TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Hellinger stability of function-space posteriors under data perturbations

Statement

Let $X, Y$ be separable real Banach spaces, let $D \subseteq X$ be Borel, and let $μ_{0}$ be a Borel probability measure on $X$ with $μ_{0} (D) = 1$ . Assume there is a bounded Borel subset $E \subseteq D$ with $μ_{0} (E) > 0$ . Give $D$ the relative norm topology and trace Borel sigma-algebra $B (D) = {D \cap A : A \in B (X)}$ . Let $R > 0$ and let $Φ : D \times Y \to R$ be jointly continuous and measurable from $(D \times Y, B (D) \otimes B (Y))$ to the Borel sigma-algebra of the extended real line after including its finite real values there. Suppose $M_{1}, M_{2} : [0, \infty)^{2} \to [0, \infty)$ are Borel and nondecreasing in each argument, and for all $u \in D$ and $y, y^{'} \in B_{Y} (0, R) : = {v \in Y : ∥ v ∥_{Y} \leq R}$ , $Φ (u; y) \geq - M_{1} (R, ∥ u ∥_{X}), ∣ Φ (u; y) - Φ (u; y^{'}) ∣ \leq M_{2} (R, ∥ u ∥_{X}) ∥ y - y^{'} ∥_{Y} .$ Assume $J_{R} : = \int_{D} e^{M_{1} (R, ∥ u ∥_{X})} (1 + M_{2} (R, ∥ u ∥_{X})^{2}) μ_{0} (d u) < \infty .$ For each fixed datum $y \in B_{Y} (0, R)$ , define the evidence and posterior by $Z (y) : = \int_{D} e^{- Φ (u; y)} μ_{0} (d u), Π^{y} (d u) : = Z (y)^{- 1} e^{- Φ (u; y)} μ_{0} (d u) .$ Then there is $z_{R} > 0$ such that $z_{R} \leq Z (y) \leq J_{R}$ for every $y \in B_{Y} (0, R)$ , and, with the library root-L2 Hellinger distance $h$ , $h (Π^{y}, Π^{y^{'}}) \leq C_{R} ∥ y - y^{'} ∥_{Y}, C_{R} : = \frac{\sqrt{J_{R}}}{2 \sqrt{z_{R}}} + \frac{J_{R}}{2 z_{R}},$ for all $y, y^{'} \in B_{Y} (0, R)$ . This is a pointwise fixed-data statement; no observation law or conditional-law version is asserted. In a separate branch, if AC is assumed for the library density/total-variation comparison convention, then $∥ Π^{y} - Π^{y^{'}} ∥_{T V} \leq h (Π^{y}, Π^{y^{'}}) \leq C_{R} ∥ y - y^{'} ∥_{Y},$ where $∥ P - Q ∥_{T V} : = \sup_{A \in B (D)} ∣ P (A) - Q (A) ∣$ .

Statement references

A prior is a Borel probability with full mass on its declared forward domain.

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 and data spaces.

The Borel sigma-algebra is generated by the open sets.

A probability measure has total mass one.

Continuity means inverse images of open sets are open.

Joint measurability is with respect to the product sigma-algebra.

A map is measurable when inverse images of measurable sets are measurable.

The extended real line adds two infinite endpoints to the real line.

The extended-real Borel sigma-algebra uses the order topology and restricts to the real Borel sets.

A product-measurable function has measurable fixed-coordinate sections.

Fixed-data evidence and the normalized potential update are defined fiberwise when evidence is positive and finite.

A common integrable lower envelope and continuity at support points imply positive finite evidence.

The support of a Borel probability on a separable Banach space is Borel and has full measure.

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

Under AC, library total variation is at most the library root-L2 Hellinger distance.

Arithmetic and absolute-value operations preserve measurability of real-valued functions.

The nonnegative integral is additive on sums of nonnegative measurable functions.

The root integral of a square defines the norm for square-integrable functions.

