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Exact conditional law for the finite-modal fluid filter

Physical scope

Conditional result for the adopted finite-horizon, finite-modal, linear-Gaussian velocity model and specified weighted spatial sensors in a fixed nonrelativistic Cartesian inertial chart. It identifies the exact conditional law within that model; it does not assert that an actual flow, sensor, or noise source follows the adopted model.

Statement

Assume the Axiom of Choice (AC), meaning every family of nonempty sets has a choice function, and the finite-modal observation model described here. Let $I \subseteq R$ be a nonempty open interval, let $E$ be a three-dimensional oriented Euclidean affine space, and use an orientation-preserving Cartesian inertial chart $x : E \to R^{3}$ with absolute time $t$ . Let $D \subset R^{3}$ be bounded and Lebesgue measurable with $0 < ∣ D ∣ < \infty$ , and set $H = L^{2} (D; R^{3})$ , with its norm topology and Borel sigma-algebra. Fix an integer horizon $T \geq 0$ and times $t_{0} < t_{1} < \dots < t_{T}$ in $I$ , positive integer dimensions $m_{k}, p_{k}$ , finite-dimensional modal maps $Φ_{k} (a) = \sum_{j = 1}^{m_{k}} a_{j} ϕ_{k, j}$ with $ϕ_{k, j} \in H$ , and weighted sensor maps $L_{k}$ given by $(L_{k} v)_{i} = \int_{D} v (x) \cdot ψ_{k, i} (x) d^{3} x$ for weights $ψ_{k, i} \in H$ . Assign $ϕ_{k, j}$ dimensionless units, coefficients and velocity fields units m s $^{- 1}$ , sensor weights units m $^{- 3}$ , and observations and sensor noise units m s $^{- 1}$ ; then $F_{k}$ and $H_{k}$ are dimensionless. The coefficient observation matrices satisfy $[H_{k}]_{i j} = (L_{k} ϕ_{k, j})_{i}$ , so $L_{k} (Φ_{k} (a)) = H_{k} a$ .

On a common probability space $(Ω, F, P)$ , let $a_{0} \sim N_{m_{0}} (μ_{0}, P_{0})$ , $q_{k} \sim N_{m_{k}} (0, Q_{k})$ , and $r_{k} \sim N_{p_{k}} (0, R_{k})$ be mutually independent, with symmetric positive-semidefinite covariance matrices, where $C ⪰ 0$ means $z^{T} C z \geq 0$ for every vector $z$ of matching dimension and $N_{d} (μ, C)$ denotes the $d$ -dimensional Gaussian law with mean $μ$ and covariance $C$ , including singular $C$ . Assume $a_{k} = F_{k} a_{k - 1} + q_{k}, y_{k} = H_{k} a_{k} + r_{k} (1 \leq k \leq T) .$ Initialize $m_{0 ∣ 0} = μ_{0}$ and $P_{0 ∣ 0} = P_{0}$ . For $1 \leq k \leq T$ , define $m_{k ∣ k - 1} = F_{k} m_{k - 1 ∣ k - 1}, P_{k ∣ k - 1} = F_{k} P_{k - 1 ∣ k - 1} F_{k}^{T} + Q_{k},$ $v_{k} = y_{k} - H_{k} m_{k ∣ k - 1}, S_{k} = H_{k} P_{k ∣ k - 1} H_{k}^{T} + R_{k} .$ Here $μ_{0} \in R^{m_{0}}$ , $F_{k} \in R^{m_{k} \times m_{k - 1}}$ , $H_{k} \in R^{p_{k} \times m_{k}}$ , $P_{0} \in R^{m_{0} \times m_{0}}$ , $Q_{k} \in R^{m_{k} \times m_{k}}$ , and $R_{k} \in R^{p_{k} \times p_{k}}$ . Assume every $S_{k}$ to be inverted is positive definite, meaning $z^{T} S_{k} z > 0$ for every nonzero $z \in R^{p_{k}}$ . Set $K_{k} = P_{k ∣ k - 1} H_{k}^{T} S_{k}^{- 1}, m_{k ∣ k} = m_{k ∣ k - 1} + K_{k} v_{k}, P_{k ∣ k} = P_{k ∣ k - 1} - K_{k} S_{k} K_{k}^{T} .$ Write $y_{1 : k} = (y_{1}, \dots, y_{k})$ , let $y_{1 : 0}$ be the singleton empty history, and let $ν_{k}$ be the probability law of $y_{1 : k}$ . For every $0 \leq k \leq T$ , the Gaussian kernel $π_{k} (y_{1 : k}, \cdot) = N_{m_{k}} (m_{k ∣ k}, P_{k ∣ k})$ is a probability kernel from histories to coefficient-state laws and satisfies the regular-conditional-law identities: for every Borel state set $A \subseteq R^{m_{k}}$ and Borel history set $B \subseteq \prod_{j = 1}^{k} R^{p_{j}}$ , $P (a_{k} \in A, y_{1 : k} \in B) = \int_{B} π_{k} (y, A) ν_{k} (d y)$ . For $k = 0$ , the history space is a singleton and $π_{0} = N_{m_{0}} (μ_{0}, P_{0})$ . If $U_{k} = Φ_{k} (a_{k})$ is the represented velocity field, its regular conditional law is the pushforward kernel $ρ_{k} (y_{1 : k}, C) = π_{k} (y_{1 : k}, Φ_{k}^{- 1} (C)) (C \in B (H)) .$ Both kernels are defined at every observation history, including histories with probability zero; this selects a version and asserts no pointwise uniqueness there. The result is exact for the declared finite model and does not assert that an untruncated or nonlinear fluid flow, an actual sensor, or actual noise follows that model.

