Pith. sign in

REVIEW 3 major objections 5 minor 77 references

Grid Particle Gibbs with Ancestor Sampling for State-Space Models

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A coarse hidden Markov model on a deterministic grid steers SMC particles into high-posterior regions, giving particle Gibbs with ancestor sampling lower estimation error per unit computation time.

desk verdict Solid incremental method with a plausible efficiency gain on simulated SV, but the abstract overclaims and the reporting needs more rigor before it can be recommended unconditionally. read the letter →

arxiv 2501.03395 v1 pith:6ILILCBW submitted 2025-01-06 stat.CO

Mary Llewellyn , Ruth King , Víctor Elvira , Gordon Ross This is my paper · ORCID
classification stat.CO MSC 62M0565C0562F15
keywords particleGibbswithancestorsamplingstate-spacemodelshiddenMarkovmodelapproximationdeterministicgridproposalsampleimpoverishmentregime-switchingsequentialMonteCarloBayesianinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the most expensive part of particle Gibbs with ancestor sampling—the sequential Monte Carlo sweeps that propose new latent-state paths—can be made much cheaper by proposing from a coarse, deterministic grid instead of from the model's raw transition noise. The authors partition the state space into equally sized intervals, evaluate the model's transition and observation densities at the interval midpoints, and use the resulting approximate hidden Markov model to sample each particle from a grid cell that the approximate filter favors. Because the proposals land in regions of high posterior mass, fewer particles are wasted in resampling, so the same accuracy is reached with far fewer particles. On a simulated regime-switching stochastic volatility model and on a post-COVID tourism demand model, they report lower estimation error per unit computation time than standard particle Gibbs with ancestor sampling, with the best results coming from a hybrid that applies the grid only to states with sufficiently narrow posterior support.

What carries the argument

The engine of the method is the deterministic-grid HMM approximation. The state space is divided into $N$ equally sized intervals $I^{(1)}, \dots, I^{(N)}$ with two infinite outer cells; each interval has midpoint $\xi^{(n)}$ and length $L^{(n)}$. Midpoint integration turns the state-space model into a discrete HMM with $\hat P(B_1 = n \mid \theta) \propto L^{(n)} p(\xi^{(n)} \mid \theta)$, $\hat P(B_t = n \mid B_{t-1} = k, \theta) \propto L^{(n)}L^{(k)} p(\xi^{(n)} \mid \xi^{(k)}, \theta)$, and $\hat p(y_t \mid B_t = n, \theta) \propto L^{(n)} p(y_t \mid \xi^{(n)}, \theta)$, each normalized over cells. From these, the approximate optimal importance distribution $\hat P(B_t = n \mid y_t, B_{t-1} = k, \theta) \propto \hat P(B_t = n \mid B_{t-1} = k, \theta)\, \hat p(y_t \mid B_t = n, \theta)$ is formed only for grid cells occupied at the previous time point, so the per-iteration cost drops from $O(N^2 T)$ to $O(N^2 + N \sum_t \tilde N_{t-1})$. This object carries the argument: it directs particles toward high-posterior regions while keeping the computation near-linear in the number of occupied cells.

What would settle it

Run GPGAS and conditional PGAS on the regime-switching stochastic volatility model of Section 4.1, but let a new regime appear after the pilot iterations whose latent-state values lie outside the pilot-tuned grid range $[-12, 12]$. If the frozen equal-sized grid cannot direct particles into the new high-posterior region and GPGAS's error per unit computation time is no better than conditional PGAS, the approximation-quality assumption, and with it the claimed computational gain, fails. The paper already reports this failure mode for the wide-support trend state $u_t$ in the tourism model (Table 1).

Watch

Extended reading notes

Core claim

The central claim is that a cheap hidden Markov model (HMM) approximation of a general state-space model can serve as the proposal inside particle Gibbs with ancestor sampling (PGAS, the MCMC scheme that updates the whole latent-state path with a conditional sequential Monte Carlo sweep), and that this removes much of the sample impoverishment—the loss of particle diversity from resampling—that forces PGAS to use many particles. The approximation partitions the continuous state space into $N$ equal cells with midpoints $\xi^{(n)}$ and lengths $L^{(n)}$, then replaces the intractable integrals in the HMM's initial, transition, and observation probabilities by one-point midpoint quadrature: $\hat P(B_t = n \mid B_{t-1} = k, \theta) \propto L^{(n)}L^{(k)} p(\xi^{(n)} \mid \xi^{(k)}, \theta)$ and $\hat p(y_t \mid B_t = n, \theta) \propto L^{(n)} p(y_t \mid \xi^{(n)}, \theta)$. At each sequential Monte Carlo step a grid cell index is drawn from the discrete filtered distribution $\hat P(B_t = n \mid y_t, B_{t-1} = k, \theta) \propto \hat P(B_t = n \mid B_{t-1} = k, \theta)\, \hat p(y_t \mid B_t = n, \theta)$, and a continuous particle is then drawn uniformly inside that cell, or from a truncated Gaussian in the outer cells. This tractable proposal mimics the optimal importance distribution and reduces sample impoverishment; the paper reports 11-50% fewer states left unupdated in the SMC sweep and clear improvements in mean relative absolute error per unit time across the regime-switching examples.

