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A two-mode linear-Gaussian Rao–Blackwellized filter

Example

Let $Z_{0}$ be uniform on ${0, 1}$ and let $Z_{1} = Z_{0}$ . Conditional on $Z_{0} = z$ , take $U_{0} \sim N (0, 1)$ and $U_{1} = a_{z} U_{0} + ξ_{1}, Y_{1} = U_{1} + ε_{1},$ where $a_{0} = 0$ , $a_{1} = 1$ , and $ξ_{1}, ε_{1}$ are independent $N (0, 1)$ variables, independent of $U_{0}$ . Observe $Y_{1} = 0$ . The exact conditional posterior is the two-component mixture $L (U_{1}, Z_{1} ∣ Y_{1} = 0) = \frac{\sqrt{3}}{\sqrt{3} + \sqrt{2}} N (0, 1 / 2) \otimes δ_{0} + \frac{\sqrt{2}}{\sqrt{3} + \sqrt{2}} N (0, 2 / 3) \otimes δ_{1} .$ These are the component moments and mode weights that the Rao–Blackwellized filter represents analytically; a finite particle cloud still approximates the mode mixture.

Facts & Assumptions

Given: The finite model and observation in the Example.

[F1]

A conditionally linear-Gaussian state-space model has a Markov nonlinear component and a linear-Gaussian component conditional on its nonlinear path. Conditionally linear-Gaussian state-space model

[A1]

AC is inherited from the Gaussian-law and finite-horizon conditional-kernel conventions. The Axiom of Choice

[F2]

Conditional on a nonlinear path, the Kalman filter gives the exact Gaussian state filter and the marginal observation density. The Kalman filter is exact conditional on a nonlinear path

[F3]

The Rao–Blackwellized particle filter retains those conditional Gaussian components and weights nonlinear paths by their conditional observation densities. Rao–Blackwellized particle filter

[F4]

A positive-definite one-dimensional Gaussian covariance gives density $(2 π S)^{- 1 / 2} \exp (- v^{2} / (2 S))$ . A positive-definite multivariate normal law has its Gaussian density

Verification

Given: $Z_{0} \sim Bernoulli (1 / 2)$ , $Z_{1} = Z_{0}$ , the conditional Gaussian model stated in the Example, and $Y_{1} = 0$ .

1.1A1F1F2given

The state and observation noises are independent centered Gaussians, the conditional initial covariance is $1$ , the process covariance is $1$ , and the observation covariance is $1 > 0$ . Thus the model satisfies [F1] and the exact conditional Kalman interface [F2].

1.2F2givenalgebra

Conditional on $Z_{1} = z$ , prediction gives $m_{1 ∣ 0} = 0$ and $P_{1 ∣ 0} = a_{z}^{2} + 1$ . Hence $P_{1 ∣ 0} = 1$ for $z = 0$ and $P_{1 ∣ 0} = 2$ for $z = 1$ . The innovation is zero and the innovation variances are $S_{0} = 2$ and $S_{1} = 3$ .

2.1A1F1F2F4step 1.2algebra

By [F4], the two conditional observation densities at zero are $ℓ_{0} = (4 π)^{- 1 / 2}$ and $ℓ_{1} = (6 π)^{- 1 / 2}$ . Multiplying by the equal prior masses and normalizing gives $P (Z_{1} = 0 ∣ Y_{1} = 0) = \sqrt{3} / (\sqrt{3} + \sqrt{2})$ and $P (Z_{1} = 1 ∣ Y_{1} = 0) = \sqrt{2} / (\sqrt{3} + \sqrt{2})$ .

3.1A1F2F3step 2.1algebra∎

The Kalman covariance update is $P_{1 ∣ 1} = P_{1 ∣ 0} R / (P_{1 ∣ 0} + R)$ here, so the two conditional posterior variances are $1 / 2$ and $2 / 3$ ; both conditional means remain zero. Mixing these exact conditional Gaussians with the mode probabilities from step 2.1 gives the displayed posterior. The filter in [F3] uses these same per-mode likelihoods and Gaussian components; only its finite particle approximation to the mode weights is random.

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Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 7, §7.5 (standard reference, not scraped)
Fredrik Lindsten, Pete Bunch, Simo Särkkä, Thomas B. Schön, and Simon J. Godsill, Rao-Blackwellized particle smoothers for conditionally linear Gaussian models, §3.2 (standard reference, not scraped)