Minkowski’s inequality gives the triangle inequality for square-integrable functions.

Dashti and Stuart, full lecture notes, §4.1

Latz, full paper, §§3.1 and 3.4

Facts & Assumptions

Given: The separable Banach spaces, Borel full-measure prior domain, positive prior mass on a bounded subset, product-Borel measurability and joint continuity, common lower envelope, data-Lipschitz bound, and weighted integrability in the Statement.

[F1]

A prior on a forward domain is a Borel probability $μ_{0}$ on $X$ with $μ_{0} (D) = 1$ . Product measurability as an extended-real-valued function makes each fixed-data potential section measurable; the evidence and posterior are $Z (y) = \int_{D} e^{- Φ (u; y)} d μ_{0}$ and $d Π^{y} / d μ_{0} = e^{- Φ (u; y)} / Z (y)$ when $0 < Z (y) < \infty$ . A Borel prior concentrated on a forward-map domain, Probability measures and probability spaces, The Borel sigma-algebra of a topological space, Misfit evidence and the potential posterior law, The product sigma-algebra and its finite iterates, A measurable function between measurable spaces, The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined, The Borel sigma-algebra on the extended real line, Every section of a product-measurable function is measurable

[F2]

A continuous section on $D$ is continuous in the relative topology; for a finite continuous potential with a common integrable lower envelope, the evidence is finite and strictly positive on every asserted data fiber. Continuity of a map of topological spaces at a point and globally, Function-space posterior propriety from a common lower envelope

[F3]

The support of a Borel probability on a separable Banach space is Borel and has full measure; here Banach means complete for the norm metric and separability supplies a countable dense set. The support of a Borel probability on a separable Banach space has full measure, Banach space, Separability: the existence of an at most countable dense subset, The Borel sigma-algebra of a topological space

[F4]

With the explicit posterior densities $p_{y} = e^{- Φ (\cdot; y)} / Z (y)$ relative to $μ_{0}$ , $h (Π^{y}, Π^{y^{'}}) = ∥ \sqrt{p_{y}} - \sqrt{p_{y^{'}}} ∥_{L^{2} (μ_{0})}$ . The displayed definition requires no further density choice when these representatives are supplied. Root-L2 Hellinger distance for dominated laws

[A1]

Assume the Axiom of Choice only for the optional generic density/TV comparison branch. The Axiom of Choice

[F5]

Under [A1], for two probability laws with a common sigma-finite dominator, $∥ P - Q ∥_{T V} = \sup_{A} ∣ P (A) - Q (A) ∣ \leq h (P, Q)$ in the library conventions. Hellinger invariance and comparison with library total variation, Total variation distance for probability laws

[F6]

The derivative of $t \mapsto e^{- t / 2}$ is $- \frac{1}{2} e^{- t / 2}$ and that of $t \mapsto e^{- t}$ is $- e^{- t}$ ; the real mean value theorem bounds an increment by the supremum of the derivative magnitude times the interval length. The exponential function is smooth and $(\exp)^{'} = \exp$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Mean value theorem on a closed real interval

[F7]

The nonnegative integral is monotone and positively homogeneous. Monotonicity and nonnegative homogeneity of the nonnegative integral

[F8]

For fixed $R$ , the map $s \mapsto M_{i} (R, s)$ is Borel: the slice map $s \mapsto (R, s)$ is continuous in the usual product topology, and preimages under a continuous map preserve the sigma-algebra generated by open sets. The norm map and the real exponential are continuous, so composing these maps gives measurable radius integrands. A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, The reverse triangle inequality $∣ d (x, z) - d (y, z) ∣ \leq d (x, y)$ in any metric space, The exponential function is strictly increasing, Composition with a Borel measurable outer map preserves measurability

[F9]

Sums, differences, and absolute values of measurable real-valued functions are measurable. Arithmetic and lattice operations preserve measurability whenever they are defined

[F10]

