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A nonintegrable error envelope destroys the linear Hellinger rate

Statement refuted

For every separable real Banach space $X$ , Borel set $D \subseteq X$ , and Borel probability $μ_{0}$ with $μ_{0} (D) = 1$ and positive mass on a bounded subset of $D$ . Give $D$ the relative norm topology. Let $Φ, Φ_{N} : D \to R$ be finite continuous potentials indexed by integers $N \geq 1$ , let $M_{1}, M_{2} : [0, \infty) \to [0, \infty)$ be Borel and nondecreasing, and let $ψ_{N} \geq 0$ tend to zero. If $Φ, Φ_{N} \geq - M_{1} (∥ u ∥_{X})$ and $∣ Φ_{N} (u) - Φ (u) ∣ \leq M_{2} (∥ u ∥_{X}) ψ_{N}$ for every $u, N$ , and their evidence normalizers are positive and finite, then there are $C < \infty$ and $N_{0}$ such that the library root-L2 Hellinger distance satisfies $h (Π, Π_{N}) \leq C ψ_{N}$ for every $N \geq N_{0}$ , even without assuming $\int_{D} e^{M_{1} (∥ u ∥_{X})} (1 + M_{2} (∥ u ∥_{X})^{2}) d μ_{0} < \infty$ .

Statement references

The valid linear Hellinger-rate theorem assumes weighted integrability of the lower and error envelopes.

The library Hellinger distance is the root-L2 norm of the difference of square-root densities.

A Banach space is complete for its norm metric.

A separable space has an at most countable dense subset.

A norm induces the metric used on the parameter space.

The Borel sigma-algebra is generated by open sets.

A probability measure has total mass one.

The metric topology declares a set open when every point has an open ball inside it.

The relative topology consists of intersections of ambient open sets with the subspace.

Continuity means inverse images of open sets are open.

Dashti and Stuart, full lecture notes, §4.2

Cotter, Dashti, and Stuart, full paper, §2

Facts & Assumptions

Given: A counterexample is required to the no-integrability claim in Statement refuted.

[F1]

For a fixed-data potential, $Z = \int e^{- Φ} d μ_{0}$ and the posterior density relative to $μ_{0}$ is $e^{- Φ} / Z$ on positive finite evidence fibers. Misfit evidence and the potential posterior law

[F2]

For supplied densities $p, q$ relative to the same measure, the library Hellinger distance satisfies $h^{2} = \int (\sqrt{p} - \sqrt{q})^{2} d μ_{0}$ . Root-L2 Hellinger distance for dominated laws

[F3]

The series $\sum_{n \geq 1} n^{- 2}$ converges, while $\sum_{n \geq 1} n^{- 1}$ diverges. For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$

[F4]

The valid potential-discretization result assumes $\int e^{M_{1}} (1 + M_{2}^{2}) d μ_{0} < \infty$ to conclude a linear Hellinger rate. Hellinger error bound for posterior potential discretization

[F5]

The real exponential is continuous and strictly increasing, with $e^{- x} = 1 / e^{x}$ and $e^{1} > 1$ . The exponential function is strictly increasing, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$

[F6]

The real line with its absolute-value metric is a separable Banach space. For each $n \geq 1$ , the intervals $(x - ∣ x - n ∣ / 2, x + ∣ x - n ∣ / 2)$ , indexed by $x \neq n$ , cover $R ∖ {n}$ and omit $n$ , so ${n}$ is closed. Hence $D = ⋃_{n \geq 1} {n}$ is Borel. Also $(n - \frac{1}{2}, n + \frac{1}{2}) \cap D = {n}$ for each $n \geq 1$ , so every singleton is relatively open, $D$ is discrete, and every real-valued function on $D$ is continuous. Banach space, Separability: the existence of an at most countable dense subset, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, The Borel sigma-algebra of a topological space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Continuity of a map of topological spaces at a point and globally

[F7]

The assigned values of $μ_{0}$ are nonnegative, and the empty set has measure zero by the empty sum. For pairwise disjoint Borel sets $A_{j} \subseteq R$ , the nonnegative series in the definition of $μ_{0}$ partitions across the disjoint sets $A_{j} \cap D$ , so $μ_{0} (⋃_{j} A_{j}) = \sum_{j} μ_{0} (A_{j})$ . By [F3], $c > 0$ is finite and $c \sum_{n \geq 1} n^{- 2} = 1$ , hence $μ_{0}$ is a Borel probability measure with $μ_{0} (D) = 1$ . Probability measures and probability spaces

