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Linear Gaussian inverse posteriors and Tikhonov — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

A rank-one prior gives a singular Gaussian inverse posterior

Example

Assume AC. Let $X \sim N_{2} (0, diag (1, 0))$ , $η \sim N_{2} (0, I_{2})$ independently, $A = I_{2}$ , and observe $y = (2, 3)^{T}$ in the model $Y = A X + η$ . The prior support is $S_{0} = {(t, 0)^{T} : t \in R}$ . The Gaussian posterior version has mean $(1, 0)^{T}$ and covariance $diag (1 / 2, 0)$ . On $S_{0}$ , the support-restricted objective is $J_{y} (x) = (x_{1} - 2)^{2} + (x_{2} - 3)^{2} + x_{1}^{2},$ and its unique minimizer is $(1, 0)^{T}$ .

Example references

The posterior formula allows a singular state covariance when the observed covariance is positive definite.

The MAP, posterior mean, and support-restricted minimizer agree for the finite Gaussian model.

Facts & Assumptions

Given: AC, the model and datum in the Example.

[A1]

AC is inherited from the Gaussian-law and finite Gaussian-conditioning interfaces. The Axiom of Choice

[F1]

The finite linear Gaussian posterior mean and covariance are given by the displayed conditioning formulas, even for singular prior covariance. Finite linear Gaussian inverse posterior with singular prior covariance

[F2]

The posterior MAP and mean equal the unique support-restricted Tikhonov minimizer. Finite Gaussian MAP, posterior mean, and support-restricted Tikhonov minimizer coincide

[F3]

A rank-one Gaussian prior with covariance $diag (1, 0)$ has support ${(t, 0)^{T} : t \in R}$ and pseudoinverse $diag (1, 0)$ . Support and pseudoinverse coordinates of a finite singular Gaussian

Verification

Given: The hypotheses and facts in the Example and Facts & Assumptions.

1.1A1F1givenalgebra

Here $C_{0} = diag (1, 0)$ and $S = A C_{0} A^{T} + Γ = diag (2, 1)$ . Thus $C_{0} S^{- 1} y = (1, 0)^{T}$ and $C_{0} - C_{0} S^{- 1} C_{0} = diag (1 / 2, 0)$ , which gives the posterior mean and covariance in the Example by [F1].

2.1F2F3step 1.1algebra∎

On $S_{0}$ , $x_{2} = 0$ and $C_{0}^{†} = diag (1, 0)$ by [F3]. The objective becomes $(x_{1} - 2)^{2} + 9 + x_{1}^{2} = 2 (x_{1} - 1)^{2} + 11$ , so it has the unique minimizer $(1, 0)^{T}$ . By [F2] this point is also the posterior MAP and mean, as asserted.

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Prior whitening turns an anisotropic Gaussian posterior into Tikhonov form

Example

Assume AC. Let $X \sim N_{2} (0, diag (4, 1))$ , $η \sim N_{2} (0, diag (1, 4))$ independently, $A = I_{2}$ , and observe $y = (2, 4)^{T}$ in $Y = A X + η$ . Put $L = diag (2, 1)$ and $W = diag (1, 1 / 2)$ . The coordinate change $x = L z$ makes the prior on $z$ standard Gaussian, and $W$ whitens the observation noise in that $W Γ W^{T} = I_{2}$ . The preconditioned Tikhonov objective is $∥ W A L z - W y ∥_{2}^{2} + ∥ z ∥_{2}^{2} = ∥ diag (2, 1 / 2) z - (2, 2)^{T} ∥_{2}^{2} + ∥ z ∥_{2}^{2} .$ The posterior mean, MAP point, and minimizer in the original parameter coordinates are $(8 / 5, 4 / 5)^{T}$ , and the posterior covariance is $(4 / 5) I_{2}$ .

Example references

The posterior formula handles general positive-definite observation covariance.

Prior whitening gives the support-coordinate Tikhonov objective whose unique minimizer is the MAP and posterior mean.

Facts & Assumptions

Given: AC, the model and datum in the Example.

[A1]

AC is inherited from the Gaussian-law and finite Gaussian-conditioning interfaces. The Axiom of Choice

[F1]

The finite linear Gaussian posterior formula is valid for singular or positive-definite prior covariance and positive-definite noise covariance. Finite linear Gaussian inverse posterior with singular prior covariance

[F2]

The finite Gaussian MAP and posterior mean equal the support-restricted Tikhonov minimizer. Finite Gaussian MAP, posterior mean, and support-restricted Tikhonov minimizer coincide

[F3]

For $λ > 0$ , the Tikhonov objective is $∥ D z - b ∥_{2}^{2} + λ ∥ z ∥_{2}^{2}$ , with unique minimizer $(D^{T} D + λ I)^{- 1} D^{T} b$ . The Tikhonov regularised least-squares objective $∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}$ for $λ > 0$ , For every $λ > 0$ , the Tikhonov objective is strictly convex and has the unique minimiser $(A^{*} A + λ I)^{- 1} A^{*} b$

Verification

Given: The hypotheses and facts in the Example and Facts & Assumptions.

