Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 2 results · all verified · 2 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 2 also cleared it.

Fixed-interval Gaussian smoothing and RTS recursion — Examples

1 · Prerequisites

2 · Summary

Examples

The first example carries out both passes exactly in a two-transition scalar random walk and identifies the smoothed marginals at the selected data. The second separates forecast-covariance invertibility from Kalman-filter admissibility: the filter remains defined with a zero forecast covariance, and the smoothing law exists, but the ordinary RTS inverse does not. It does not claim a pseudoinverse formula.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Exact two-time scalar Rauch–Tung–Striebel smoothing calculation

Example

Consider the scalar random walk $X_{0} \sim N (0, 1)$ , $X_{j} = X_{j - 1} + q_{j}$ , $Y_{j} = X_{j} + r_{j}$ for $j = 1, 2$ , with mutually independent $q_{j}, r_{j} \sim N (0, 1)$ . Fix the data $y_{1} = 1, y_{2} = 2$ . Every innovation and one-step forecast covariance is positive, so both forward filtering and the RTS backward recursion are defined.

Facts & Assumptions

Given: The scalar model and observations in the Example.

[A1]

AC is used by the multivariate-normal model and conditional-law suppliers. The Axiom of Choice.

[F1]

The model has mutually independent Gaussian initial, process, and measurement variables with the displayed state and observation equations. Linear-Gaussian state-space models with independent noises.

[F2]

The Kalman filter computes the prediction, innovation, gain, and update by the stated recursions. Exact linear-Gaussian Kalman filter.

[F3]

The RTS procedure initializes at the final filter and applies the backward gain, mean, and covariance updates. Rauch–Tung–Striebel fixed-interval Gaussian smoother.

[F4]

Under positive-definite innovation and forecast covariances, the RTS outputs are fixed-interval conditional-law versions. The Rauch–Tung–Striebel recursion gives fixed-interval Gaussian smoothing laws.

Verification

Given: $X_{0} \sim N (0, 1)$ , unit process and measurement variances, unit transition and observation coefficients, and $(y_{1}, y_{2}) = (1, 2)$ .

1.1A1F1F2givenalgebra

At time $1$ , the filter predicts $m_{1 ∣ 0} = 0$ and $P_{1 ∣ 0} = 2$ . Hence $S_{1} = 3$ , $v_{1} = 1$ , and $K_{1} = 2 / 3$ , so $m_{1 ∣ 1} = 2 / 3$ and $P_{1 ∣ 1} = 2 - (2 / 3) 3 (2 / 3) = 2 / 3$ . In particular $S_{1} > 0$ .

2.1F2step 1.1algebra

At time $2$ , the prediction is $m_{2 ∣ 1} = 2 / 3$ and $P_{2 ∣ 1} = 5 / 3$ . Thus $S_{2} = 8 / 3$ , $v_{2} = 4 / 3$ , and $K_{2} = 5 / 8$ , giving $m_{2 ∣ 2} = 2 / 3 + (5 / 8) (4 / 3) = 3 / 2$ and $P_{2 ∣ 2} = 5 / 3 - (5 / 8) (8 / 3) (5 / 8) = 5 / 8$ . Both $S_{2}$ and $P_{2 ∣ 1}$ are positive.

3.1F3F4step 1.1step 2.1algebra∎

The first backward gain is $C_{1} = (2 / 3) / (5 / 3) = 2 / 5$ . Therefore $m_{1 ∣ 2} = 2 / 3 + (2 / 5) (3 / 2 - 2 / 3) = 1$ and $P_{1 ∣ 2} = 2 / 3 + (2 / 5) (5 / 8 - 5 / 3) (2 / 5) = 1 / 2$ . The next gain is $C_{0} = 1 / 2$ , so $m_{0 ∣ 2} = 0 + (1 / 2) (1 - 0) = 1 / 2$ and $P_{0 ∣ 2} = 1 + (1 / 2) (1 / 2 - 2) (1 / 2) = 5 / 8$ . The theorem [F4] identifies the resulting smoothing kernels as $N (1 / 2, 5 / 8)$ for $X_{0}$ , $N (1, 1 / 2)$ for $X_{1}$ , and $N (3 / 2, 5 / 8)$ for $X_{2}$ at the selected data history.

ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

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.

Fixed-interval Gaussian smoothing and RTS recursion — Examples

1 · Prerequisites

2 · Summary

Examples

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Exact two-time scalar Rauch–Tung–Striebel smoothing calculation

Example

Facts & Assumptions

Verification

Singular forecast covariance lies outside the ordinary RTS inverse domain

Example

Facts & Assumptions

Verification

Sources