Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks

Arnaud DoucetNando de FreitasKevin MurphyStuart Russell

article2000UAI1,531 citations

Proposes Rao-Blackwellised particle filtering for dynamic Bayesian networks, showing how to combine sequential Monte Carlo sampling with exact analytical filtering to achieve significantly higher estimation accuracy in complex applications like robot mapping and non-stationary regression.

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Real-time state estimation and tracking are essential across modern engineering disciplines, including robotics, computer vision, and speech processing. While standard particle filters provide robust simulation-based inference for non-linear and non-Gaussian dynamic systems, they suffer from significant computational inefficiency when applied to high-dimensional state spaces. Traditional exact filters, conversely, are computationally restricted to simpler linear or small discrete systems. The article demonstrates how exploiting the internal structure of dynamic Bayesian networks via Rao-Blackwellisation drastically improves estimation accuracy and computational efficiency by analytically marginalizing out tractable sub-components and sampling only the remaining variables.

The authors develop a generalized theoretical framework for Rao-Blackwellised particle filters and evaluate their performance on dynamic models, combining sequential importance sampling, selection steps, and Markov chain Monte Carlo diversity mechanisms. They validate this methodology across two practical domains: online non-stationary regression using adaptive neural networks and simultaneous robot localization and grid-based mapping under sensor and motion noise.

The core findings establish that analytical marginalization systematically reduces the variance of importance weights and posterior estimates compared to standard particle filtering. The greatest variance reduction occurs when the marginalized variables exhibit high conditional variance. In robot navigation benchmarks, the method achieved estimation accuracy matching exact Bayesian inference using as few as 50 to 100 particles, successfully maintaining spatial correlations where alternative factorized approximations failed. Furthermore, the convergence rate remains independent of the state-space dimension under mild regularity conditions.

These results provide a scalable foundation for deploying high-accuracy tracking in resource-constrained, real-time environments, substantially reducing computational costs and memory overhead. Decision-makers in autonomous navigation and adaptive signal processing should prioritize hybrid architectures that analytically solve linear or finite discrete states while reserving particle sampling for non-linear parent states. Future development must focus on automating the structural identification of analytically tractable variables, as current implementations rely on manual model partitioning.

University of California Berkeley University of Cambridge dynamic bayesian networks inference kalman filters Rao-Blackwellised particle filtering Dynamic Bayesian Networks sequential Monte Carlo methods

arXiv: 1301.3853

Paper: CONDENSATION—Conditional Density Propagation for Visual Tracking, MICHAEL ISARD et al. (1998). Introduces the Condensation algorithm, establishing the fundamental particle filtering / sequential Monte Carlo framework for dynamical state estimation upon which Rao-Blackwellisation is built.
Paper: An Introduction to the Kalman Filter, Greg Welch et al. (1995). Provides the foundational equations and principles of the Kalman filter, which serves as one of the exact sub-filters used to marginalize out conditionally linear Gaussian variables in Rao-Blackwellised particle filtering.
Paper: An Introduction to Variational Methods for Graphical Models, MICHAEL I. JORDAN et al. (1999). Presents exact and approximate inference methods for graphical models, including the junction tree algorithm leveraged directly by the source paper to marginalize tractable sub-structures.
Paper: A Tutorial on Learning with Bayesian Networks, David Heckerman (1999). Offers essential background on representing, learning, and performing probabilistic inference over dynamic and static Bayesian networks.
Paper: Estimating uncertain spatial relationships in robotics, Randall Smith et al. (1986). Establishes the stochastic mapping framework for mobile robot localization and map building, which forms one of the central real-world application domains demonstrated in the source paper.

Paper: The Unscented Particle Filter, Rudolph van der Merwe et al. (2000). Extends sequential Monte Carlo methods by integrating the unscented Kalman filter to generate advanced proposal distributions, directly complementing structural variance-reduction techniques like Rao-Blackwellisation.
Paper: Color-Based Probabilistic Tracking, P. Pérez et al. (2002). Applies sequential Monte Carlo particle filtering to continuous visual state estimation using multi-modal color distributions in dynamic environments.
Paper: Incremental Learning for Robust Visual Tracking, David A. Ross et al. (2008). Employs particle filtering alongside online adaptive subspace learning for visual tracking under complex, non-stationary conditions.