Load-bearing premise

The load-bearing premise is that one coarse, fixed, equal-sized grid, with probabilities computed by one-point midpoint integration and frozen after a short pilot run, continues to approximate the posterior propagation well enough to steer particles into high-probability regions for every time step of the whole MCMC run; the paper's own tourism result shows this can fail when a latent state's posterior is spread over a wide range, where pure GPGAS is less efficient than conditional PGAS.

Editorial extensions

If this is right

  • GPGAS can reach a given level of latent-state estimation error with substantially fewer particles than PGAS: in the stochastic volatility experiments, the average number of states not updated in the SMC sweep falls by 11-50%, so the accuracy gain is not bought with extra particles.
  • The per-iteration cost scales with the number of occupied grid cells rather than with $N^2$ per time point, so adding grid resolution to cover the posterior is much cheaper than adding particles.
  • On the tourism demand model, the efficient configuration is hybrid: apply GPGAS to the regime labels and the narrow-support level state, and keep standard PGAS for the wide-support trend state; this raises the effective sample size to roughly 5800 versus 3100-3300 for pure PGAS variants, with more iterations completed per hour.
  • Because the grid only changes the SMC proposal and not the target distribution, the resulting MCMC samples still target the joint posterior $p(x_{1:T}, \theta \mid y_{1:T})$, so the gains carry over without changing the interpretation of the posterior estimates.
  • The method's advantage is largest when the posterior mass is concentrated enough for an equal-sized grid to cover it with moderate $N$; the paper documents that pure GPGAS loses to conditional PGAS on the wide-support trend state $u_t$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I expect the grid-proposal idea to transfer beyond MCMC to plain particle filtering with fixed parameters, where the HMM matrices are computed once rather than per iteration, making the overhead even smaller; the paper itself notes filtering as a natural extension.
  • The equal-sized grid is a convenience rather than a necessity: a quantile-informed grid that allocates more cells where the posterior has mass could remove the wide-support failure mode the paper reports for $u_t$, at the cost of some simplicity.
  • Because the HMM is frozen after pilot iterations, GPGAS should be sensitive to posterior movement later in the run; a testable extension is to compare the fixed version against one that recomputes the grid on a slow schedule, weighing robustness against wall-clock gains.
  • The reported 11-50% reductions in unupdated states suggest the gain scales with switching frequency; a natural experiment is to sweep the regime-persistence parameter $\pi_{11}$ and measure when the GPGAS advantage over PGAS disappears.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes Grid Particle Gibbs with Ancestor Sampling (GPGAS), an extension of particle Gibbs with ancestor sampling for general state-space models. The method partitions the continuous state space into a fixed, equal-sized grid and constructs an approximate hidden Markov model using midpoint-integrated transition and observation probabilities (Section 3.1). The approximate HMM is used to form SMC importance proposal distributions that first sample a grid cell and then draw a continuous particle within that cell (Section 3.2). The resulting algorithm, Algorithm 6, is applied to a simulated regime-switching stochastic volatility model and to a real post-COVID tourism demand model for Edinburgh. The paper claims substantial computational gains over standard PGAS and reports favorable error-versus-time results in both settings, with the real-data study carried out on a hybrid GPGAS+PGAS update scheme.

