Bayesian inverse problems and posterior contraction

For observations y=Ax+δξ with ξ standard Gaussian, the likelihood is proportional to exp(−||Ax−y||²/(2δ²)). A prior N(0,τ²I), with fixed τ>0, supplies another quadratic exponent. Completing the square yields a Gaussian posterior. Its mean equals the minimizer of a Tikhonov objective with α=δ²/τ², and its covariance quantifies the remaining spread under this model. This equality of a mean and a regularized solution uses Gaussian linear structure; it is not a general identity between Bayesian averages and arbitrary penalized optimizers.

In function spaces there is generally no flat Lebesgue reference measure and no Gaussian probability measure with identity covariance on an infinite-dimensional Hilbert space. A valid prior must live on an appropriate space, with covariance and support chosen consistently. Bayes' formula is then a Radon–Nikodym derivative relative to that prior, under conditions making the likelihood integrable with positive normalizing constant. Formal finite-dimensional density expressions cannot simply be copied into the continuum without this measure-theoretic step.

For a scalar observation y=σx+δξ with prior variance τ², the posterior variance is (τ⁻²+σ²/δ²)⁻¹. A large σ relative to δ adds substantial precision; a small ratio adds little. If σ=0, the likelihood is independent of x, and both posterior mean and variance remain those of the prior. In a higher-dimensional model with correlated prior components, data may update an unobserved coordinate through prior correlations, but that transfer still depends on the prior rather than a direct measurement of the coordinate.

A proper posterior can therefore coexist with nonidentifiability of the forward map. A concentrated prior may produce a narrow posterior in an invisible direction, yet the data would not distinguish a different admissible truth there. Posterior credible sets state probabilities conditional on the chosen model and prior. They are not automatically repeated-experiment confidence sets with the advertised coverage for every fixed truth. Misspecified noise scales, forward models, or priors can make apparently small posterior uncertainty misleading.

Posterior contraction asks whether posterior mass outside shrinking neighborhoods of a fixed data-generating truth tends to zero as noise decreases or information increases. In fixed finite dimension, if A has full column rank, the Gaussian prior above is fixed and nondegenerate, and y=Ax†+δξ, posterior covariance is of order δ² and the posterior mean's noise error is of order δ in probability. Its prior-induced bias is of order δ² for fixed x†. Thus the expected posterior mass outside radius Mδδ tends to zero for any deterministic Mδ tending to infinity; choosing Mδδ→0 makes these neighborhoods shrink.

This finite-dimensional argument uses a positive smallest singular value and does not give a dimension-independent inverse-problem rate. In an infinite sequence problem, increasing resolution exposes ever smaller singular values; contraction depends on their decay, prior regularity, and how the truth aligns with the prior. A fixed nullspace prevents data-driven contraction to arbitrary truths in that direction under an independent unchanged prior. General contraction theorems need explicit support, smoothness, and information conditions, so no universal rate is claimed here from Gaussianity alone.