Unknown operators, model mismatch, and robustness limits

Let $(X,d_X)$ and $(\mathsf{Y},d_Y)$ be metric spaces, let $U\subseteq X$ be a nonempty class of admissible truths, and let $q:U\to Q$ assign each truth its recovery target in a metric space $(Q,d_Q)$. Let $\mathfrak{A}$ index forward maps $F_A:U\to \mathsf{Y}$; in this definition an operator means such a forward map, and linearity is not assumed. A specified set $\mathcal{C}\subseteq U\times \mathfrak{A}$ records which truth/operator pairs are admissible; no full-product assumption is made. Equip $\mathsf{Y}$ with a sigma-algebra $\mathcal{Y}$ and an addition operation such that every translation ${\tau }_z(e)=z+e$ is measurable. A noise law is a probability measure $\nu$ on $(\mathsf{Y},\mathcal{Y})$. For each admissible pair $(x,A)\in \mathcal{C}$, the observation random element on the probability space $(\mathsf{Y},\mathcal{Y},\nu )$ is $e\mapsto F_A(x)+e$, and its sampling law is $P_{x,A,\nu }(B)=\nu (\{e\in \mathsf{Y}:F_A(x)+e\in B\}),B\in \mathcal{Y}\mathrm{.}$ The translation measurability assumption makes this law a probability measure.

For a true specification $(x_{\star },A_{\star },{\nu }_{\star })$ with $(x_{\star },A_{\star })\in \mathcal{C}$ and a specified working operator $\tilde{A}\in \mathfrak{A}$ and noise law $\tilde{\nu }$, there is operator mismatch at the truth when $F_{A_{\star }}(x_{\star })\mathrm{\ne }{F}_{\tilde{A}}(x_{\star })$, and there is noise-law mismatch when ${\nu }_{\star }\mathrm{\ne }\tilde{\nu }$. A working statistical model is a metric index space $(\mathrm{\Xi },d_{\mathrm{\Xi }})$, a family $(Q_{\xi }{)}_{\xi \in \mathrm{\Xi }}$ of sampling laws on the same data space, and a prior probability measure $\mathrm{\Pi }$ on a sigma-algebra containing every open metric ball. The statistical model is misspecified for true observation law $P_{\star }=P_{x_{\star },A_{\star },{\nu }_{\star }}$ when $P_{\star }\mathrm{\notin }\{Q_{\xi }:\xi \in \mathrm{\Xi }\}$. If a true model index ${\xi }_{\star }\in \mathrm{\Xi }$ is specified, there is a prior-support mismatch at ${\xi }_{\star }$ when some open ball $B_{\mathrm{\Xi }}({\xi }_{\star },r)$, $r>0$, has $\mathrm{\Pi }(B_{\mathrm{\Xi }}({\xi }_{\star },r))=0$. These are distinct properties; an operator or noise-law mismatch need not by itself make the statistical model misspecified or exclude the true index from the prior's support. This definition makes no assertion about posterior existence or convergence.

Metric parameter and data spaces, admissible truths, forward maps, and metric recovery targets

Metric spaces and metric balls

Measurable data and index spaces

Noise laws and priors as probability measures

Sampling laws indexed by parameters

Measurable translations

An observation random element is a measurable map

The law of a random element

The law is a probability measure

Baek, Aquino, and Mukherjee, Generalized Bayes Approach to Inverse Problems with Model Misspecification

Cvetković et al., Choosing Observation Operators to Mitigate Model Error in Bayesian Inverse Problems

Dashti and Stuart, The Bayesian Approach to Inverse Problems

InverseBench's reported algorithm comparisons and prior-shift observations are empirical for its selected tasks, datasets, and tuning. Its reported setup assumes exact forward models and lists robustness to forward-model mismatch as future work; these observations do not establish a general robustness theorem.

Zheng et al., InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences, full 29-page paper

Let $(Q,d_Q)$ be a metric target space and let $\mathcal{G}$ be a sigma-algebra on $Q$. Let ${\tau }_i=(x_i,A_i,{\nu }_i)$, $i\in \{0,1\}$, be two admissible truth/operator/noise specifications on one measurable data space $(\mathsf{Y},\mathcal{Y})$, with sampling laws $P_i$ and targets $q_i=q(x_i)$. Assume the metric-loss sections $a\mapsto d_Q(q_i,a)$ are $\mathcal{G}$-measurable. Let $\delta$ be one randomized decision rule from $(\mathsf{Y},\mathcal{Y})$ to $(Q,\mathcal{G})$, used under both specifications, and define its expected metric risk by $R_i(\delta )={\int }_{\mathsf{Y}}({\int }_Q{d}_Q(q_i,a)\delta (y,da))P_i(dy)\mathrm{.}$ If $P_0=P_1$ and $\mathrm{\Delta }=d_Q(q_0,q_1)$, then $\max \{R_0(\delta ),R_1(\delta )\}\ge \frac{\mathrm{\Delta }}{2}\mathrm{.}$ Each risk is an extended nonnegative integral; deterministic measurable estimators are included as point-mass decision kernels. This theorem makes no assertion when the observation laws differ, and quantitative robustness requires a separately established stability modulus.

Maxim Raginsky, Minimax lower bounds, §1, Lemma 1 and complete proof

John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 8, §§8.1–8.3