Significance. If the efficiency gains are reproducible, GPGAS is a useful contribution to the toolkit for fitting nonlinear and non-Gaussian state-space models, especially regime-switching models where sample impoverishment is severe. Strengths of the paper include the clear algorithmic presentation, the practical computational strategies listed in Section 3.4, and the use of high-particle PGAS ground-truth runs in both the simulation and the real-data study. However, the empirical evidence is more qualified than the abstract suggests: the headline real-data gains come from a hybrid that applies GPGAS to only part of the state vector, and the simulated study is reported without uncertainty bars and without a stated convergence-exclusion criterion. The central efficiency claim is plausible but is not yet established at the level of generality claimed.

major comments (3)
  1. [Section 4.1.2 / Figures 3-4] The captions of Figures 3 and 4 state that non-convergent implementations are excluded, but no convergence criterion is defined anywhere in the manuscript. If exclusion is based on the error relative to the ground truth, the comparison can be biased in favor of whichever algorithm survives the filter. Please report the exact diagnostic used, the number of excluded runs per configuration, and ideally show results with all runs included as a sensitivity check. This is load-bearing because the simulated-study efficiency claim rests on these plots.
  2. [Section 4.2.2 / Table 1] The real-data results show that the pure GPGAS algorithm is not more efficient than conditional PGAS: GPGAS has MRAE 0.044 and ESS 1100, compared with conditional PGAS MRAE 0.039 and ESS 3300. The best result in Table 1, GPGAS+PGAS, replaces the GPGAS update for u_{1:T} with a standard PGAS update. The abstract and the discussion nevertheless claim substantial computational gains of the proposed method without this qualification. The claim should be restricted to the setting where the fixed equal-spaced grid is a reasonable approximation, or to the hybrid algorithm, and the paper should provide a practical criterion (for example, the ratio of posterior range to transition scale) for when pure GPGAS is expected to be efficient. Without such a criterion, the generality of the headline claim is not established.
  3. [Section 4.1.2 and Section 4.2.2] The efficiency comparisons are reported as single points per configuration from 10 runs, with no Monte Carlo error bars, and each comparison uses a single ground-truth run (around 89 hours for the simulation and 97 hours for the tourism data). Since the central claim is an efficiency claim, the absence of uncertainty quantification makes it difficult to assess whether the observed differences are systematic or within Monte Carlo noise. Please add error bars or interval estimates for the reported MRAE/ESS values and, if feasible, quantify the Monte Carlo error of the ground-truth estimates.
minor comments (5)
  1. [Section 3.3] In the displayed weight formula, the denominator for the time-1 weight is written as q(x_m^1, B_1 = b_m^1 | y_t, θ); it should condition on y_1 rather than y_t.
  2. [Section 2.1 / Equation (3)] The denominator of the normalized weight is typeset as 'Pm_{k=1} w_{1:t}' which should be a summation symbol; please correct the typesetting and clarify the notation for the sets w_{1:t}^{1:M} and W_{1:t}^{1:M}.
  3. [Appendix B] The text contains duplicated words: 'Figure 6 shows the the results' and 'Figure 7 shows the the results' should be 'shows the results'.
  4. [Table 1] The caption says the metric is the mean relative absolute error (MRAE) of the posterior predictive mean and variance, then reports 'average RRMSE' for credible intervals; please define MRAE and RRMSE and make the notation consistent, since RRMSE is not defined elsewhere.
  5. [Section 3.4(a) / Sections 4.1.1 and 4.2.1] The HMM approximation is fixed after a pilot-chosen iteration tilde_s (2000 in the simulation, 1500 in the tourism example) using posterior means, but no sensitivity analysis is provided for this choice or for the grid ranges. A short robustness discussion or table would help readers judge how sensitive the method is to these tuning decisions.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the GPGAS efficiency claim is validated empirically against standard PGAS and a high-particle ground truth, not derived from the paper's own inputs.

full rationale

The paper's central derivation is the construction of grid-based HMM proposal densities (Sections 3.1 and 3.2, Equations (8) and (9)) and their insertion into conditional SMC with ancestor sampling (Algorithm 6). This construction is fully specified in the paper; the grid midpoint approximation is an explicit approximation choice, not a hidden input. The claimed computational gains are assessed by comparing GPGAS with standard PGAS (Lindsten et al., 2014) and with a PGAS ground truth using M = 5000 particles (Sections 4.1.2 and 4.2.2), i.e., against externally defined baselines. Self-citations to Llewellyn et al. (2023a) supply the HMM-grid idea and a competing PMPMH algorithm, and Llewellyn et al. (2023b) supplies the tourism data; none of these citations is used as the evidence for the efficiency conclusion. The only notable weakness is that the paper's own Table 1 shows pure GPGAS underperforming conditional PGAS on the u_t state, which qualifies the abstract's 'substantial computational gains' claim, but that is an approximation-quality and correctness issue rather than a circularity: no equation or fitted parameter is equal by construction to the claimed result.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The paper's efficiency claim depends on standard PGAS validity plus several tuning choices. The main domain assumption is that midpoint-integrated HMM probabilities on a fixed equal-sized grid produce a proposal that tracks the posterior; the tourism example shows this assumption can fail for wide-range states. No new physical entities are introduced.