The nonnegative integral is additive, monotone, and positively homogeneous. Additivity of the nonnegative Lebesgue integral, Monotonicity and nonnegative homogeneity of the nonnegative integral

[F11]

For square-integrable real functions, the root integral of the square is the $L^{2}$ norm and satisfies the triangle inequality. The function space $L^{p} (μ)$ for $0 < p < \infty$ , Minkowski's inequality for integrals, including $p = \infty$

[F12]

If a Borel set $A$ has $μ_{0} (A) = 1$ , then every Borel $B$ satisfies $μ_{0} (B \cap A) = μ_{0} (B)$ : the complement of $A$ is null by countable additivity and $μ_{0} (B ∖ A) \leq μ_{0} (X ∖ A)$ by monotonicity. Probability measures and probability spaces, Measures on sigma-algebras, Measures are monotone

Proof

Given: The hypotheses and fixed-data convention in the Statement.

1.1F1F2F7F8F9givenalgebra

The fixed-radius sections $s \mapsto M_{i} (R, s)$ are Borel and nondecreasing by [F8] and the hypotheses. Since $M_{2} \geq 0$ , $e^{M_{1} (R, ∥ u ∥_{X})} \leq e^{M_{1} (R, ∥ u ∥_{X})} (1 + M_{2} (R, ∥ u ∥_{X})^{2})$ , so [F7] gives $\int_{D} e^{M_{1} (R, ∥ u ∥_{X})} d μ_{0} \leq J_{R} < \infty$ . Thus the hypotheses of [F2] hold for every $y \in B_{Y} (0, R)$ , and all stated pointwise updates are probability measures with $0 < Z (y) < \infty$ .

1.2F1F3F12givenchoose

By the Statement choose a bounded Borel set $E \subseteq D$ with $μ_{0} (E) > 0$ , and choose $K < \infty$ with $∥ u ∥_{X} \leq K$ for all $u \in E$ . By [F3], the support has full measure; applying [F12] to $E$ gives $μ_{0} (E \cap supp μ_{0}) = μ_{0} (E) > 0$ , so choose $u_{*} \in E \cap supp μ_{0}$ .

1.3F6F7F8F9F10F11givenalgebra

Fix $u \in D$ and $y, y^{'} \in B_{Y} (0, R)$ , and write $a = Φ (u; y)$ , $b = Φ (u; y^{'})$ , and $δ = ∥ y - y^{'} ∥_{Y}$ . Both $a, b \geq - M_{1} (R, ∥ u ∥_{X})$ , so every real between them has the same lower bound. By [F6] and the data-Lipschitz hypothesis, $∣ e^{- a / 2} - e^{- b / 2} ∣ \leq \frac{1}{2} e^{M_{1} (R, ∥ u ∥_{X}) / 2} ∣ a - b ∣ \leq \frac{1}{2} e^{M_{1} (R, ∥ u ∥_{X}) / 2} M_{2} (R, ∥ u ∥_{X}) δ .$ Squaring and integrating, using [F7], gives $∥ e^{- Φ (\cdot; y) / 2} - e^{- Φ (\cdot; y^{'}) / 2} ∥_{L^{2} (μ_{0})} \leq \frac{1}{2} \sqrt{J_{R}} δ$ by the definition of the $L^{2}$ norm [F11].

1.4F6F7F8F9F10givenalgebra

By [F6], the same lower envelope bounds the derivative of $t \mapsto e^{- t}$ between $a$ and $b$ by $e^{M_{1} (R, ∥ u ∥_{X})}$ . Hence $∣ Z (y) - Z (y^{'}) ∣ \leq δ \int_{D} e^{M_{1} (R, ∥ u ∥_{X})} M_{2} (R, ∥ u ∥_{X}) d μ_{0} \leq J_{R} δ,$ where $M_{2} \leq 1 + M_{2}^{2}$ and [F7] was used in the last inequality. For the first inequality, monotonicity and additivity apply to the pointwise bounds $e^{- Φ (u; y)} \leq e^{- Φ (u; y^{'})} + ∣ e^{- Φ (u; y)} - e^{- Φ (u; y^{'})} ∣$ and its reverse.

