Gaussian Smoothing in CLG Models

• Gaussian smoothing in CLG models estimates full state trajectories by combining analytical Kalman updates with Monte Carlo methods for nonlinear components.
• Key methodologies include Rao-Blackwellized particle smoothers, hybrid two-filter approaches, and turbo smoothing that balance computational cost and accuracy.
• Advanced techniques like Gaussian mixture reduction and particle rejuvenation significantly reduce computational complexity while improving RMSE by 20–40% or more.

A conditionally linear Gaussian (CLG) model describes stochastic dynamical systems where the state evolution and observation equations are linear and Gaussian in a subset of states, conditioned on the remaining (potentially nonlinear or discrete) states. Smoothing in this context refers to estimating the full trajectory of hidden states using all available observations over a fixed interval. Advanced CLG smoothing methods exploit conditional structure to combine analytical (Kalman-style) updates for linear subsystems and Monte Carlo or mixture techniques for the nonlinear/discrete components. This article provides a comprehensive treatment of Gaussian smoothing in CLG models, encompassing both the theoretical formulation and state-of-the-art smoothing algorithms.

1. Structure of Conditionally Linear Gaussian Models

A discrete-time CLG state-space model partitions the hidden state $x_k = [x_k^{(L)}; \, x_k^{(N)}]$ into a linear substate $x_k^{(L)}$ evolving via linear-Gaussian dynamics conditioned on the nonlinear or discrete substate $x_k^{(N)}$:

$$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$

$$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$

where $w_k^{(L)}$ and $w_k^{(N)}$ are mutually independent zero-mean Gaussians. The nonlinear/“regime” component $x_k^{(N)}$ may be discrete or continuous; for example, in a Jump Markov Linear System (JMLS), the regime $r_k$ is discrete and $x_k$ is fully linear-Gaussian conditioned on $x_k^{(L)}$0 (Balenzuela et al., 2020, Vitetta et al., 2017, Lindsten et al., 2015, Nguyen et al., 2017).

2. Smoothing Formulation in CLG Models

Given observations $x_k^{(L)}$1, the fixed-interval smoothing objective is to compute $x_k^{(L)}$2 or the joint $x_k^{(L)}$3. The CLG structure enables marginalization of the linear subspace conditioned on a realization of the nonlinear/discrete trajectory, reducing variance and computational complexity—a technique known as Rao-Blackwellization (Lindsten et al., 2015, Vitetta et al., 2017, Nguyen et al., 2017).

The general solution exploits the factorization: $x_k^{(L)}$4 where $x_k^{(L)}$5 is available from forward filtering and $x_k^{(L)}$6 from backward information filtering. In fully linear Gaussian cases, these are propagated analytically; for general CLG models, Rao-Blackwellized particle smoothers and mixture-based approaches are employed.

3. Two-Filter and Hybrid Smoothing Algorithms

Several classes of algorithms target the CLG smoothing problem:

3.1. Exact Two-Filter Gaussian Mixture Smoother (JMLS, Discrete Regime CLG)

In Jump Markov Linear Systems, the smoother applies Kalman-style forward filtering and an information-form backward filter. The hybrid state $x_k^{(L)}$7 admits a closed-form Gaussian mixture parameterization:

• Forward recursion: $x_k^{(L)}$8 is a mixture over regimes.
• Backward information recursion: $x_k^{(L)}$9 is propagated as a Gaussian sum in information form.
• Smoother fusion: The product yields a mixture with component count growing as $x_k^{(N)}$0 (regime multiplicity).

Without approximation, mixture cardinality becomes computationally intractable (Balenzuela et al., 2020).

3.2. Rao-Blackwellized Particle Smoothing (General CLG)

For nonlinear regime chains or high-dimensional continuous $x_k^{(N)}$1, Rao-Blackwellized particle smoothers (RBPS) are dominant:

• Represent trajectories of $x_k^{(N)}$2 with a particle system.
• For each particle, propagate analytic Kalman filtering/smoothing for $x_k^{(N)}$3.
• Backward smoothing is implemented via forward-filter-backward-simulation (FFBS) or two-filter decompositions, merging forward and backward messages to obtain smoothed estimates and sample trajectories (Lindsten et al., 2015, Vitetta et al., 2017, Nguyen et al., 2017).

3.3. Variants: Turbo Smoothing and Particle Rejuvenation

Turbo smoothing leverages parallel concatenation of forward and backward Bayesian filters on a factor graph, iteratively exchanging pseudo-measurement messages between extended Kalman filter (EKF) and particle filter (PF) modules in forward and backward passes. This framework can achieve tight complexity-accuracy trade-offs and significant memory savings (Vitetta et al., 2019). Particle rejuvenation introduces additional backward sampling or resampling steps to mitigate path degeneracy in the discrete state, enlarging the support of the backward trajectories and reducing estimation variance (Nguyen et al., 2017).