Statement references

Classical frames, primitive quantities and SI units

Finite-second-moment random velocity sensor law

The L2 velocity sensor is well-defined

Finite modal velocity fields inherit finite second moments

Finite coefficient observation and innovation interface

Exact filtering for the declared linear Gaussian model

Filtering, predictive, and smoothing laws in a finite-horizon state-space model

Finite-modal fluid observation and noise model

The Axiom of Choice

Law, Stuart, and Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 2 §2.4 and Chapter 4 §4.1

Särkkä, Bayesian Filtering and Smoothing, Chapter 4 §§4.1–4.3

Facts & Assumptions

Given: AC; the finite-horizon coefficient and observation equations; mutually independent Gaussian initial, process, and observation vectors with positive-semidefinite covariances; positive-definite innovation covariances; finite modal maps; and weighted spatial sensors as in the Statement.

[A1]

AC is the axiom that every family of nonempty sets has a choice function; it is assumed for the Gaussian-law construction. The Axiom of Choice

[F1]

The physical framework uses absolute time and an adapted Cartesian inertial chart on Newtonian spacetime; its Eulerian velocity has units m s $^{- 1}$ . Classical frames, primitive quantities and SI units

[F2]

The sensor is the weighted spatial integral $S_{ψ} (v) = \int_{D} v \cdot ψ d^{3} x$ on almost-everywhere classes in $H = L^{2} (D; R^{3})$ ; it is not a point evaluation. Finite-second-moment random velocity sensor law

[F3]

Each sensor pairing is finite, representative-independent, linear, and bounded on $H$ . The L2 velocity sensor is well-defined

[F4]

Under AC, if a measurable coefficient vector has finite second moment, the finite modal lemma gives a strongly measurable field and a finite-second-moment pushforward law; its Statement does not claim modal-map continuity. Finite modal velocity fields inherit finite second moments

[F5]

The physical postulate adopts the finite-dimensional coefficient state equation, weighted sensor observation equation, and independent Gaussian noise assumptions; it does not assert empirical adequacy. Finite-modal fluid observation and noise model

[F6]

The finite coefficient interface accepts finite-dimensional real vector spaces with specified ordered bases, linear observation maps, and coefficient equations $a_{k} = F_{k} a_{k - 1} + q_{k}$ , $y_{k} = H_{k} a_{k} + r_{k}$ with mutually independent Gaussian initial and noise vectors and positive-semidefinite covariances, including singular ones. Finite coefficient observation and innovation interface

[F7]

Under AC and positive-definite solved innovations, the declared finite coefficient model has the stated exact Gaussian conditional law at every observation history, including singular state and noise covariances. Exact filtering for the declared linear Gaussian model

[F8]

A filtering kernel is an everywhere-defined probability kernel whose composition with the observation history is a regular conditional distribution of the state; it satisfies the conditional event identities on every Borel history set. Filtering, predictive, and smoothing laws in a finite-horizon state-space model

[F9]

For an ordered basis $(b_{k, 1}, \dots, b_{k, m_{k}})$ and linear observation map $M_{k}$ , the coefficient interface defines its observation matrix by $[H_{k}]_{i j} = [M_{k} (b_{k, j})]_{i}$ . Finite coefficient observation and innovation interface

[F10]

The space $H = L^{2} (D; R^{3})$ has its stated norm, which satisfies the norm triangle inequality. Finite-second-moment random velocity sensor law

Proof

Given: The hypotheses and notation of the Statement.

