1$ , and $ϵ_{n}^{2} = \frac{\sqrt{t_{n}}}{n}, n ϵ_{n}^{2} = \sqrt{t_{n}} ⟶ \infty .$ The integral formula [F5] gives $\log n \leq 2 \sqrt{n}$ for $n \geq 1$ , hence $ϵ_{n}^{2} \leq \sqrt{2} n^{- 3 / 4} \to 0$ . Thus $ϵ_{n} \to 0$ and the rate conditions in [F1] hold.

1.2F2F5givenalgebraconstruct

For $j = 0, \dots, ⌊ 1 / ϵ_{n} ⌋$ , set $θ_{j} = j ϵ_{n}$ . These parameters lie in $[0, 1]$ , and $d_{1} (p_{θ_{j}}, p_{θ_{k}}) = 2 ∣ θ_{j} - θ_{k} ∣ > ϵ_{n} (j \neq k) .$ Therefore $D (ϵ_{n}, F_{n}, d_{1}) \geq ⌊ 1 / ϵ_{n} ⌋ + 1 > 1 / ϵ_{n}$ and $\log D (ϵ_{n}, F_{n}, d_{1}) > \log (1 / ϵ_{n}) = \frac{1}{2} t_{n} - \frac{1}{4} \log t_{n} .$ For $t \geq 1$ , [F5] gives $\log t \leq 2 \sqrt{t}$ . If $t_{n} \geq 64$ , then $\frac{1}{2} t_{n} - \frac{1}{4} \log t_{n} \geq \frac{1}{2} t_{n} - \frac{1}{2} \sqrt{t_{n}} > \sqrt{t_{n}} = n ϵ_{n}^{2}$ . Hence the entropy premise fails for all sufficiently large $n$ .

1.3A1F3given

The sieve is the full parameter space, so $Π (Θ_{n}^{c}) = 0$ . Its local KL/log-square set contains $θ_{0}$ , where $K L (P_{0} ∥ P_{θ_{0}}) = 0$ and $\log (p_{θ_{0}} / p_{0}) = 0$ ; consequently its prior mass is $1$ , which exceeds $e^{- C n ϵ_{n}^{2}}$ for every $C > 0$ .

2.1F1F4F6given∎

For any data vector $x_{1 : n}$ , the likelihood at the sole prior atom is $p_{θ_{0}}^{\otimes n} (x_{1 : n}) = 2^{- n} > 0$ , so the evidence is also $2^{- n}$ . The normalization in [F1] therefore gives $Π (\cdot ∣ x_{1 : n}) = δ_{θ_{0}}$ . For every $M > 0$ , its mass on ${θ : d_{1} (p_{θ}, p_{0}) > M ϵ_{n}}$ is exactly zero for every sample, and thus it contracts in $P_{0}^{n}$ -probability under the IID data law [F4, F6] despite the failed entropy condition.

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Sources

Subhashis Ghosal, Jayanta K. Ghosh and Aad W. van der Vaart, Convergence rates of posterior distributions, Theorem 2.1 (standard reference, not scraped)
Subhashis Ghosal, The Dirichlet Process, Related Priors and Posterior Asymptotics, §2.5.2 (standard reference, not scraped)