ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Singular forecast covariance lies outside the ordinary RTS inverse domain

Example

Take $T = 1$ , $X_{0} \sim N (0, 0)$ , $F_{1} = 1$ , $Q_{1} = 0$ , $H_{1} = 0$ , and $R_{1} = 1$ , with independent noises and observation $y_{1} = 0$ . Then the forward Kalman innovation covariance is positive, but the state forecast covariance is zero. The fixed-interval smoothing law still exists and is the point mass at zero; the ordinary RTS gain formula is outside its stated domain.

Facts & Assumptions

Given: The scalar model and observation in the Example.

[A1]

AC is the Axiom of Choice assumed by the Gaussian model and normal-law definition. The Axiom of Choice.

[F1]

A linear-Gaussian state-space model permits positive-semidefinite covariances, including zero covariance. Linear-Gaussian state-space models with independent noises.

[F2]

Under the multivariate-normal convention, $N (0, 0)$ is the point mass at zero. Multivariate normal law, including singular covariance.

[F3]

The forward filter requires positive-definite innovation matrices and updates by the stated Kalman recursions. Exact linear-Gaussian Kalman filter.

[F4]

The ordinary RTS procedure requires a positive-definite one-step forecast covariance wherever it solves for a backward gain. Rauch–Tung–Striebel fixed-interval Gaussian smoother.

[F5]

A fixed-interval smoothing kernel is an everywhere probability kernel whose composition with the observation is an RCD of the state. Fixed-interval smoothing law in a linear-Gaussian model.

Verification

Given: $X_{0} \sim N (0, 0)$ , $q_{1} \sim N (0, 0)$ , $r_{1} \sim N (0, 1)$ , $X_{1} = X_{0} + q_{1}$ , $Y_{1} = r_{1}$ , and $y_{1} = 0$ .

1.1A1F1F2givenalgebra

By [F2], $X_{0} = 0$ and $q_{1} = 0$ almost surely, so $X_{1} = 0$ almost surely. The observation is $Y_{1} = r_{1}$ and the model covariances are positive semidefinite.

2.1F2F3F4step 1.1algebra

The forward recursion gives $P_{1 ∣ 0} = 1 \cdot 0 \cdot 1 + 0 = 0$ and $m_{1 ∣ 0} = 0$ . Since $H_{1} = 0$ and $R_{1} = 1$ , its innovation covariance is $S_{1} = 1 > 0$ , gain $K_{1} = 0$ , and filtered state remains $N (0, 0)$ . Thus the forward filter is admissible, while the RTS step would require the ordinary inverse of $P_{1 ∣ 0} = 0$ , which does not exist by [F4]. No pseudoinverse substitution follows from that algorithm.

3.1F5step 1.1step 2.1construct∎

Define $ψ_{0 ∣ 1} (y, \cdot) = δ_{0}$ for every $y \in R$ . For any Borel sets $A, B \subseteq R$ , $P (X_{0} \in A, Y_{1} \in B) = 1_{{0 \in A}} P (Y_{1} \in B) = \int_{B} δ_{0} (A) P_{Y_{1}} (d y)$ by step 1.1. Hence this everywhere-defined kernel satisfies the RCD identity in [F5]; it is the fixed-interval smoothing law even though the ordinary RTS recursion is undefined.

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Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 8, §8.2 (standard reference, not scraped)
Kody Law, Andrew M. Stuart, and Konstantinos Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 3, §3.1 (standard reference, not scraped)