free parameters (6)
  • Number of grid cells N = 10, 25, 50, 100 (SV); 25 to 400 (tourism)
    Chosen by pilot tuning; larger N improves HMM approximation quality at higher cost.
  • Finite grid cell ranges = [-12,12] for SV; [-5,20] for mu; [-300,1000] for u
    Set by pilot tuning to cover high posterior mass; affects proposal coverage and accuracy.
  • Within-cell truncated Gaussian variance = 2.4 (SV), 2.5 (mu), 130 (u)
    Set to 10% of the finite grid cell range; an ad hoc choice not derived from theory.
  • HMM fixing iteration tilde_s = 2000 (SV); 1500 (tourism)
    HMM matrices are frozen using posterior means from pilot samples after this iteration; a practical approximation.
  • Resampling threshold psi = 25% of ESS (SV); 50% of ESS (tourism)
    Called case-optimal; selected by pilot runs and affects SMC variance and computational cost.
  • Random walk proposal scales = Various, e.g., 0.15, 1, 0.01, 1e-6
    Tuning parameters for Metropolis-within-Gibbs updates of parameters without closed-form conditionals.
assumptions (6)
  • standard math Particle Gibbs and PGAS correctness: CSMC targets an extended distribution that admits p(x1:T|y1:T,theta) as a marginal, so proposed latent states are always accepted and the chain converges to p(x1:T,theta|y1:T).
    Relied on in Sections 2.2 and 2.3; established by Andrieu et al. (2010) and Chopin and Singh (2015), not re-proven here.
  • domain assumption Deterministic midpoint integration over grid cells gives a sufficiently accurate HMM approximation of the SSM.
    Used in Section 3.1, Equation (8); if the grid is too coarse or the range too small, the proposal misses posterior mass, as the tourism u_t case shows.
  • domain assumption The state space is one-dimensional for the grid construction; higher-dimensional cases require conditional updates or other extensions.
    Stated in Section 3.1; Section 4.2 avoids a 3D grid by updating (s,mu) and u separately.
  • ad hoc to paper Fixing the HMM approximation after a pilot-chosen iteration using posterior means keeps the proposal adequate for later MCMC iterations.
    Section 3.4(a); a pragmatic choice with no diagnostic or theoretical guarantee.
  • standard math Within-cell importance distributions can be chosen independent of theta and data without breaking SMC validity.
    Section 3.2; valid as long as the proposal is normalized and has support covering the state space, but efficiency is heuristic.
  • domain assumption Regime labels s_t are discrete and can be integrated exactly in the joint HMM transition probabilities.
    Used in Sections 4.1.1 and 4.2.1; exact for the discrete regime process.
invented entities (1)
  • Grid cell index process B_t
    purpose: Auxiliary discrete variable used to construct the HMM-based importance proposal and map sampled cells to continuous particles.
    An algorithmic device introduced in Section 3.2; it is not a physical entity and has no external falsifiable prediction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Grid Particle Gibbs with Ancestor Sampling for State-Space Models." pith.science (2026). https://pith.science/paper/6ILILCBW

@misc{pith2026250103395,
  author       = {Pith},
  title        = {Pith review of: Grid Particle Gibbs with Ancestor Sampling for State-Space Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ILILCBW}},
  note         = {Machine review of arXiv:2501.03395}
}
read the original abstract

We consider the challenge of estimating the model parameters and latent states of general state-space models within a Bayesian framework. We extend the commonly applied particle Gibbs framework by proposing an efficient particle generation scheme for the latent states. The approach efficiently samples particles using an approximate hidden Markov model (HMM) representation of the general state-space model via a deterministic grid on the state space. We refer to the approach as the grid particle Gibbs with ancestor sampling algorithm. We discuss several computational and practical aspects of the algorithm in detail and highlight further computational adjustments that improve the efficiency of the algorithm. The efficiency of the approach is investigated via challenging regime-switching models, including a post-COVID tourism demand model, and we demonstrate substantial computational gains compared to previous particle Gibbs with ancestor sampling methods.

Figures

Figures reproduced from arXiv: 2501.03395 by the authors.

