Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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A rank-one prior gives a singular Gaussian inverse posterior

Example

Assume AC. Let X∼N2(0,diag⁡(1,0)), η∼N2(0,I2) independently, A=I2, and observe y=(2,3)T in the model Y=AX+η. The prior support is S0={(t,0)T:t∈R}. The Gaussian posterior version has mean (1,0)T and covariance diag⁡(1/2,0). On S0, the support-restricted objective is Jy(x)=(x1−2)2+(x2−3)2+x12, and its unique minimizer is (1,0)T.

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∈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 C0=diag⁡(1,0) and S=AC0AT+Γ=diag⁡(2,1). Thus C0S−1y=(1,0)T and C0−C0S−1C0=diag⁡(1/2,0), which gives the posterior mean and covariance in the Example by [F1].

2.1F2F3step 1.1algebra∎

On S0, x2=0 and C0†=diag⁡(1,0) by [F3]. The objective becomes (x1−2)2+9+x12=2(x1−1)2+11, so it has the unique minimizer (1,0)T. By [F2] this point is also the posterior MAP and mean, as asserted.

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