2.1F1F3F7F12givenstep 1.2chooseconstructalgebra

Continuity of $u \mapsto Φ (u; 0)$ at $u_{*}$ gives $ρ > 0$ such that $Φ (u; 0) \leq Φ (u_{*}; 0) + 1$ for $u \in D \cap B_{X} (u_{*}, ρ)$ . The support property gives $μ_{0} (B_{X} (u_{*}, ρ)) > 0$ , and [F12] with $μ_{0} (D) = 1$ gives $m_{*} : = μ_{0} (D \cap B_{X} (u_{*}, ρ)) = μ_{0} (B_{X} (u_{*}, ρ)) > 0$ . Every such $u$ satisfies $∥ u ∥_{X} < K + ρ$ . For any $y \in B_{Y} (0, R)$ , the data-Lipschitz bound between $y$ and $0$ , followed by monotonicity of $M_{2}$ in its norm argument, gives $Φ (u; y) \leq Φ (u_{*}; 0) + 1 + M_{2} (R, K + ρ) R .$ By monotonicity and positive homogeneity of the nonnegative integral [F7], this yields $Z (y) \geq m_{*} \exp [- Φ (u_{*}; 0) - 1 - M_{2} (R, K + ρ) R] = : z_{R} > 0$ , uniformly on the data ball.

3.1F7step 1.1step 2.1givenalgebra

The lower envelope gives $e^{- Φ (u; y)} \leq e^{M_{1} (R, ∥ u ∥_{X})}$ , and [F7] therefore yields $Z (y) \leq \int_{D} e^{M_{1} (R, ∥ u ∥_{X})} d μ_{0} \leq J_{R}$ . Together with step 2.1 this proves the asserted uniform normalizer bounds.

3.2F4F11step 2.1step 1.3step 1.4givenalgebra

Put $a_{y} = e^{- Φ (\cdot; y) / 2}$ and $q_{y} = a_{y} / \sqrt{Z (y)}$ , so [F4] gives $h (Π^{y}, Π^{y^{'}}) = ∥ q_{y} - q_{y^{'}} ∥_{2}$ . Add and subtract $a_{y^{'}} / \sqrt{Z (y)}$ and use the triangle inequality in $L^{2}$ to obtain $∥ q_{y} - q_{y^{'}} ∥_{2} \leq \frac{∥ a_{y} - a_{y^{'}} ∥_{2}}{\sqrt{Z (y)}} + \frac{∣ \sqrt{Z (y^{'})} - \sqrt{Z (y)} ∣}{\sqrt{Z (y)}} .$ The first term is at most $\sqrt{J_{R}} δ / (2 \sqrt{z_{R}})$ by steps 2.1 and 1.3. Since $∥ a_{y^{'}} ∥_{2}^{2} = \int_{D} e^{- Φ (u; y^{'})} d μ_{0} = Z (y^{'})$ , the second term equals $∣ Z (y^{'}) - Z (y) ∣ / [\sqrt{Z (y)} (\sqrt{Z (y^{'})} + \sqrt{Z (y)})]$ , hence is at most $J_{R} δ / (2 z_{R})$ by steps 2.1 and 1.4. This proves the stated $C_{R}$ bound.

4.1A1F5step 3.2given∎

Under the separate assumption [A1], apply [F5] to the two explicit posterior densities relative to the common probability measure $μ_{0}$ . Thus $∥ Π^{y} - Π^{y^{'}} ∥_{T V} \leq h (Π^{y}, Π^{y^{'}})$ , and step 3.2 gives the optional TV estimate without adding an axiom to the Hellinger estimate.

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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)
J. Latz, On the Well-posedness of Bayesian Inverse Problems, SIAM/ASA Journal on Uncertainty Quantification 8 (2020) (standard reference, not scraped)