Sequential Gaussian Learning and Filtering

Conceptual map

III-18.01Gaussian conditioning
III-18.02Rank-one precision updates
III-18.03Cumulative excitation
III-18.04Changing latent states
III-18.05Numerical stability

1. One linear Gaussian update

Gaussian learning adds precision in the observed direction. Posterior covariance records both the amount and orientation of accumulated information.

Let $\theta \sim N(m,P)$ and observe $y=h^{\top}\theta+\varepsilon$ with $\varepsilon \sim N(0,r)$ independent and $$r>0$$ . Completing the square gives

\[\begin{aligned}K=Ph/(h^{\top }Ph+r), \\ m^{+}=m+K(y-h^{\top }m), \\ P^{+}=P-Phh^{\top }P/(h^{\top }Ph+r).\end{aligned}\](1)

Proposition 1 · Rank-one precision addition

If $$P$$ is positive definite, then

\[(P^{+})^{-1}=P^{-1}+hh^{\top }/r, \quad \operatorname{det}(P)/\operatorname{det}(P^{+})=1+h^{\top }Ph/r.\](2)

Proof. Apply the Sherman–Morrison formula to the precision sum for the covariance identity and the matrix determinant lemma for the determinant ratio. Substitution gives (1). ∎

2. Directional learning and persistent excitation

For a fixed state with independent signals ( $h_{t},r_{t}$ ), sequential updating yields

\[P_{T}^{-1}=P_{0}^{-1}+\sum _{t=1}^{T}h_{t}h_{t}^{\top }/r_{t}.\](3)

With $P_{0}=I$ , unit noise, and $$T$$ repeated observations in direction $e_{1}$ , the posterior covariance is diag(1/(1+ $$T$$ ),1). The first coordinate is learned and the second remains at prior variance one. If the two coordinate directions are each observed $$T$$ times, covariance becomes diag(1/(1+ $$T$$ ),1/(1+ $$T$$ )).

Full directional learning requires the minimum eigenvalue of the cumulative information matrix to diverge. Growth of its trace alone permits unlearned directions.

3. A changing state requires prediction before updating

For $x_{t+1}=Ax_{t}+w_{t}$ and $y_{t}=Hx_{t}+v_{t}$ , with process covariance $$Q$$ and measurement covariance $$R$$ , the Kalman covariance recursion is

\[\begin{aligned}P_{t}^{-}=AP_{t-1}A^{\top }+Q, \\ P_{t}=P_{t}^{-}-P_{t}^{-}H^{\top }(HP_{t}^{-}H^{\top }+R)^{-1}HP_{t}^{-}.\end{aligned}\](4)

Failure case · Static precision accumulation in a changing-state model

Process noise injects new uncertainty between reports and persistence transports old uncertainty. Equation (3) applies to one fixed latent vector; it cannot replace the prediction–update recursion in (4).

4. Scalar Riccati fixed point and contraction

With scalar transition $$a$$ , process variance $$q$$ , direct observation, and measurement variance $$r$$ , the posterior-variance map is

\[F(P)=r(a^{2}P+q)/(a^{2}P+q+r).\](5)

Proposition 2 · Stable scalar contraction

If $\lvert a\rvert<1$ and $q\ge0$ , $$r>0$$ , then $$F$$ is a contraction on [0, $\infty$ ) with derivative bounded by $a^{2}$ . It has a unique nonnegative fixed point solving

\[a^{2}P^{2}+(q+r-ra^{2})P-rq=0.\](6)

Proof. Differentiation gives $F^{\prime}(P)=a^{2}r^{2}/(a^{2}P+q+r)^{2}\le a^{2}$ . The contraction theorem gives existence and uniqueness on the invariant interval [0, $$r$$ ]. Multiplying $$P=F(P)$$ gives (6). ∎

For $$a=0.9$$ , $$q=0.1$$ , and $$r=0.25$$ , the positive fixed point is 0.106825. The derivative there is about 0.26567.

5. Implementation, exercises, and sources

Use a Joseph-form covariance update when floating-point roundoff threatens symmetry or positive semidefiniteness. Track innovation residuals and their predicted variance, symmetrize only after recording the asymmetry error, and compare covariance and square-root implementations on long sequences.

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Exercises

Derive (1) by completing the square and verify both identities in (2).
Compare repeated and alternating measurement directions in the two-dimensional example for $$T=10$$ , 100, and 1,000.
Derive the positive root of (6) and reproduce the fixed-point and derivative values for the benchmark.

Partial solutions

1. Collect the quadratic terms in $\theta$ to obtain precision $P^{-1}+hh^{\top}/r$ ; the linear terms give the posterior mean. Sherman–Morrison converts the inverse precision to (1). 2. At $$T$$ observations, repeated-direction covariance is diag(1/(1+ $$T$$ ),1), while balanced directions give diag(1/(1+ $$T$$ ),1/(1+ $$T$$ )). The minimum cumulative-information eigenvalue distinguishes the two designs.

Thomas Kailath, Ali Sayed, and Babak Hassibi, Linear Estimation, Chapters 9–12.Gaussian updates and filtering.
Brian D. O. Anderson and John B. Moore, Optimal Filtering, Chapters 2–4.Riccati recursions and stability.
Roger Horn and Charles Johnson, Matrix Analysis.Rank-one inverse and determinant identities.

6. Audit checkpoint

State whether the latent state is fixed or changing, the prior, signal directions, noise covariances, excitation eigenvalues, prediction–update order, PSD residuals, innovation diagnostics, and Riccati convergence criterion.

7. Scope boundary

The chapter covers linear Gaussian updating, persistent excitation, Kalman covariance recursion, and a scalar contraction certificate. Nonlinear filtering, stochastic control, and continuous-time filtering lie outside scope.