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A pseudo-marginal sequential Monte Carlo online smoothing algorithm

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Pith's one-line read The paper establishes that online additive smoothing remains valid when transition densities are intractable but estimable, using a pseudo-marginal version of the PaRIS algorithm with linear complexity and bias/variance guarantees.

desk verdict Substantial theory for a two-level pseudo-marginal PaRIS, but the advertised diffusion scope outruns the strong-mixing assumptions the main theorems actually require. read the letter →

arxiv 1908.07254 v2 pith:MUPLTSCE submitted 2019-08-20 stat.CO stat.ME

Pierre Gloaguen (MIA-Paris) , Sylvain Le Corff (IP Paris , CITI , TIPIC-SAMOVAR) , Jimmy Olsson (KTH Royal Institute of Technology) This is my paper · ORCID
classification stat.COstat.ME
keywords centrallimittheoremexponentialconcentrationpartiallyobserveddiffusionsparticlesmoothingpseudo-marginalmethodssequentialMonteCarloonlineadditivefunctionals
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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 considers online estimates of expectations of additive path functionals—sums of terms along the hidden path—under general path distributions whose transition densities cannot be evaluated, but can be simulated or estimated, with or without bias. It proves that a pseudo-marginal version of the PaRIS particle smoother, which replaces every intractable density by a random estimate in both the forward and backward sampling steps, computes these estimates in $O(N)$ time per time step with constant memory. The paper establishes exponential concentration and a central limit theorem for this estimator, and shows that under strong mixing its asymptotic variance grows at most linearly in the path length $n$. If the density estimates are biased with size controlled by a parameter $\varepsilon$, the asymptotic bias is at most $O(n\varepsilon)$ for additive functionals, and $O(\varepsilon)$ for filter marginals. A sympathetic reader would care because the result covers partially observed diffusions and approximate Bayesian computation, where exact transition densities are unavailable.

What carries the argument

The load-bearing object is two-level pseudo-marginalisation of the PaRIS update. PaRIS avoids particle-path degeneracy by alternating forward sampling, which propagates particles through the model, with backward sampling, which propagates statistics $\tau_n^i$ approximating the backward-recursed additive functional $T_n h_n$. In the pseudo-marginal version, both levels use random estimates of the intractable density: the forward weights use $\ell_n^{\langle z\rangle}$ directly, and the backward index kernel $\bar\Pi_n(i,\cdot)$ samples ancestor indices with probabilities proportional to $\omega_n^j\ell_n^{\langle z\rangle}(\xi_n^j,\xi_{n+1}^i)$. The extended-target argument of pseudo-marginal methods keeps the algorithm consistent for the averaged densities $\ell_n^\varepsilon$, and the parameter $\varepsilon$ measures the bias between those averages and the true densities. Rejection or Markov-chain sampling from the extended kernel keeps the per-step cost linear in the number of particles.

What would settle it

Run Algorithm 6 with $M=2$ on a finite-state hidden Markov model that satisfies the uniform density bounds, using deliberately biased transition-density estimates of known size $\varepsilon$. Theorems 4.6 and 4.10 predict that as $N$ grows the estimator's variance increases at most linearly in the path length $n$ and its bias stays below a constant times $n\varepsilon$; observing superlinear variance growth, or bias that cannot be bounded by $C n\varepsilon$ for any fixed $C$ at large $N$, would falsify the central claims.

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Extended reading notes

Core claim

The central claim is that Algorithm 6, the pseudo-marginal PaRIS update, is a valid online smoother for general path models of the form $\varphi_{0:n}(dx_{0:n})\propto\chi(dx_0)\prod_{m=0}^{n-1}L_m(x_m,dx_{m+1})$ when the kernels $L_m$ have densities that are intractable but estimable. Each transition density $\ell_n$ is replaced by a random estimate $\ell_n^{\langle z\rangle}$ with expectation $\ell_n^\varepsilon$, and the algorithm converges to the 'skew' expectation $\varphi^\varepsilon_{0:n}h_n$ under the averaged model. The paper proves a Hoeffding-type concentration inequality and a central limit theorem for the estimator, giving an explicit asymptotic variance with an additional nonnegative term caused by the pseudo-marginal randomness. Under a strong-mixing condition (H3) and with at least $M=2$ backward samples per particle, the asymptotic variance grows at most linearly in $n$, which is optimal for path-space estimators. Under a bias-control condition (H4), the skew target is within $O(n\varepsilon)$ of the true target for additive functionals, with an $O(\varepsilon)$ filter-marginal version as a corollary.

Load-bearing premise

The long-term stability and bias bounds rely on every transition density, true and estimated, being bounded below and above by fixed positive constants, which in practice means compact state spaces; if that fails, the $O(n)$ variance and $O(n\varepsilon)$ bias guarantees are not established.

Editorial extensions

If this is right

  • Online additive smoothing becomes feasible for partially observed diffusions and approximate-Bayesian-computation models at $O(N)$ work per time step with $O(1)$ memory, instead of quadratic-cost alternatives.
  • Biased density estimates are safe when their bias is controlled: choosing $\varepsilon$ small relative to $1/n$ keeps the total asymptotic bias bounded while variance grows linearly.
  • The explicit central-limit-theorem variance, including the pseudo-marginal variance term, can guide choices of particle count $N$, backward sample size $M$, and estimation noise level.
  • The framework contains random-weight particle filters, the original PaRIS, and ABC smoothing as special cases, so their consistency and stability follow from the same theorems.

Reading between the lines

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

  • The $O(n\varepsilon)$ bias bound suggests a practical discretisation rule the paper does not state: take the density-estimation bias $\varepsilon$ of order $1/n$ or smaller so that the asymptotic bias stays bounded while the Monte Carlo variance accumulates linearly.
  • Because score functions and EM intermediate quantities are additive functionals, the same algorithm is a candidate for online parameter estimation in intractable models, though the paper does not run that experiment.
  • The central-limit-theorem variance could be used to allocate simulation effort between particles and density-estimation draws, in analogy with tuning pseudo-marginal MCMC; the paper leaves this allocation open.
  • If the uniform-density strong-mixing assumption could be relaxed to geometric ergodicity, the method might extend to noncompact state spaces such as Gaussian transition models; no such extension is proved here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a pseudo-marginal extension of the PaRIS algorithm for online smoothing of additive functionals under path measures with intractable transition densities. Algorithm 6 replaces both forward and backward sampling operations by pseudo-marginal versions, admitting unbiased or biased density estimators. The main theoretical results are exponential concentration (Theorem 4.1), a central limit theorem with an explicit variance decomposition (Theorem 4.2), an O(n) bound on asymptotic variance under a strong-mixing condition (Theorem 4.6), and an O(n epsilon) bound on asymptotic bias under an additional Lipschitz-type condition on the estimated density (Theorem 4.10). The paper also derives by-product CLTs for the random-weight particle filter and for the ideal PaRIS in a general Feynman-Kac/auxiliary particle filter setting, and it presents a short numerical example based on an Ornstein-Uhlenbeck model.

Significance. If the advertised claims are taken at face value, the paper makes a substantial contribution: it extends the PaRIS algorithm to models whose transition densities are only estimable, preserves linear-in-N complexity and constant memory in n, and provides the first O(n epsilon) bias bound of this type. The proofs are detailed and organized, the assumptions H1-H4 are explicit, and there are no hidden fitted parameters in the derivation. The variance decomposition in Theorem 4.2 is interpretable, with distinct contributions from initialization, the ideal PaRIS dynamics, and the additional pseudo-marginal randomness. The by-product time-uniform variance bound for the random-weight particle filter (Proposition 4.9) is also valuable. The principal weakness is the mismatch between the advertised range of applications and the strong-mixing assumption needed for the long-horizon and bias results, together with a numerical section that does not actually run the proposed Algorithm 6.

major comments (3)
  1. [Section 4.2 (Assumption H3); Theorems 4.6 and 4.10] The two long-horizon results that support the advertised practical scope both rest on Assumption H3, which requires uniform lower and upper bounds on the transition densities and their biased counterparts. As the authors themselves note, this is a Doeblin condition that typically requires compact state spaces. The diffusion model of Example 2 has Gaussian transition densities on R^{d_x}, and the Ornstein-Uhlenbeck example of Section 5 has a Gaussian transition density on R, so H3 fails in precisely those flagship applications. Consequently, for those models the paper provides no control on the long-run variance (Theorem 4.6) and no O(n epsilon) bound between the skew expectation and the true additive functional (Theorem 4.10); only the fixed-horizon CLT about the skew model (Theorem 4.2) applies. The abstract and introduction should qualify the claim that the algorithm applies to diffusion processes, or additional results for non-compact state spaces should be supplied.
  2. [Section 5, Figure 1] The numerical section does not demonstrate the proposed algorithm. The PaRIS results in Figure 1 are generated by Algorithm 3 (ideal PaRIS) with the optimal importance function, not by the pseudo-marginal Algorithm 6, and the epsilon-skew is encoded in the observation density rather than in a biased estimator of an intractable transition density. Thus the experiment cannot validate the pseudo-marginal machinery of Section 3, nor does it quantify the additional variance term in Theorem 4.2. In addition, the OU model violates H3, and the constant c used to verify H4 in Section 5 depends on sup_n |y_n|, which is infinite almost surely for Gaussian observations; hence the example does not satisfy the assumptions of Theorem 4.10. A simulation of Algorithm 6 on a model satisfying H3-H4, with a comparison of the bias of the pseudo-marginal estimator, is needed to support the advertised claims.
  3. [Abstract and Introduction vs. Theorem 4.2] For epsilon > 0, Theorem 4.2 establishes asymptotic normality around the skew expectation phi^epsilon_{0:n} h_n, not around the target phi_{0:n} h_n. The bridge to the true target is provided only by Theorem 4.10, which requires H3-H4. Since H3 is not satisfied by the diffusion examples, the reader is left without end-to-end theoretical support for the headline applications. The manuscript should state this limitation explicitly in the abstract and introduction whenever the 'wide range' of diffusion and intractable-likelihood models is advertised; as written, the reader can reasonably infer that the convergence, stability, and bias guarantees apply to those models.
minor comments (4)
  1. [Section 3.2.3, initialization of Algorithm 6] The initialization text says 'letting ... tau_i^n = 0', which appears to be a typo for tau_i^0 = 0; the same notation should be made consistent in Algorithms 3 and 6.
  2. [Assumption H2] The statement 'So is also w^{-1} : X0 -> dchi/dnu(x0)' is not well-formed notation; the boundedness condition should refer to the initial weight function w_0(x0) = dchi/dnu(x0) or to its reciprocal explicitly.
  3. [Reference [24]] Reference [24] is listed as Stochastic Analysis and Applications 32:397-420, 1998, but this volume and page range correspond to a much later year; the citation should be corrected to avoid bibliographic confusion.
  4. [Reproducibility] No code or data are provided for the numerical example; making a small reproducible script available would strengthen the computational claims and help readers assess the behavior of Algorithm 6.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(n ε) bias bound follows from the explicit H4 bias assumption combined with the H3 mixing condition, and the convergence theorems are proven rather than assumed.

full rationale

The paper's central advertised result is the O(n ε) bound on the asymptotic bias between the skew path model φ^ε and the true path model φ (Theorem 4.10). This is not a fitted prediction or a restatement of an input: Assumption (H4) is an explicit, externally checkable condition bounding |L^ε_n h − L_n h| uniformly by c ε ||h||_∞, and Assumption (H3) provides the Doeblin/minorization constants used in the propagation argument. The proof in Appendix D combines these assumptions with Dobrushin-coefficient bounds on forward and backward kernels; the conclusion is genuinely derived, not equivalent to the assumption by construction. Likewise, Theorems 4.1 and 4.2 prove concentration and a CLT for Algorithm 6 with respect to the skew target φ^ε. It is true by design that pseudo-marginalization replaces ℓ_n with ℓ^ε_n, but establishing the asymptotic distribution of the two-level randomized backward sampling estimator requires the nontrivial induction and variance decomposition in Appendix C; that result is not assumed as an input. The numerical section illustrates Theorem 4.10 using Kalman smoothing and the ideal PaRIS (Algorithm 3) rather than Algorithm 6, and the OU example does not satisfy H3; these are scope and support limitations, not circular reasoning. Self-citations to the original PaRIS papers [28] and [29] are used as algorithmic and proof baselines, but the present paper supplies its own proofs and the cited works are independent published results with their own derivations, so they are not load-bearing self-citations in the circularity sense. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data: N, M, and epsilon are user-set algorithmic or precision parameters, not tuned to make the theoretical results hold. The central claims rest on the four stated assumptions H1 through H4, of which H3 and H4 are the most restrictive and the most load-bearing for the advertised O(n) variance and O(n epsilon) bias results.

assumptions (5)
  • domain assumption H1: pseudo-marginal estimators ell_n^langle z rangle exist with expectation ell^epsilon_n and sampling kernel R_n.
    Introduced in Section 3.1 and used by every algorithm and theorem. It is the entry point for replacing intractable transition densities with random estimates, and it is not proved from more basic assumptions.
  • domain assumption H2: proposal ratios w_n, w^epsilon_n, the initial density ratio, and adjustment weights are uniformly bounded.
    Required for the Hoeffding inequalities and the CLT in Section 4.1. It excludes proposal kernels that are poorly matched to the target transitions.
  • domain assumption H3: all transition densities and their epsilon perturbed versions are uniformly bounded below and above by sigma_- and sigma_+.
    This strong mixing assumption powers the Dobrushin coefficient bounds used in Theorem 4.6 and Theorem 4.10. It typically requires compact state spaces and is not satisfied by the Gaussian OU example in Section 5.
  • domain assumption H4: the biased kernels satisfy |L^epsilon_n h(x) - L_n h(x)| <= c epsilon ||h||_infinity uniformly over n, h, and x.
    This uniform bias control is the input to Theorem 4.10. The theorem establishes propagation of this per-step bound to the whole path measure, so the O(n epsilon) conclusion is only as strong as this assumption.
  • standard math External CLT and martingale machinery, e.g. the Lindeberg-type theorem of Douc and Moulines referenced as [14, Theorem A.3].
    The proof of Theorem C.1 invokes a known triangular array CLT for conditionally independent variables. This is standard background and is not proved in the paper.

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Pith. "Pith review of A pseudo-marginal sequential Monte Carlo online smoothing algorithm." pith.science (2026). https://pith.science/paper/MUPLTSCE

@misc{pith2026190807254,
  author       = {Pith},
  title        = {Pith review of: A pseudo-marginal sequential Monte Carlo online smoothing algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MUPLTSCE}},
  note         = {Machine review of arXiv:1908.07254}
}
abstract

We consider online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle-based, rapid incremental smoother (PaRIS) algorithm proposed in [J.Olsson and J.Westerborn. Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Bernoulli, 23(3):1951--1996, 2017] to this setting. The resulting algorithm, which has a linear complexity in the number of particles and constant memory requirements, applies to a wide range of challenging path-space Monte Carlo problems, including smoothing in partially observed diffusion processes and models with intractable likelihood. The algorithm is furnished with several theoretical results, including a central limit theorem, establishing its convergence and numerical stability. Moreover, under strong mixing assumptions we establish a novel $O(n \varepsilon)$ bound on the asymptotic bias of the algorithm, where $n$ is the path length and $\varepsilon$ controls the bias of the density estimators.

Figures

Figures reproduced from arXiv: 1908.07254 by the authors.