1.1A1F1givenalgebra

Here $S = C_{0} + Γ = 5 I_{2}$ , so [F1] gives $m (y) = C_{0} S^{- 1} y = (8 / 5, 4 / 5)^{T}$ and $C (y) = C_{0} - C_{0} S^{- 1} C_{0} = (4 / 5) I_{2}$ .

2.1F2F3step 1.1algebra∎

The whitening matrices satisfy $W Γ W^{T} = I_{2}$ and $D = W A L = diag (2, 1 / 2)$ , while $b = W y = (2, 2)^{T}$ . Applying [F3] with $λ = 1$ gives $\hat{z} = (D^{T} D + I_{2})^{- 1} D^{T} b = (4 / 5, 4 / 5)^{T}$ . Therefore $x = L \hat{z} = (8 / 5, 4 / 5)^{T}$ . By [F2] this is the unique MAP, posterior mean, and support-restricted minimizer.

CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

A non-Gaussian posterior can have different MAP and mean

Statement refuted

For every probability density $p$ on $R$ with finite first absolute moment and at least one global maximizer, every maximum a posteriori (MAP) point, meaning every global maximizer of $p$ , equals its mean $\int_{R} t p (t) d t$ .

Statement references

The Gaussian result has explicit prior, likelihood, support, and quadratic-posterior hypotheses.

Facts & Assumptions

Given: AC; a prior uniform on $[0, 1]$ ; independent standard-normal observation noise; and the observation model in the Counterexample.

[A1]

AC is assumed by the standard-normal law. The Axiom of Choice

[F1]

The standard-normal noise has a strictly positive Gaussian density. Standard normal and normal laws, A positive-definite multivariate normal law has its Gaussian density

[F2]

A normalized potential update is the prior reweighted by its likelihood and divided by its evidence when the evidence is positive and finite. Misfit evidence and the potential posterior law

[F3]

The equality of MAP, posterior mean, and support-restricted Tikhonov minimizer proved for the finite Gaussian model uses Gaussian prior and likelihood assumptions. Finite Gaussian MAP, posterior mean, and support-restricted Tikhonov minimizer coincide

Counterexample

Given: Let $θ$ have the uniform probability density $1_{[0, 1]}$ on $R$ . Independently let $ϵ \sim N (0, 1)$ , observe $Y = θ + ϵ$ , and condition on $y = 0$ . The likelihood potential is $Φ_{0} (θ) = θ^{2} / 2$ on $[0, 1]$ and the posterior density relative to Lebesgue measure is $p (θ ∣ 0) = \frac{e^{- θ^{2} / 2}}{\int_{0}^{1} e^{- t^{2} / 2} d t} 1_{[0, 1]} (θ) .$

1.1A1F1F2givenalgebra

The evidence in [F2] is $Z = \int_{0}^{1} e^{- t^{2} / 2} d t$ . Since $e^{- 1 / 2} \leq e^{- t^{2} / 2} \leq 1$ on $[0, 1]$ , one has $0 < e^{- 1 / 2} \leq Z \leq 1$ . Thus the displayed normalized posterior density is proper.

2.1F1step 1.1algebra

On $(0, 1]$ , the derivative of $e^{- θ^{2} / 2}$ is $- θ e^{- θ^{2} / 2} < 0$ . The posterior density vanishes outside $[0, 1]$ , so its unique global maximizer is $θ = 0$ . Its support is the bounded interval $[0, 1]$ and it is not a point mass. A nondegenerate one-dimensional Gaussian has a strictly positive density on all of $R$ by [F1], whereas a zero-variance Gaussian is a point mass. Therefore this posterior is non-Gaussian.

3.1F3step 1.1step 2.1algebra∎

The posterior mean is $\frac{\int_{0}^{1} t e^{- t^{2} / 2} d t}{Z} = \frac{1 - e^{- 1 / 2}}{Z} > 0,$ which differs from the MAP point $0$ . This refutes the universal claim. It also shows why [F3] is restricted to the Gaussian model.

Linear Gaussian inverse posteriors and Tikhonov — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

A rank-one prior gives a singular Gaussian inverse posterior

Example

Example references

Facts & Assumptions

Verification

Prior whitening turns an anisotropic Gaussian posterior into Tikhonov form

Example

Example references

Facts & Assumptions

Verification

A non-Gaussian posterior can have different MAP and mean

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Sources