Uncertainty quantification and resolution limits

For a fixed linear reconstruction B applied to data y=Ax†+e, write xhat=BAx†+Be. The matrix R=BA is the resolution operator. Its difference from identity describes deterministic bias on the chosen truth, while B controls noise propagation. For identity Tikhonov, R has singular-coordinate eigenvalues σ²/(σ²+α), so weak features are systematically attenuated. A residual may be small because A itself barely sees those features. Inspecting only the reconstructed image or the residual misses this systematic part of error.

Columns or rows of a resolution matrix can describe spreading in a selected pixel basis, but their physical interpretation depends on grid spacing and the basis functions. A singular-vector shrinkage factor is not automatically a distance in millimeters. Finer discretization can add poorly resolved directions without adding information. Resolution claims therefore require the experiment's geometry and the reconstruction representation, rather than a bare count of output pixels or a visually sharp interpolation between them.

If e is zero-mean Gaussian with covariance δ²I, δ>0, and the truth is fixed, the repeated-data covariance of the linear estimator is δ²BB*. Its mean is Rx†, so mean squared error also includes the squared bias. A small covariance does not establish a small mean squared error; choosing B=0 would have zero noise variance while discarding the entire signal. This observation makes the distinction concrete without relying on asymptotic statistics.

Under the independent Gaussian prior from the Bayesian lesson, posterior covariance is instead (τ⁻²I+δ⁻²A*A)⁻¹. Even when the posterior mean equals Tikhonov with α=δ²/τ², that posterior covariance generally differs from the repeated-data covariance of the mean estimator. The former describes uncertainty in x conditional on observed y under the prior model; the latter describes how the estimated mean varies across repeated noisy observations of a fixed x†. Coverage of posterior credible sets needs its own argument, particularly when the prior introduces appreciable bias.

Suppose two admissible objects have forward data separated by at most 2δ. Their radius-δ data balls overlap: the midpoint of the two predictions is compatible with either object under the deterministic noise bound. Any single-valued estimator produces the same answer at that observed midpoint. By the triangle inequality, its error for at least one of the two truths is at least half their object-space separation. This is an information limitation independent of a particular reconstruction algorithm.

A prior or restricted truth class can exclude one alternative, and a new measurement can separate their predictions. Either change alters the premises of the impossibility statement. Repeating identical information can reduce stochastic error under independence, but it does not remove an exact nullspace. Sensitivity calculations, credible sets, and deterministic ambiguity bounds should therefore be reported with their assumptions visible. The goal is to identify which features are supported by measurements, which are selected by modeling, and which remain unresolved at the declared noise level.