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Abstract

Particle filters (PFs) are powerful sampling-based inference/learning algorithms for dynamic Bayesian networks (DBNs). They allow us to treat, in a principled way, any type of probability distribution, nonlinearity and non-stationarity. They have appeared in several fields under such names as "condensation", "sequential Monte Carlo" and "survival of the fittest". In this paper, we show how we can exploit the structure of the DBN to increase the efficiency of particle filtering, using a technique known as Rao-Blackwellisation. Essentially, this samples some of the variables, and marginalizes out the rest exactly, using the Kalman filter, HMM filter, junction tree algorithm, or any other finite dimensional optimal filter. We show that Rao-Blackwellised particle filters (RBPFs) lead to more accurate estimates than standard PFs. We demonstrate RBPFs on two problems, namely non-stationary online regression with radial basis function networks and robot localization and map building. We also discuss other potential application areas and provide references to some finite dimensional optimal filters.

1 INTRODUCTION
2 PROBLEM FORMULATION
3 IMPORTANCE SAMPLING AND RAO-BLACKWELLISATION
4 RAO-BLACKWELLISED PARTICLE FILTERS
4.1 IMPLEMENTATION ISSUES
4.1.1 Sequential importance sampling
4.1.2 Selection step
4.1.3 MCMC step
4.2 CONVERGENCE RESULTS
5 EXAMPLES
5.1 ON-LINE REGRESSION AND MODEL SELECTION WITH NEURAL NETWORKS
5.2 ROBOT LOCALIZATION AND MAP BUILDING
5.3 CONCLUSIONS AND EXTENSIONS
References

Knowls

Knowl 1 — Rao-Blackwellised State Decomposition and Marginal Recursion in Dynamic Bayesian Networks
model/method
In a state-space model or dynamic Bayesian network (DBN) with hidden state $z_t$ and observations $y_{1:t} = \{y_1, \dots, y_t\}$ , suppose the hidden state vector $z_t$ is partitioned into two components $z_t = (r_t, x_t)$ such that the transition density factors as:

$p(z_t \mid z_{t-1}) = p(x_t \mid r_{1:t}, x_{t-1}) p(r_t \mid r_{t-1})$

and the conditional posterior distribution $p(x_{0:t} \mid y_{1:t}, r_{0:t})$ can be computed analytically in closed form using a finite-dimensional filter (such as the Kalman filter for conditionally linear-Gaussian models or the HMM filter / junction tree algorithm for discrete sub-networks).

By the chain rule of probability, the joint posterior distribution factors as:

$p(r_{0:t}, x_{0:t} \mid y_{1:t}) = p(x_{0:t} \mid y_{1:t}, r_{0:t}) p(r_{0:t} \mid y_{1:t})$

The marginal posterior distribution $p(r_{0:t} \mid y_{1:t})$ over the sampled state trajectory $r_{0:t}$ satisfies the recursive updating formula:

$p(r_{0:t} \mid y_{1:t}) = \frac{p(y_t \mid y_{1:t-1}, r_{0:t}) p(r_t \mid r_{t-1}) p(r_{0:t-1} \mid y_{1:t-1})}{p(y_t \mid y_{1:t-1})}$

where the conditional marginal likelihood is:

$p(y_t \mid y_{1:t-1}, r_{0:t}) = \int p(y_t \mid x_t, r_t) p(x_t \mid y_{1:t-1}, r_{0:t}) \, dx_t$