Figure 1
Figure 1. (a) The red line is the deviation of φ ε 0:nhn from φ0:nhn for varying ε ∈ {0, 0.05, 0.1, . . ., 0.5} and fixed n = 50, computed by means of Kalman smoothing. Turquoise markers are similar errors obtained on the basis of 60 repli￾cates of the ideal PaRIS in Algorithm 3 with N = 200 particles and M = 2 back￾ward samples. (b) The red line is the same deviation for increasing n ∈ J1, 50K and fixed ε = 0.1. Then green a… view at source ↗

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Andrieu and G

    C. Andrieu and G. O. Roberts. The pseudo-marginal approach f or efficient Monte Carlo computations. Ann. Statist. , 37(2):697–725, 2009

    work page 2009

  2. [2]

    Bally and D

    V. Bally and D. Talay. The law of the Euler scheme for stochastic diff er- ential equations: II. Approximation of the density. Monte Carlo Methods and Applications, 2:93–128, 1996

    work page 1996

  3. [3]

    M. A. Beaumont. Estimation of population growth or decline in gene tically monitored populations. Genetics, 164(3):1139–1160, 2003. /A pseudo-marginal SMC online smoothing algorithm 32

    work page 2003

  4. [4]

    Beskos, O

    A. Beskos, O. Papaspiliopoulos, G.O. Roberts, and P. Fearnhead . Exact and computationally efficient likelihood-based estimation for discrete ly ob- served diffusion processes. J. R. Stat. Soc. Ser. B , 68(3):333–382, 2006. With discussions

    work page 2006

  5. [5]

    L. M. Candanedo, V. Feldheim, and D. Deramaix. A methodology ba sed on hidden Markov models for occupancy detection and a case study in a low energy residential building. Energy and Buildings , 148:327–341, 2017

    work page 2017

  6. [6]

    O. Capp´ e. Online EM algorithm for hidden Markov models. J. Comput. Graph. Statist. , 20(3):728–749, 2011

    work page 2011

  7. [7]

    Capp´ e,´E

    O. Capp´ e,´E. Moulines, and T. Ryden. Inference in Hidden Markov Models . Springer, 2005

    work page 2005

  8. [8]

    Chopin and O

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

    work page 2020

Show all 43 references
  1. [9]

    Del Moral

    P. Del Moral. Feynman-Kac Formulae. Genealogical and Interacting Par- ticle Systems with Applications . Springer, 2004

    2004

  2. [10]

    Del Moral, A

    P. Del Moral, A. Doucet, and S. Singh. A Backward Particle Inte rpretation of Feynman-Kac Formulae. ESAIM M2AN , 44(5):947–975, 2010

    2010

  3. [11]

    Del Moral, J

    P. Del Moral, J. Jacod, and P. Protter. The Monte-Carlo meth od for filter- ing with discrete-time observations. Probab. Theory Relat. Fields , 123:346– 368, 2001

    2001

  4. [12]

    Del Moral, E

    P. Del Moral, E. Moulines, J. Olsson, and C. Verg´ e. Convergen ce proper- ties of weighted particle islands with application to the double bootstr ap algorithm. Stochastic Systems , 2(6):367–419, 2016

    2016

  5. [13]

    R. Douc, A. Garivier, ´E. Moulines, and J. Olsson. Sequential Monte Carlo smoothing for general state space hidden Markov models. Ann. Appl. Probab., 21(6):2109–2145, 2011

    2011

  6. [14]

    Douc and E

    R. Douc and E. Moulines. Limit theorems for weighted samples with appli- cations to sequential monte carlo methods. Ann. Statist. , 36(5):2344–2376, 2008

    2008

  7. [15]

    Doucet, S

    A. Doucet, S. Godsill, and C. Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and computing , 10(3):197–208, 2000

    2000

  8. [16]

    G. B. Durham and A. R. Gallant. Numerical techniques for maximu m likelihood estimation of continuous-time diffusion processes. Journal of Business & Economic Statistics , 20(3):297–316, 2002

    2002

  9. [17]