Figure 1
Figure 1. Partition of the state space into equally-sized grid cells, the same for each time [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Simulated data from the stochastic volatility model: [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Mean relative absolute errors versus computational time for the (a) posterior [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Mean relative absolute errors versus computational time for the (a) posterior [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: Plots of (a) the aggregate weekly revenue of hotels in Edinburgh in Great British [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: Mean relative absolute errors for the (a) posterior mean and (b) posterior variance [PITH_FULL_IMAGE:figures/full_fig_p044_6.png]
Figure 7
Figure 7. Figure 7: Mean relative absolute errors for the (a) posterior mean and (b) posterior variance [PITH_FULL_IMAGE:figures/full_fig_p045_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages

  1. [1]

    Andrieu, C., Davy, M., and Doucet, A. (2003). Efficient particle filtering for jump Markov systems. Application to time-varying autoregressions. IEEE Transactions on Signal Processing , 51(7):1762--1770

    work page 2003

  2. [2]

    Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B , 72(3):269--342

    work page 2010

  3. [3]

    B., Cole, D., Empacher, F., Gryba, R., King, A

    Auger-M \'e th \'e , M., Newman, K. B., Cole, D., Empacher, F., Gryba, R., King, A. A., Leos-Barajas, V., Mills Flemming, J., Nielsen, A., Petris, G., and Thomas, L. (2021). A guide to state--space modeling of ecological time series. Ecological Monographs , 91(4):1--38

    work page 2021

  4. [4]

    Bar‐Shalom, Y., Li, X., and Kirubarajan, T. (2002). Estimation with Applications to Tracking and Navigation : Theory , Algorithms and Software . Wiley

    work page 2002

  5. [5]

    and Di Cairano, S

    Berntorp, K. and Di Cairano, S. (2017). Particle Gibbs with ancestor sampling for identification of tire - friction parameters . 20th IFAC World Congress , 50(1):14849--14854

    work page 2017

  6. [6]

    Bishop, C. (1998). Bayesian PCA . In Proceedings of the 11th International Conference on Neural Information Processing Systems , pages 382 -- 388

    work page 1998

  7. [7]

    and King, R

    Borowska, A. and King, R. (2023). Semi-complete data augmentation for efficient state-space model fitting. Journal of Computational and Graphical Statistics , 32(1):19--35

    work page 2023

  8. [8]

    and Elvira, V

    Branchini, N. and Elvira, V. (2021). Optimized auxiliary particle filters: Adapting mixture proposals via convex optimization. In Proceedings of the Thirty - Seventh Conference on Uncertainty in Artificial Intelligence , pages 1289--1299

    work page 2021

Show all 77 references
  1. [9]

    Bucy, R. S. and Senne, K. D. (1971). Digital synthesis of non-linear filters. Automatica , 7(3):287--298

    1971

  2. [10]

    Carpenter, J., Cliffordy, P., and Fearnhead, P. (2000). An improved particle filter for non -linear problems . IEE Proceedings - Radar, Sonar and Navigation , 146(1):2--7

    2000

  3. [11]

    Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models . Biometrika , 81(3):541--553

    1994

  4. [12]

    and Papaspiliopoulos, O

    Chopin, N. and Papaspiliopoulos, O. (2020). An Introduction to Sequential Monte Carlo . Springer

    2020

  5. [13]

    and Singh, S

    Chopin, N. and Singh, S. S. (2015). On particle Gibbs sampling. Bernoulli , 21(3):1855--1883

    2015

  6. [14]

    and Míguez, J

    Crisan, D. and Míguez, J. (2018). Nested particle filters for online parameter estimation in discrete-time state-space Markov models. Bernoulli , 24(4):3039--3086

    2018

  7. [15]

    and Hastings, A

    de Valpine, P. and Hastings, A. (2002). Fitting population models incorporating process noise and observation error. Ecological Monographs , 72(1):57--76

    2002

  8. [16]

    and Robin, S

    Donnet, S. and Robin, S. (2017). Using deterministic approximations to accelerate SMC for posterior sampling. arXiv . https://doi.org/10.48550/arXiv.1707.07971

    2017 doi arXiv

  9. [17]

    Doucet, A., Gordon, N., and Krishnamurthy, V. (2001). Particle filters for state estimation of jump Markov linear systems. IEEE Transactions on Signal Processing , 49(3):613--624

    2001

  10. [18]

    and Johansen, A

    Doucet, A. and Johansen, A. (2009). A tutorial on particle filtering and smoothing: Fifteen years later. volume 12. Oxford University Press

    2009

  11. [19]

    and Boers, Y

    Driessen, H. and Boers, Y. (2004). An efficient particle filter for jump M arkov nonlinear systems. IEE Target Tracking 2004: Algorithms and Applications , pages 19--22

    2004

  12. [20]

    Duník, J., Soták, M., Veselý, M., Straka, O., and Hawkinson, W. (2019). Design of Rao – Blackwellized point-mass filter with application in terrain aided navigation. IEEE Transactions on Aerospace and Electronic Systems , 55(1):251--272

    2019

  13. [21]

    and Koopman, S

    Durbin, J. and Koopman, S. J. (2012). Time Series Analysis by State Space Methods : Second Edition . Oxford University Press

    2012

  14. [22]

    El-Laham, Y., Yang, L., Djuric, P., and Bugallo, M. (2021). Particle filtering under general regime switching. 28th European Signal Processing Conference , pages 2378--2382

    2021

  15. [23]

    F., and Djurić, P

    Elvira, V., Martino, L., Bugallo, M. F., and Djurić, P. M. (2018). In search for improved auxiliary particle filters. In 26th European Signal Processing Conference , pages 1637--1641

    2018

  16. [24]

    Elvira, V., Martino, L., Luengo, D., and Bugallo, M. F. (2019). Generalized multiple importance sampling. Statistical Science , 34(1):129--155

    2019

  17. [25]

    Fearnhead, P. (2011). MCMC for state - space models . In Brooks, S., Gelman, A., Jones, G. L. J., and Meng, X.-L., editors, Handbook of Markov Chain Monte Carlo , pages 513--529. Chapman & Hall

    2011

  18. [26]

    Fr\" u hwirth-Schnatter, S. (2004). Efficient Bayesian parameter estimation. In Harvey, A., Koopman, S. J., and Shephard, N., editors, State Space and Unobserved Component Models: Theory and Applications , pages 123--151. Cambridge University Press

    2004

  19. [27]

    Frühwirth-Schnatter, S. (2001). Fully Bayesian analysis of switching Gaussian state space models. Annals of the Institute of Statistical Mathematics , 53(1):31--49

    2001

  20. [28]

    J., Salmond, D

    Gordon, N. J., Salmond, D. J., and Smith, A. F. M. (1993). Novel approach to nonlinear/non- Gaussian Bayesian state estimation. In IEE Proceedings - Radar, Sonar and Navigation , volume 140, pages 107--113

    1993

  21. [29]

    and Hartl, T

    Haimerl, P. and Hartl, T. (2023). Modeling COVID -19 infection rates by regime-switching unobserved components models. Econometrics , 11(2)

    2023

  22. [30]

    Hamilton, J. D. (1989). A new approach to the economics analysis of nonstationary time series and the business cycle. Econometrica , 57(2):357--384

    1989

  23. [31]

    He, M., Das, P., Hotan, G., and Purdon, P. L. (2023). Switching state-space modeling of neural signal dynamics. PLOS Computational Biology , 19(8)

    2023

  24. [32]

    B., and Ninness, B

    Henriksen, S., Wills, A., Schön, T. B., and Ninness, B. (2012). Parallel implementation of particle MCMC methods on a GPU . In 16th IFAC Symposium on System Identification , pages 1143--1148

    2012

  25. [33]

    Kalman, R. E. (1960). A new approach to linear filtering and prediction problems . Journal of Basic Engineering , 82(1):35--45

    1960

  26. [34]

    S., Maciejowski, J., and Chopin, N

    Kantas, N., Doucet, A., Singh, S. S., Maciejowski, J., and Chopin, N. (2014). On particle methods for parameter estimation in state - space models . Statistical Science , 30(3):328--351

    2014

  27. [35]

    and Bergman, N

    Karlsson, R. and Bergman, N. (2000). Auxiliary particle filters for tracking a maneuvering target. Proceedings of the 39th IEEE Conference on Decision and Control , 4:3891--3895

    2000

  28. [36]

    Kim, C. J. and Nelson, C. R. (1999). State- Space Models with Regime Switching : Classical and Gibbs - Sampling Approaches with Applications . MIT Press

    1999

  29. [37]

    Kim, J. (2015). Bayesian inference in a non-linear/non- Gaussian switching state space model: Regime -dependent leverage effect in the U.S. stock market. MPRA Paper , (67153)

    2015

  30. [38]

    Kim, J. R. and Cho, S. (2022). Developing a regime-switching present value model: switching fundamentals and bubbles. International Economic Journal , 36(4):477--490