Rao-Blackwellised particle filtering (RBPF) uses sequential Monte Carlo to sample only the lower-dimensional trajectory $r_{0:t}$ , while maintaining the exact distribution $p(x_{0:t} \mid y_{1:t}, r_{0:t})$ analytically for each particle.
Knowl 2 — Generic Rao-Blackwellised Particle Filter Algorithm
algorithm
The generic Rao-Blackwellised particle filter (RBPF) recursively updates a particle approximation of the marginal posterior distribution $p(r_{0:t-1} \mid y_{1:t-1})$ to $p(r_{0:t} \mid y_{1:t})$ upon receiving the observation $y_t$ , while maintaining the analytical conditional filter state $p(x_t \mid y_{1:t}, r_{0:t}^{(i)})$ for each particle.
Input: Particles $\{r_{0:t-1}^{(i)}\}_{i=1}^N$ at time $t-1$ , observation $y_t$ , proposal distribution $q(r_t \mid r_{0:t-1}, y_{1:t})$
Output: Particles $\{r_{0:t}^{(i)}\}_{i=1}^N$ and analytical filtering states for $p(x_t \mid y_{1:t}, r_{0:t}^{(i)})$ at time $t$
for $i = 1$ to $N$ do
    Sample candidate state $\tilde{r}_t^{(i)} \sim q(r_t \mid r_{0:t-1}^{(i)}, y_{1:t})$
    Form extended trajectory $\tilde{r}_{0:t}^{(i)} = (\tilde{r}_t^{(i)}, r_{0:t-1}^{(i)})$
    Update analytical filter (e.g., Kalman filter or HMM filter) conditional on $\tilde{r}_{0:t}^{(i)}$ to compute $p(y_t \mid y_{1:t-1}, \tilde{r}_{0:t}^{(i)})$ and $p(x_t \mid y_{1:t}, \tilde{r}_{0:t}^{(i)})$
    Evaluate unnormalized importance weight:
         $w_t^{(i)} = \frac{p(y_t \mid y_{1:t-1}, \tilde{r}_{0:t}^{(i)}) p(\tilde{r}_t^{(i)} \mid r_{t-1}^{(i)})}{q(\tilde{r}_t^{(i)} \mid r_{0:t-1}^{(i)}, y_{1:t})}$
end for
for $i = 1$ to $N$ do
    Normalize importance weights:
         $\tilde{w}_t^{(i)} = \frac{w_t^{(i)}}{\sum_{j=1}^N w_t^{(j)}}$
end for
Resample (Selection step):
    Multiply or suppress particles $\{\tilde{r}_{0:t}^{(i)}\}_{i=1}^N$ according to normalized weights $\{\tilde{w}_t^{(i)}\}_{i=1}^N$ to obtain $N$ equally-weighted particles $\{\bar{r}_{0:t}^{(i)}\}_{i=1}^N$
MCMC step (optional):
    for $i = 1$ to $N$ do
        Apply a Markov transition kernel with invariant distribution $p(r_{0:t} \mid y_{1:t})$ to particle $\bar{r}_{0:t}^{(i)}$ to obtain $r_{0:t}^{(i)}$
    end for
return $\{r_{0:t}^{(i)}\}_{i=1}^N$
The selection step can be performed via multinomial, residual, or stratified resampling with computational complexity $\mathcal{O}(N)$ . The MCMC step mitigates sample impoverishment by adding particle diversity after resampling, without requiring kernel ergodicity.
Knowl 3 — Variance Reduction and Central Limit Theorem for Rao-Blackwellised Importance Sampling
theoretical result
Let $z_{0:t} = (r_{0:t}, x_{0:t})$ denote the hidden state trajectories, $y_{1:t}$ the observed sequence, and $f_t(r_{0:t}, x_{0:t})$ a test function whose posterior expectation $I(f_t) = \mathbb{E}_{p(r_{0:t}, x_{0:t} \mid y_{1:t})}[f_t(r_{0:t}, x_{0:t})]$ is estimated using proposal distribution $q(r_{0:t}, x_{0:t} \mid y_{1:t}) = q(r_{0:t} \mid y_{1:t}) p(x_{0:t} \mid y_{1:t}, r_{0:t})$ .

Define the standard importance sampling estimator:

$\bar{I}_N(f_t) = \frac{\sum_{i=1}^N f_t(r_{0:t}^{(i)}, x_{0:t}^{(i)}) w(r_{0:t}^{(i)}, x_{0:t}^{(i)})}{\sum_{i=1}^N w(r_{0:t}^{(i)}, x_{0:t}^{(i)})}$

where $w(r_{0:t}, x_{0:t}) = \frac{p(r_{0:t}, x_{0:t} \mid y_{1:t})}{q(r_{0:t}, x_{0:t} \mid y_{1:t})}$ , and the Rao-Blackwellised importance sampling estimator:

$\bar{I}_N^{\mathrm{RB}}(f_t) = \frac{\sum_{i=1}^N \mathbb{E}_{p(x_{0:t} \mid y_{1:t}, r_{0:t}^{(i)})}[f_t(r_{0:t}^{(i)}, x_{0:t})] w(r_{0:t}^{(i)})}{\sum_{i=1}^N w(r_{0:t}^{(i)})}$

where $w(r_{0:t}) = \frac{p(r_{0:t} \mid y_{1:t})}{q(r_{0:t} \mid y_{1:t})} = \int w(r_{0:t}, x_{0:t}) p(x_{0:t} \mid y_{1:t}, r_{0:t}) \, dx_{0:t}$ .