1.1F1F2F3F5givenalgebra

In the fixed Cartesian inertial chart of [F1], for each $k$ and coefficient vector $a$ , the weighted sensor maps in [F2] and postulate [F5] use finite linearity of the pairings [F3] to give $(L_{k} (Φ_{k} (a)))_{i} = S_{ψ_{k, i}} ⁣ (\sum_{j = 1}^{m_{k}} a_{j} ϕ_{k, j}) = \sum_{j = 1}^{m_{k}} a_{j} S_{ψ_{k, i}} (ϕ_{k, j}) = \sum_{j = 1}^{m_{k}} [H_{k}]_{i j} a_{j} .$ All terms are defined because $ϕ_{k, j}, ψ_{k, i} \in H$ and [F3] makes each pairing finite. Thus $L_{k} (U_{k}) = H_{k} a_{k}$ , identifying the sensor signal in the stated observation equation.

1.2F10givenalgebra

For each $k$ , put $C_{k} = \sum_{j = 1}^{m_{k}} ∥ ϕ_{k, j} ∥_{H}$ . For $a, b \in R^{m_{k}}$ , the norm triangle inequality [F10] gives $∥ Φ_{k} (a) - Φ_{k} (b) ∥_{H} \leq \sum_{j = 1}^{m_{k}} ∣ a_{j} - b_{j} ∣ ∥ ϕ_{k, j} ∥_{H} \leq C_{k} ∣ a - b ∣,$ because each coordinate satisfies $∣ a_{j} - b_{j} ∣ \leq ∣ a - b ∣$ . Thus $Φ_{k}$ is continuous. This estimate is derived here; the finite-second-moment lemma [F4] does not assert continuity.

2.1A1F5F6F7F9givenstep 1.1constructalgebra

To instantiate [F6], take the abstract trial space $V_{k} = R^{m_{k}}$ with its standard ordered basis $(e_{k, 1}, \dots, e_{k, m_{k}})$ , where $e_{k, j}$ has coordinate $j$ equal to one and all other coordinates zero. This list is linearly independent and spans $V_{k}$ , regardless of any linear dependence among the physical fields $ϕ_{k, j}$ . For $1 \leq k \leq T$ , give this abstract space the linear observation map $M_{k} : V_{k} \to R^{p_{k}}$ defined by $M_{k} (b) = L_{k} (Φ_{k} (b))$ . Its coefficient observation matrix has entries $[M_{k} (e_{k, j})]_{i} = [L_{k} (ϕ_{k, j})]_{i} = [H_{k}]_{i j}$ by [F9] and step 1.1. Its trial vector $\sum_{j} a_{k, j} e_{k, j}$ is exactly $a_{k}$ ; the physical velocity is obtained separately by the synthesis map $Φ_{k}$ . Use the given $F_{k}$ , Gaussian initial state and noises, and covariances on this abstract coefficient realization. By [F5] and [F6], their dimensions, mutual independence, and covariance conventions match the interface. The explicit AC assumption [A1] and positive-definite $S_{k}$ hypotheses supply its remaining admissibility conditions, so [F7] applies to this coefficient model without requiring $Φ_{k}$ to be injective.

3.1F4F7F8step 2.1step 1.2givenalgebra∎

Apply [F7] to obtain the coefficient regular conditional kernel $π_{k}$ at every history. By step 1.2, $Φ_{k}^{- 1} (C)$ is Borel for every $C \in B (H)$ . For each history $y$ , $C \mapsto π_{k} (y, Φ_{k}^{- 1} (C))$ is a probability measure, and for each Borel $C$ its value is measurable in $y$ by the kernel property; hence $ρ_{k}$ is a probability kernel. For any Borel observation-history set $B$ , the defining conditional-law identity in [F8] therefore gives $P (U_{k} \in C, y_{1 : k} \in B) = P (a_{k} \in Φ_{k}^{- 1} (C), y_{1 : k} \in B) = \int_{B} π_{k} (y, Φ_{k}^{- 1} (C)) ν_{k} (d y) .$ The last integrand is $ρ_{k} (y, C)$ , so the conditional event identity holds for $U_{k}$ and $ρ_{k}$ is its regular conditional law by [F8]. Because $π_{k}$ is defined at every history, so is its pushforward $ρ_{k}$ , including on null histories.

Exact conditional law for the finite-modal fluid filter

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources