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Tests and a sieve tail imply posterior contraction from a denominator bound

Statement

For each $n$ , let $P_{θ, n}$ and $P_{0, n}$ have jointly measurable densities $p_{θ, n}$ and $p_{0, n}$ relative to a common sigma-finite measure, and let $Π$ be a prior. Let $Θ_{n}$ be a measurable sieve, let $A_{n} = {θ : d_{n} (θ, θ_{0}) > r_{n}},$ and let $D_{n} = \int p_{θ, n} (Y) / p_{0, n} (Y) Π (d θ)$ be the likelihood-ratio denominator on the $P_{0, n}$ -full set where $p_{0, n} (Y) > 0$ . Suppose $ϵ_{n} > 0$ , $n ϵ_{n}^{2} \to \infty$ , and constants $A < B$ , $T > A$ satisfy $P_{0, n} {D_{n} < e^{- A n ϵ_{n}^{2}}} \to 0, Π (Θ_{n}^{c}) \leq e^{- B n ϵ_{n}^{2}},$ and there are measurable tests $ϕ_{n}$ with $P_{0, n} (ϕ_{n} = 1) \leq e^{- T n ϵ_{n}^{2}}, \sup_{θ \in A_{n} \cap Θ_{n}} P_{θ, n} (ϕ_{n} = 0) \leq e^{- T n ϵ_{n}^{2}} .$ Then the posterior mass of $A_{n}$ tends to zero in $P_{0, n}$ -probability.

Facts & Assumptions

Given: The displayed dominated model, measurable sieve and tested alternative, denominator lower bound, sieve-tail bound, and test errors.

[F1]

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

[F2]

Markov's inequality for random variables: for nonnegative $X$ and $u > 0$ , $P (X > u) \leq E X / u$ .

Proof

1.1F1givenalgebra

Define $N_{n} = \int_{A_{n} \cap Θ_{n}} \frac{p_{θ, n} (Y)}{p_{0, n} (Y)} Π (d θ), T_{n} = \int_{A_{n} \cap Θ_{n}^{c}} \frac{p_{θ, n} (Y)}{p_{0, n} (Y)} Π (d θ) .$ By change of density on ${p_{0, n} > 0}$ and Tonelli [F1], $E_{0, n} [(1 - ϕ_{n}) N_{n}] \leq \int_{A_{n} \cap Θ_{n}} P_{θ, n} (ϕ_{n} = 0) Π (d θ) \leq e^{- T n ϵ_{n}^{2}} .$ For every $θ$ , $E_{0, n} [p_{θ, n} (Y) / p_{0, n} (Y)] = \int_{{p_{0, n} > 0}} p_{θ, n} \leq 1$ , so Tonelli also gives $E_{0, n} T_{n} \leq Π (Θ_{n}^{c}) \leq e^{- B n ϵ_{n}^{2}}$ . Integrating the same bound over all $Θ$ gives $E_{0, n} D_{n} \leq 1$ , hence $D_{n} < \infty$ almost surely.

2.1F2step 1.1givenalgebra∎

Fix $δ > 0$ . By step 1.1 the denominator is finite almost surely. On fibers with $D_{n} = 0$ , assign the posterior mass any value in $[0, 1]$ ; these fibers lie in ${D_{n} < e^{- A n ϵ_{n}^{2}}}$ . On the event $D_{n} \geq e^{- A n ϵ_{n}^{2}}$ and $ϕ_{n} = 0$ , posterior mass $Π (A_{n} ∣ Y) > δ$ implies $N_{n} + T_{n} > δ e^{- A n ϵ_{n}^{2}}$ . The union bound and Markov [F2] therefore give $P_{0, n} {Π (A_{n} ∣ Y) > δ} \leq P_{0, n} {D_{n} < e^{- A n ϵ_{n}^{2}}} + e^{- T n ϵ_{n}^{2}} + δ^{- 1} ⁣ (e^{- (T - A) n ϵ_{n}^{2}} + e^{- (B - A) n ϵ_{n}^{2}}) .$ Every term tends to zero because $T > A$ , $B > A$ and $n ϵ_{n}^{2} \to \infty$ . Since this holds for every $δ > 0$ , posterior mass on $A_{n}$ converges to zero in probability.

Depends on

Used by

Gaussian-process posterior contraction in density and regression models Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Subhashis Ghosal and Aad W. van der Vaart, Convergence rates of posterior distributions for noniid observations, §2, Theorems 1 and 3 (standard reference, not scraped)
Bas J. K. Kleijn, Aad W. van der Vaart and Harry van Zanten, Lectures on Nonparametric Bayesian Statistics, Chapter 9 (standard reference, not scraped)