For any sample size $N$ , the variances satisfy:

$\mathrm{var}_{q(r_{0:t} \mid y_{1:t})}(w(r_{0:t})) \le \mathrm{var}_{q(r_{0:t}, x_{0:t} \mid y_{1:t})}(w(r_{0:t}, x_{0:t}))$ $\mathrm{var}_{q(r_{0:t} \mid y_{1:t})}\left(\frac{1}{N}\sum_{i=1}^N \mathbb{E}_{p}[f_t] w(r_{0:t}^{(i)})\right) \le \mathrm{var}_{q(r_{0:t}, x_{0:t} \mid y_{1:t})}\left(\frac{1}{N}\sum_{i=1}^N f_t(r_{0:t}^{(i)}, x_{0:t}^{(i)}) w(r_{0:t}^{(i)}, x_{0:t}^{(i)})\right)$ $\mathrm{var}_{q(r_{0:t} \mid y_{1:t})}\left(\frac{1}{N}\sum_{i=1}^N w(r_{0:t}^{(i)})\right) \le \mathrm{var}_{q(r_{0:t}, x_{0:t} \mid y_{1:t})}\left(\frac{1}{N}\sum_{i=1}^N w(r_{0:t}^{(i)}, x_{0:t}^{(i)})\right)$

Furthermore, both estimators satisfy a Central Limit Theorem as $N \to \infty$ :

$\sqrt{N}\left(\bar{I}_N(f_t) - I(f_t)\right) \Longrightarrow \mathcal{N}(0, \sigma_1^2)$ $\sqrt{N}\left(\bar{I}_N^{\mathrm{RB}}(f_t) - I(f_t)\right) \Longrightarrow \mathcal{N}(0, \sigma_2^2)$

with $\sigma_2^2 \le \sigma_1^2$ . The asymptotic variance difference is:

$\sigma_1^2 - \sigma_2^2 = \mathbb{E}_{q(r_{0:t} \mid y_{1:t})}\left[\mathrm{var}_{q(x_{0:t} \mid y_{1:t}, r_{0:t})}\left((f_t(r_{0:t}, x_{0:t}) - I(f_t)) w(r_{0:t}, x_{0:t})\right)\right] \ge 0$

which shows that Rao-Blackwellisation provides large variance reduction when the conditional variance of $x_{0:t}$ given $r_{0:t}$ and $y_{1:t}$ is high.
Knowl 4 — Optimal Importance Proposal Distribution and Incremental Weights for RBPF
theoretical result
When importance distributions of the sequential form:

$q(r_{0:t} \mid y_{1:t}) = q(r_0) \prod_{k=1}^t q(r_k \mid y_{1:k}, r_{0:k-1})$

are used in a Rao-Blackwellised particle filter, the proposal distribution $q(r_t \mid r_{0:t-1}, y_{1:t})$ that minimizes the variance of the importance weights conditional upon $r_{0:t-1}$ and $y_{1:t}$ is the one-step true marginal posterior distribution:

$p(r_t \mid r_{0:t-1}, y_{1:t}) = \frac{p(y_t \mid y_{1:t-1}, r_{0:t}) p(r_t \mid r_{t-1})}{p(y_t \mid y_{1:t-1}, r_{0:t-1})}$

The corresponding incremental importance weight $w_t$ , which is independent of the sampled value $r_t$ , is given by:

$w_t = p(y_t \mid y_{1:t-1}, r_{0:t-1}) = \int p(y_t \mid y_{1:t-1}, r_{0:t}) p(r_t \mid r_{t-1}) \, dr_t$

where $p(y_t \mid y_{1:t-1}, r_{0:t})$ is the predictive likelihood obtained by marginalizing over $x_t$ via the exact filter conditional on $r_{0:t}$ .
Knowl 5 — Monotonic Increase of Unconditional Importance Weight Variance in SIS
theoretical result
For sequential importance sampling (SIS) schemes where the proposal distribution factorizes sequentially according to:

$q(r_{0:t} \mid y_{1:t}) = q(r_0) \prod_{k=1}^t q(r_k \mid y_{1:k}, r_{0:k-1})$

the unconditional variance of the importance weights $w(r_{0:t}) = \frac{p(r_{0:t} \mid y_{1:t})}{q(r_{0:t} \mid y_{1:t})}$ , treating the observations $y_{1:t}$ and the state trajectory $r_{0:t}$ as random variables, increases monotonically over time $t$ .

This stochastic variance growth causes SIS weight degeneracy: in practice, after a small number of iterations, the normalized weight of a single particle approaches 1 while all other particle weights vanish, necessitating selection/resampling mechanisms.
Knowl 6 — Mean Square Convergence Bound for Rao-Blackwellised Particle Filters
theoretical result
Let $B(\mathbb{R}^{n_r(t+1)})$ denote the space of bounded, Borel measurable functions on the trajectory space $\mathbb{R}^{n_r(t+1)}$ of $r_{0:t}$ , with supremum norm $\|f_t\| = \sup_{r} |f_t(r)|$ .

If the incremental importance weights $w_t$ are upper bounded and a valid selection scheme is used (such as multinomial, residual, or stratified sampling where $\mathbb{E}[N_i] = N \tilde{w}_t^{(i)}$ ), then for all $t \ge 0$ , there exists a constant $c_t < \infty$ independent of the particle count $N$ such that for any $f_t \in B(\mathbb{R}^{n_r(t+1)})$ :

$\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^N f_t(r_{0:t}^{(i)}) - \int f_t(r_{0:t}) p(r_{0:t} \mid y_{1:t}) \, dr_{0:t}\right)^2\right] \le c_t \frac{\|f_t\|^2}{N}$

where the expectation is taken with respect to the random simulation operations introduced by the particle filter.

The convergence rate $\mathcal{O}(1/\sqrt{N})$ in $L_2$ norm is independent of the dimension of the state space, although the constant $c_t$ typically grows exponentially with time $t$ for general continuous state-space models.
Knowl 7 — RBPF for Online Regression and Model Selection with Dynamic RBF Neural Networks
model/method
In non-stationary online regression using radial basis function (RBF) neural networks, the function output at time $t$ is modeled as a mixture of $k_t$ basis functions plus linear terms and additive Gaussian observation noise $\mathcal{N}(0, \sigma_t^2)$ :

$y_t = \sum_{j=1}^{k_t} \alpha_{j,t} \phi(\|x_t - \mu_{j,t}\|) + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, \sigma_t^2)$

where the number of basis functions $k_t$ , basis centers $\mu_t = \{\mu_{1,t}, \dots, \mu_{k_t,t}\}$ , regression coefficients $\alpha_t = (\alpha_{1,t}, \dots, \alpha_{k_t,t})^T$ , and noise variance $\sigma_t^2$ are all time-varying latent variables.

Because the network output is linear in $\alpha_t$ conditional on the basis parameters $(k_t, \mu_t)$ and variance $\sigma_t^2$ , the system forms a conditionally linear Gaussian state-space model (CLGSSM). The RBPF algorithm:
1. Samples the non-linear discrete and continuous latent variables $r_t = (k_t, \mu_t, \sigma_t^2)$ using sequential Monte Carlo combined with reversible jump MCMC transitions to handle dimension changes in $k_t$ .
2. Marginalizes and estimates the linear weight vector $x_t = \alpha_t$ analytically and exactly for each particle using standard Kalman filters.
Knowl 8 — Rao-Blackwellised Particle Filtering for Concurrent Robot Localization and Map Learning
model/method
For a mobile robot navigating a discrete grid of $N_L$ cells with $N_C$ possible cell colors, concurrent localization and map learning is formulated as Bayesian state estimation over the robot location $L_t \in \{1, \dots, N_L\}$ and the grid map $M_t = (M_t(1), \dots, M_t(N_L))$ , where each $M_t(i) \in \{1, \dots, N_C\}$ . The full joint state space has size $\mathcal{O}(N_L N_C^{N_L})$ .