    Fearnhead, O

    P. Fearnhead, O. Papaspiliopoulos, and G. O. Roberts. Particle filters for partially observed diffusions. J. Roy. Statist. Soc. B , 70(4):755–777, 2008

    2008

  10. [18]

    Fearnhead, O

    P. Fearnhead, O. Papaspiliopoulos, G. O. Roberts, and A. Stua rt. Random weight particle filtering of continuous time processes. J. Roy. Statist. Soc. B, 72(4):497–512, 2010

    2010

  11. [19]

    Gloaguen, M.-P

    P. Gloaguen, M.-P. Etienne, and S. Le Corff. Online sequential Mo nte Carlo smoother for partially observed diffusion processes. EURASIP Journal on Advances in Signal Processing , 2018(1):9, 2018

    2018

  12. [20]

    S. J. Godsill, A. Doucet, and M. West. Monte Carlo smoothing for non- linear time series. J. Amer. Statist. Assoc. , 50:438–449, 2004

    2004

  13. [21]

    Le Gland and L

    F. Le Gland and L. Mevel. Recursive estimation in HMMs. In Proc. IEEE /A pseudo-marginal SMC online smoothing algorithm 33 Conf. Decis. Control , pages 3468–3473, 1997

    1997

  14. [22]

    Le Gland and N

    F. Le Gland and N. Oudjane. Stability and uniform approximation o f nonlinear filters using the Hilbert metric and application to particle filte rs. Ann. Appl. Probab. , 14:144–187, 2004

    2004

  15. [23]

    J. M. Marin, P. Pudlo, C. P. Robert, and R. J. Ryder. Approxima te Bayesian computational methods. Stat. Comput. , 22:1167–1180, 2012

    2012

  16. [24]

    J. S. Martin, A. Jasra, S. S. Singh, N. Whiteley, P. Del Moral, an d E. Mc- Coy. Approximate Bayesian Computation for smoothing. Stochastic Anal- ysis and Applications , 32:397–420, 1998

    1998

  17. [25]

    J. M. Mcgree, C. C. Drovandi, G. White, and A. N. Pettitt. A pse udo- marginal sequential Monte Carlo algorithm for random effects mode ls in Bayesian sequential design. Stat. Comput. , 26:1121–1136, 2016

    2016

  18. [26]

    Michelot, R

    T. Michelot, R. Langrock, and T. A. Patterson. moveHMM: an R package for the statistical modelling of animal movement data using hidden Ma rkov models. Methods in Ecology and Evolution , 7(11):1308–1315, 2016

    2016

  19. [27]

    Olsson and J

    J. Olsson and J. Str¨ ojby. Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators . Electron. J. Statist. , 5:1090–1122, 2011

    2011

  20. [28]

    Olsson and J

    J. Olsson and J. Westerborn. Efficient particle-based online smo oth- ing in general hidden Markov models: The PaRIS algorithm. Bernoulli, 23(3):1951–1996, 2017

    1951

  21. [29]

    Olsson and J

    J. Olsson and J. Westerborn Alenl¨ ov. Particle-based online est imation of tangent filters with application to parameter estimation in nonlinear s tate- space models. Annals of the Institute of Statistical Mathematics , 72(2):545– 576, 2018

    2018

  22. [30]

    Papavasiliou

    A. Papavasiliou. Parameter estimation and asymptotic stability in stochas- tic filtering. Stoch. Proc. Appl. , 116(7):1048–1065, 2006

    2006

  23. [31]

    M. K. Pitt and N. Shephard. Filtering via simulation: Auxiliary partic le filters. J. Am. Statist. Assoc. , 94(446):590–599, 1999

    1999

  24. [32]

    L. R. Rabiner. A tutorial on hidden Markov models and selected a ppli- cations in speech recognition. In Proceedings of the IEEE , pages 257–286, 1989

    1989

  25. [33]

    H. E. Rauch, F. Tung, and C. T. Striebel. Maximum likelihood estima tes of linear dynamic systems. AIAA journal , 3(8):1445–1450, 1965