    2022

  31. [39]

    King, R. (2011). Statistical ecology. In Brooks, S., Gelman, A., Jones, G. L. J., and Meng, X.-L., editors, Handbook of Markov Chain Monte Carlo , pages 419--447. Chapman & Hall

    2011

  32. [40]

    King, R. (2014). Statistical ecology. Annual Review of Statistics and its Application , 1(1):401--426

    2014

  33. [41]

    Kitagawa, G. (1987). Non- Gaussian state - space modeling of non-stationary time series . Journal of the American Statistical Association , 82(400):1032--1041

    1987

  34. [42]

    Koopman, S. J. and Bos, C. S. (2004). State space models with a common stochastic variance . Journal of Business and Economic Statistics , 22(3):346--357

    2004

  35. [43]

    W., Skaug, H., and Bell, B

    Kristensen, K., Nielsen, A., Berg, C. W., Skaug, H., and Bell, B. M. (2016). TMB : automatic differentiation and Laplace approximation . Journal of Statistical Software , 70(5):1--21

    2016

  36. [44]

    Langrock, R. (2011). Some applications of nonlinear and non- Gaussian state–space modelling by means of hidden Markov models. Journal of Applied Statistics , 38(12):2955--2970

    2011

  37. [45]

    and King, R

    Langrock, R. and King, R. (2013). Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. Ann. Appl. Stat. , 7(3):1709--1732

    2013

  38. [46]

    L., and Zucchini, W

    Langrock, R., MacDonald, I. L., and Zucchini, W. (2012). Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. Journal of Empirical Finance , 19(1):147--161

    2012

  39. [47]

    Lawrence, P. (2020). Tourism and Hospitality Sector Recovery Plan - Follow up. Policy and Sustainability Comittee, City of Edinburgh Council . https://democracy.edinburgh.gov.uk/documents/s24704/Item\

    2020

  40. [48]

    Liang-qun, L., Wei-xin, X., Jing-xiong, H., and Jianjun, H. (2009). Multiple model Rao - Blackwellized particle filter for manoeuvring target tracking. Defence Science Journal , 59(3):197--204

    2009

  41. [49]

    A., Tye, K

    Lin, A., Zhang, Y., Heng, J., Allsop, S. A., Tye, K. M., Jacob, P. E., and Ba, D. (2019). Clustering time series with nonlinear dynamics : a Bayesian non - parametric and particle - based approach . Proceedings of the Twenty-Second International Conference on Artificial Intell...

    2019

  42. [50]

    I., and Schön, T

    Lindsten, F., Jordan, M. I., and Schön, T. B. (2014). Particle Gibbs with ancestor sampling . Journal of Machine Learning Research , 15(63):2145--2184

    2014

  43. [51]

    and Schön, T

    Lindsten, F. and Schön, T. B. (2013). Backward simulation methods for Monte Carlo statistical inference. Foundations and Trends in Machine Learning , 6(1):1--143

    2013

  44. [52]

    Llewellyn, M., King, R., Elvira, V., and Ross, G. J. (2023a). A point mass proposal method for Bayesian state-space model-fitting. Statistics and Computing , 33(111)

    2023

  45. [53]

    Llewellyn, M., Ross, G., and Ryan-Saha, J. (2023b). COVID -era forecasting: Google trends and window and model averaging. Annals of Tourism Research , 103:103660

    2023

  46. [54]

    Martino, L., Read, J., Elvira, V., and Louzada, F. (2017). Cooperative parallel particle filters for online model selection and applications to urban mobility. Digital Signal Processing , 60:172--185

    2017

  47. [55]

    Matousek, J., Dunik, J., and Straka, O. (2019). Point- mass filter : density specific grid design and implementation . 15th European Workshop on Advanced Control and Diagnosis , pages 1093--1115

    2019

  48. [56]

    D., Doucet, A., and Jasra, A

    Moral, P. D., Doucet, A., and Jasra, A. (2012). On adaptive resampling strategies for sequential Monte Carlo methods. Bernoulli , 18(1):252--278

    2012

  49. [57]

    Newman, K. B. (1998). State- space modeling of animal movement and mortality with application to salmon . Biometrics , 54(4):1290--1314

    1998

  50. [58]

    B., King, R., Elvira, V., de Valpine, P., McCrea, R

    Newman, K. B., King, R., Elvira, V., de Valpine, P., McCrea, R. S., and Morgan, B. J. T. (2023). State-space models for ecological time series data: Practical model-fitting. Methods in Ecology and Evolution , 14(1):26--42