The observation $Y_t = f(M_t(L_t))$ is a noisy sensor readout of the color of cell $L_t$ . Conditional on the entire robot trajectory $L_{1:t}$ , all grid map cells $M_t(i)$ are mutually independent:

$P(M_t(1), \dots, M_t(N_L) \mid y_{1:t}, L_{1:t}) = \prod_{i=1}^{N_L} P(M_t(i) \mid y_{1:t}, L_{1:t})$

The RBPF samples only the low-dimensional robot trajectory $L_{1:t}$ via a particle filter, and tracks the exact discrete marginal posterior distributions $P(M_t(i) \mid y_{1:t}, L_{1:t})$ independently for each cell $i \in \{1, \dots, N_L\}$ analytically per particle using discrete HMM filter updates.
Knowl 9 — Empirical Comparison of RBPF, Exact Bayesian Inference, and Boyen-Koller Algorithm for Robot Localization and Mapping
empirical result
In a 1D corridor of 8 cells where a robot moves back and forth and gets stuck in cell 4 for two consecutive time steps on the outgoing leg, RBPF using $N=50$ particles was compared against exact Bayesian inference and the fully-factorized Boyen-Koller (BK) algorithm:
- Exact inference: The robot maintains multimodal uncertainty while moving through cells 4 to 8, and upon reaching the end of the corridor at step 9, successfully relocalizes and resolves past position and map uncertainty.
- RBPF (50 particles): Closely replicates the exact posterior distribution across all time steps, successfully resolving trajectory and map state at step 9.
- Fully-factorized BK: Represents the belief state as a fully-factored product $P(L_t, M_t(1), \dots, M_t(N_L) \mid y_{1:t}) \approx P(L_t \mid y_{1:t}) \prod_{i=1}^{N_L} P(M_t(i) \mid y_{1:t})$ . It fails and gets confused because discarding the correlations between map cells and robot location prevents successful relocalization.
In a 2D grid world of size $10 \times 10$ (state space size $\mathcal{O}(2^{100})$ ) with a $3 \times 3$ local neighborhood sensor, RBPF successfully learns the map using 100 particles.

Coverage note — None was omitted; the extracted knowls comprehensively cover the theoretical formulation of Rao-Blackwellised particle filtering, variance reduction theorems, optimal proposal construction, SIS weight degeneracy, convergence bounds, the generic RBPF algorithm, and its application and empirical validation on neural network regression and robot SLAM.

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Citation

MLA

Doucet, A., et al. “Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks”. arXiv, 2013, http://arxiv.org/abs/1301.3853v1.

APA

Doucet, A., Freitas, N. de ., Murphy, K., & Russell, S. (2013). Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks. arXiv. http://arxiv.org/abs/1301.3853v1

Chicago

Doucet, A., N. de . Freitas, K. Murphy, and S. Russell. 2013. “Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks”. arXiv. http://arxiv.org/abs/1301.3853v1.

Harvard

Doucet, A. et al. (2013) “Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1301.3853v1.

Vancouver

1. Doucet A, Freitas N de, Murphy K, Russell S (2013) Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks. arXiv

BibTeX

@article{doucet2013rao,
  title = {Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks},
  author = {Doucet, Arnaud and Freitas, Nando de and Murphy, Kevin and Russell, Stuart},
  year = {2013},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1301.3853v1},
  eprint = {1301.3853}
}

Metadata:arXiv

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Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks

Abstract

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Table of Contents

Knowls

Knowl 1 — Rao-Blackwellised State Decomposition and Marginal Recursion in Dynamic Bayesian Networks

Knowl 2 — Generic Rao-Blackwellised Particle Filter Algorithm

Knowl 3 — Variance Reduction and Central Limit Theorem for Rao-Blackwellised Importance Sampling

Knowl 4 — Optimal Importance Proposal Distribution and Incremental Weights for RBPF

Knowl 5 — Monotonic Increase of Unconditional Importance Weight Variance in SIS

Knowl 6 — Mean Square Convergence Bound for Rao-Blackwellised Particle Filters

Knowl 7 — RBPF for Online Regression and Model Selection with Dynamic RBF Neural Networks

Knowl 8 — Rao-Blackwellised Particle Filtering for Concurrent Robot Localization and Map Learning

Knowl 9 — Empirical Comparison of RBPF, Exact Bayesian Inference, and Boyen-Koller Algorithm for Robot Localization and Mapping

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