    1965

  26. [34]

    S¨ arkk¨ a.Bayesian Filtering and Smoothing

    S. S¨ arkk¨ a.Bayesian Filtering and Smoothing . Cambridge University Press, New York, 2013

    2013

  27. [35]

    S¨ arkk¨ a, A

    S. S¨ arkk¨ a, A. Vehtari, and J. Lampinen. Rao-Blackwellized pa rticle filter for multiple target tracking. Information Fusion, 8(1):2–15, 2007

    2007

  28. [36]

    E Uhlenbeck and L

    G. E Uhlenbeck and L. S Ornstein. On the theory of the Brownian motion. Phys. rev. , 36(5):823, 1930

    1930

  29. [37]

    C. Yau, O. Papaspiliopoulos, G. O. Roberts, and C. Holmes. Baye sian non- parametric hidden Markov models with applications in genomics. 73:1–2 1, 2011

    2011

  30. [38]

    Yonekura and A

    S. Yonekura and A. Beskos. Online smoothing for diffusion proce sses ob- served with noise, 2020. ArXiv:2003.12247

    2020 arXiv

  31. [39]

    Zucchini, I

    W. Zucchini, I. L. Mac Donald, and R. Langrock. Hidden Markov Models /A pseudo-marginal SMC online smoothing algorithm 34 for Time Series: an Introduction using R . CRC Press, 2017. Appendix A: Proof of Lemma 2.2 To show (i), write, using the definition ( 2.10), the right-hand ...

    2017

  32. [40]

    Moreover, note that E [ ω1 n+1|G N n+1 ] = ∫ wn(ξ I 1 n+1 n ,ξ 1 n+1,z ) Rn(ξ I 1 n+1 n ,ξ 1 n+1,dz ) = wε n(ξ I 1 n+1 n ,ξ 1 n+1), (B.1) where wn and wε n are defined in ( H2)(ii), and E [ τ 1 n+1|G N n+1 ] = N∑ i=1 ωi nℓε n(ξi n,ξ 1 n+1) ∑ N i′=1ωi′ nℓε n(ξi′ n,ξ 1 n+1) ( τ i...

  33. [41]

    provides the bound ⏐ ⏐ϕε m,nh− ⃗ϕε m,nh ⏐ ⏐≤ 2cε‖Lε m,n1Xn‖∞‖h‖∞ φε m−1Lε m−1,n1Xn (where c is the constant in ( H 4)). Now, since for all xm∈ Xm, Lε m,n1Xn (xm) = ∫ ℓm(xm,x m+1)Lε m+1,n1Xn (xm+1)µm+1(dxm+1) ≤σ+µm+1Lε m+1,n1Xn /A pseudo-marginal SMC online smoothing algorithm ...

  34. [42]

    We turn to (ii)

    and ( D.6). We turn to (ii). By definition ( D.1), Lε m+1,n¯hm|n(xm+1) = ∫ ˜ hm(xm,x m+1) Bε m(xm+1,dx m) Lε m+1,n1Xn (xm+1); thus, by applying Lemma 2.2(i) to the skew model, φε m+1Lε m+1,n¯hm|n φε m+1Lε m+1,n1Xn (D.7) = ∫∫ φε mLε m(dxm+1) ˜hm(xm,x m+1) Bε m(xm+1,dx m) Lε m+1,...

  35. [43]

    /A pseudo-marginal SMC online smoothing algorithm 55 Proof of Theorem 4.10

    and ( D.6). /A pseudo-marginal SMC online smoothing algorithm 55 Proof of Theorem 4.10. Write φε 0:nhn−φ0:nhn = n−1∑ k=0 ( φε 0:n¯hk|n−φ0:n¯hk|n ) , where each term can be decomposed according to φε 0:n¯hk|n−φ0:n¯ hk|n = n∑ m=1 (φε 0:mLm,n−1¯hk|n φε 0:mLm,n−11Xn − φε 0:m−1Lm−1...

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