    2023

  51. [59]

    Nonejad, N. (2015). Particle Gibbs with ancestor sampling for stochastic volatility models with: heavy tails, in mean effects, leverage, serial dependence and structural breaks. Studies in Nonlinear Dynamics and Econometrics , 19(5):561--584

    2015

  52. [60]

    Mitigating the impact of COVID -19 on tourism and supporting recovery

    OECD (2020). Mitigating the impact of COVID -19 on tourism and supporting recovery. In OECD Tourism Papers . https://doi.org/10.1787/23071672

    2020 doi

  53. [61]

    Onizuka, T., Hashimoto, S., and Sugasawa, S. (2023). Fast and locally adaptive Bayesian quantile smoothing using calibrated variational approximations . Statistics and Computing , 34(15):1--16

    2023

  54. [62]

    Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: auxiliary particle filters. Journal of the American Statistical Association , 94(446):590--599

    1999

  55. [63]

    Pitt, M. K. and Shephard, N. (2001). Auxiliary variable based particle filters. In Doucet, A., de Freitas, N., and Gordon, N., editors, Sequential Monte Carlo Methods in Practice , pages 273--293. Springer

    2001

  56. [64]

    and Míguez, J

    Pérez-Vieites, S. and Míguez, J. (2021). Nested Gaussian filters for recursive Bayesian inference and nonlinear tracking in state space models. Signal Processing , 189:108295

    2021

  57. [65]

    Rabiner, L. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE , 77(2):257--286

    1989

  58. [66]

    A., Lindsten, F., Paige, B., van de Meent, J

    Rainforth, T., Naesseth, C. A., Lindsten, F., Paige, B., van de Meent, J. W., Doucet, A., and Wood, F. (2016). Interacting particle Markov chain Monte Carlo . In Proceedings of the 33rd International Conference on Machine Learning , pages 2616--2625

    2016

  59. [67]

    and Pitt, M

    Shephard, N. and Pitt, M. K. (1997). Likelihood analysis of non - Gaussian measurement time series . Biometrika , 84(3):653--667

    1997

  60. [68]

    and Gasperin, M

    Smidl, V. and Gasperin, M. (2013). Rao- Blackwellized point mass filter for reliable state estimation. Proceedings of the 16th International Conference on Information Fusion , pages 312--318

    2013

  61. [69]

    So, M. K. P., Lam, K., and Li, W. K. (1998). A stochastic volatility model with Markov switching. Journal of Business & Economic Statistics , 16(2):244--253

    1998

  62. [70]

    Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation . Journal of the American Statistical Association , 82(398):528--540

    1987

  63. [71]

    and Schön, T

    Tidefelt, H. and Schön, T. B. (2009). Robust point-mass filters on manifolds. In 15th IFAC Symposium on System Identification , pages 540--545

    2009

  64. [72]

    Tourism in Scotland : The economic contribution of the sector

    Tourism Leadership Group (2018). Tourism in Scotland : The economic contribution of the sector . In Scottish Government . https://www.gov.scot/publications/tourism-scotland-economic-contribution-sector/pages/4/

    2018

  65. [73]

    F., and Djurić, P

    Urteaga, I., Bugallo, M. F., and Djurić, P. M. (2016). Sequential Monte Carlo methods under model uncertainty. 2016 IEEE Statistical Signal Processing Workshop , pages 1--5

    2016

  66. [74]

    van der Merwe , R., Wan, E., and Julier, S. (2004). Sigma-point Kalman filters for nonlinear estimation and sensor-fusion applications to integrated navigation. Proceedings of the AIAA Guidance, Navigation & Control Conference , 3

    2004

  67. [75]

    Vergé, C., Dubarry, C., Del Moral, P., and Moulines, E. (2013). On parallel implementation of sequential Monte Carlo methods: The island particle model. Statistics and Computing , 25(2):243--260

    2013

  68. [76]

    Whiteley, N., Andrieu, C., and Doucet, A. (2010). Efficient Bayesian inference for switching state-space models using discrete particle Markov chain Monte Carlo methods. arXiv . https://doi.org/10.48550/arXiv.1011.2437

    2010 doi arXiv

  69. [77]

    S., Murray, L

    Wigren, A., Risuleo, R. S., Murray, L. M., and Lindsten, F. (2019). Parameter elimination in particle Gibbs sampling. In Proceedings of the 33rd International Conference on Neural Information Processing Systems

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.