4. Gaussian Mixture Explosion and Reduction

In mixture-based (e.g., JMLS) smoothers, the number of Gaussian mixture terms explodes exponentially with time. Specifically, $x_k^{(N)}$4 at time $x_k^{(N)}$5, leading to $x_k^{(N)}$6 computational and storage complexity (Balenzuela et al., 2020).

To make smoothing tractable:

• Gaussian mixture reduction merges or prunes components. The preferred method (KL-merge) merges the pair of Gaussians with minimal increase in KL divergence, using explicit moment-matching. This controls computational cost at the price of approximation error, tunable by the maximum mixture size $x_k^{(N)}$7 (Balenzuela et al., 2020).
• Trade-off: Larger $x_k^{(N)}$8 decreases smoothing error and approaches the exact solution; smaller $x_k^{(N)}$9 increases speed at the expense of approximation accuracy.

5. Computational Complexity and Accuracy Considerations

The complexity of exact smoothers is exponential in the length $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$0 for mixture-based approaches on discrete regimes. With mixture reduction capped at $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$1 components per mode, cost becomes $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$2 (Balenzuela et al., 2020).

For Rao-Blackwellized particle smoothers handling continuous nonlinear states:

• Forward filtering: $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$3 for $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$4 particles and linear subspace of dimension $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$5.
• Backward smoothing: Comparable cost per sample trajectory.
• Total complexity: $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$6 for marginal smoothing, or $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$7 for $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$8 joint trajectories (Vitetta et al., 2017, Lindsten et al., 2015).

Empirical results indicate that, for moderate mixture or particle budgets, smoothed estimates can achieve RMSE improvements of $$\begin{aligned} x_{k+1}^{(L)} &= A_k^{(L)}(x_k^{(N)})\,x_k^{(L)} + f_k^{(L)}(x_k^{(N)}) + w_k^{(L)} \ x_{k+1}^{(N)} &= f_k^{(N)}(x_k^{(N)}) + A_k^{(N)}(x_k^{(N)})\,x_k^{(L)} + w_k^{(N)} \end{aligned}$$9 over filtering. Particle rejuvenation further reduces MSE for the discrete regime by up to $$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$0 in some settings (Nguyen et al., 2017). Turbo smoothing demonstrates accuracy comparable to RBPS at reduced computation and memory (Vitetta et al., 2019).

6. Continuous-Time Linear Gaussian Smoothing

In continuous-time settings, the smoothing problem admits a pathwise characterization. The smoothing error evolves as an Ornstein-Uhlenbeck (OU) process driven by a backward Riccati equation for the error covariance, enabling exact Monte Carlo sampling of smoothed trajectories. Both Kalman-Bucy filtering and Rauch-Tung-Striebel (RTS) smoothing appear as marginals in this framework, and the Bryson-Frazier information-form smoother arises naturally as a corollary (Kurisaki, 5 Jan 2026).

Key features include:

• The smoothed state trajectory is Gaussian with mean given by the backward-propagated filter and covariance determined by the OU smoothing error.
• Pathwise sampling of conditioned state processes is immediate once the Riccati equation is solved.

7. Practical Aspects and Algorithmic Summary

The following table summarizes key CLG smoothing approaches:

Method	Linearized States	Nonlinear/Discrete States	Complexity
Gaussian Mixture (JMLS) (Balenzuela et al., 2020)	Kalman	Markov chain (regime)	$$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$1 (exact), $$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$2 with reduction
RB Particle Smoother (Lindsten et al., 2015, Vitetta et al., 2017)	Kalman	Particle representation	$$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$3 (filter), $$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$4 (M samples)
Turbo Smoother (Vitetta et al., 2019)	EKF	Particle (PF)	$$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$5
Particle Rejuvenation (Nguyen et al., 2017)	Kalman	Particle (with resampling)	$$y_k = h_k(x_k^{(N)}) + B_k(x_k^{(N)})\,x_k^{(L)} + e_k$$6 per step with full support
Pathwise OU (continuous-time) (Kurisaki, 5 Jan 2026)	Kalman-Bucy	— (fully linear-Gaussian)	Solve Riccati + pathwise sampling

Algorithm selection in practice is governed by trade-offs between model structure, required accuracy, memory and computational budget, and the size and nature of nonlinear/discrete subsystems.

• "A New Smoothing Algorithm for Jump Markov Linear Systems" (Balenzuela et al., 2020)
• "Rao-Blackwellized Particle Smoothing as Message Passing" (Vitetta et al., 2017)
• "Rao-Blackwellized particle smoothers for conditionally linear Gaussian models" (Lindsten et al., 2015)
• "Particle rejuvenation of Rao-Blackwellized Sequential Monte Carlo smoothers for Conditionally Linear and Gaussian models" (Nguyen et al., 2017)
• "A New Smoothing Technique based on the Parallel Concatenation of Forward/Backward Bayesian Filters: Turbo Smoothing" (Vitetta et al., 2019)
• "Pathwise Representation of the Smoothing Distribution in Continuous-Time Linear Gaussian Models" (Kurisaki, 5 Jan 2026)