Counterexample

Let $X = R$ with its usual norm, $D = N_{\geq 1}$ with the trace Borel sigma-algebra, and $c = (\sum_{n = 1}^{\infty} n^{- 2})^{- 1}$ . For Borel $A \subseteq R$ , define $μ_{0} (A) : = c \sum_{n \in A \cap D} n^{- 2}$ , so $μ_{0} ({n}) = c / n^{2}$ . Let $Φ (n) = 0$ , $Φ_{N} (n) = n / N$ for integers $N \geq 1$ , and set $M_{1} (s) = 0$ , $M_{2} (s) = s$ , and $ψ_{N} = 1 / N$ . The potentials have common lower envelope zero and satisfy $∣ Φ_{N} (n) - Φ (n) ∣ = M_{2} (∣ n ∣) ψ_{N}$ , but the weighted error envelope is not prior-integrable.

1.1F3F4F6F7givenconstructalgebra

By [F3], $0 < \sum_{n \geq 1} n^{- 2} < \infty$ , so $c$ is finite and positive and [F7] makes $μ_{0}$ a probability with $μ_{0} (D) = 1$ . By [F6], $D$ is Borel and has the discrete relative topology, so every real-valued function on $D$ , including $Φ, Φ_{N}$ , is continuous. The real line with its usual norm is a separable Banach space by [F6]. Thus all $Φ, Φ_{N}$ are finite continuous functions on $D$ . They satisfy $Φ, Φ_{N} \geq 0$ and $∣ Φ_{N} (n) - Φ (n) ∣ = n / N = M_{2} (∣ n ∣) ψ_{N}$ . But $\int_{D} e^{M_{1} (∣ u ∣)} (1 + M_{2} (∣ u ∣)^{2}) μ_{0} (d u) = \sum_{n \geq 1} \frac{c}{n^{2}} (1 + n^{2}) = + \infty,$ since the second summand contributes $c$ for each $n$ . This is precisely the integrability hypothesis used by [F4], and it fails here.

2.1F1F3F5step 1.1givenchoosealgebra

The base evidence is $Z = 1$ . For each $N$ , the approximate evidence $Z_{N} = \sum_{n \geq 1} \frac{c}{n^{2}} e^{- n / N}$ is positive and at most one, so both posterior updates exist. Moreover $Z_{N} \to 1$ : given $ϵ > 0$ , choose $m$ with $\sum_{n > m} c / n^{2} < ϵ / 2$ by [F3]. The finite sum $\sum_{n = 1}^{m} (c / n^{2}) (1 - e^{- n / N})$ tends to zero by continuity of the exponential [F5], while the remaining tail of $1 - Z_{N}$ is at most $\sum_{n > m} c / n^{2} < ϵ / 2$ .

3.1F5step 2.1givenalgebra

Since $e^{- 1} < 1$ , the number $α : = (1 + e^{- 1}) / 2$ is less than one. From $Z_{N} \to 1$ , for all sufficiently large $N$ we have $Z_{N} \geq α$ . For $n \geq N$ , $\frac{e^{- n / N}}{Z_{N}} \leq \frac{e^{- 1}}{α} = \frac{2}{e + 1} < 1,$ so $1 - \sqrt{e^{- n / N} / Z_{N}} \geq κ : = 1 - \sqrt{2 / (e + 1)} > 0$ .

4.1F2step 3.1givenalgebra∎

The posterior density of the base law relative to $μ_{0}$ is $1$ , and that of the approximation is $e^{- n / N} / Z_{N}$ . By [F2] and step 3.1, $h (Π, Π_{N})^{2} = \sum_{n \geq 1} \frac{c}{n^{2}} {(1 - \sqrt{\frac{e^{- n / N}}{Z_{N}}})}^{2} \geq κ^{2} \sum_{n = N}^{2 N - 1} \frac{c}{n^{2}} \geq \frac{κ^{2} c}{4 N},$ because the block has $N$ terms and each $n$ in it satisfies $n < 2 N$ . Therefore $h (Π, Π_{N}) \geq (κ \sqrt{c} / 2) N^{- 1 / 2}$ , while $ψ_{N} = N^{- 1}$ . The ratio $h (Π, Π_{N}) / ψ_{N}$ is at least $(κ \sqrt{c} / 2) \sqrt{N}$ and is unbounded, so the claimed $O (ψ_{N})$ rate fails.

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Sources

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems (full lecture notes, arXiv:1302.6989) (standard reference, not scraped)
S. L. Cotter, M. Dashti, and A. M. Stuart, Approximation of Bayesian Inverse Problems for PDEs (standard reference, not scraped)