A Predictive View on Streaming Hidden Markov Models
Gerardo Duran-Martin Oxford-Man Institute, University of Oxford
arXiv:2604.09208v1 [stat.ML] 10 Apr 2026
Abstract We develop a predictive-first optimisation framework for streaming hidden Markov models. Unlike classical approaches that prioritise full posterior recovery under a fully specified generative model, we assume access to regime-specific predictive models whose parameters are learned online while maintaining a fixed transition prior over regimes. Our objective is to sequentially identify latent regimes while maintaining accurate stepahead predictive distributions. Because the number of possible regime paths grows exponentially, exact filtering is infeasible. We therefore formulate streaming inference as a constrained projection problem in predictivedistribution space: under a fixed hypothesis budget, we approximate the full posterior predictive by the forward-KL optimal mixture supported on S paths. The solution is the renormalised top-S posterior-weighted mixture, providing a principled derivation of beam search for HMMs. The resulting algorithm is fully recursive and deterministic, performing beam-style truncation with closed-form predictive updates and requiring neither EM nor sampling. Empirical comparisons against Online EM and Sequential Monte Carlo under matched computational budgets demonstrate competitive prequential performance.
recovery or parameter estimation, typically in offline settings (Bishop and Nasrabadi, 2006). In many modern applications, however, data arrive sequentially, requiring HMMs to be deployed in an online or streaming setting (Gama, 2010). In predictive applications, the quantity of interest is the one-step-ahead posterior predictive distribution. However, exact maintenance of this predictive in an HMM requires tracking all possible latent regime paths, whose number grows exponentially in time, rendering exact filtering computationally infeasible. To address this tension between predictive accuracy and computational tractability, we adopt a predictivefirst perspective: rather than treating regime identification as the primary objective, we directly target sequential predictive performance under a fixed hypothesis budget. We formalise this as a constrained optimisation problem over predictive mixtures under a fixed hypothesis budget. Specifically, we approximate the full posterior predictive by a mixture supported on at most S latent paths and minimise the forward Kullback–Leibler divergence. We show that the optimal solution retains the S largest posterior path weights and renormalises them, thereby recovering beam search as the solution of an explicit projection problem. The resulting algorithm is fully recursive and deterministic. It combines beam search with closed-form regime-specific updates, yielding analytic predictive mixtures without EM iterations or sampling. Code can be found at https://github.com/gerdm/ streaming-hmm.
1
Introduction
Hidden Markov models (HMMs) provide a classical framework for modelling regime changes in sequential data. Traditional formulations emphasise latent-state
Preprint.
2
Related work
Hidden Markov and Markov-switching models are widely used for regime analysis in sequential data. In macroeconomics, regime-switching models were popularised by Hamilton (1989). HMMs remain central in speech recognition and signal processing (Rabiner, 2002); a comprehensive methodological review is pro-
A Predictive View on Streaming Hidden Markov Models
vided by Mor et al. (2021). Sequential inference in HMMs typically proceeds via expectation–maximisation (EM) and its online variants (Cappé, 2011), or via sequential Monte Carlo (SMC) methods such as Rao–Blackwellised particle filters and particle learning (Murphy and Russell, 2001; Carvalho et al., 2010). EM performs recursive likelihood optimisation, whereas particle methods approximate the path posterior through stochastic sampling. Both approaches target posterior estimation under a fully specified generative model. Beam search, or top-S truncation, is a classical complexity-control strategy in sequential decoding for HMMs and related models (Ney et al., 1992; Graves, 2012). It is commonly introduced as a heuristic approximation to MAP (Viterbi) decoding (Viterbi, 2003), retaining only the highest-scoring partial paths at each step. In contrast, we maintain a normalised empirical distribution over retained paths rather than a single MAP trajectory. Our work relates to predictive-first approaches to Bayesian inference, where sequential predictive performance, rather than posterior recovery, defines the objective (Fong et al., 2023; Holmes and Walker, 2023; McLatchie et al., 2025). This includes both Bayesian models and predictive methods that do not maintain explicit finite-dimensional parameter posteriors, such as Gaussian-process and martingale posterior predictives. In particular, Gaussian processes have previously been combined with switching structures for change-point detection (Garnett et al., 2009; Saatçi et al., 2010; Altamirano et al., 2023) and in variational GP state-space models (Frigola et al., 2014). In contrast, we use regime-specific GPs directly as predictive components within a fully recursive decoding scheme (Section 5.1).
3
Streaming hidden Markov model
We present a streaming HMM (SHMM) that combines beam-style pruning over latent paths with recursive Bayesian updates of regime parameters along each retained path. While these components are individually standard, they are rarely coupled in a single fully recursive procedure aimed at maintaining predictive mixtures. In the following section, we show that this coupling arises naturally as a predictive KL projection, yielding a deterministic streaming algorithm with explicit hypothesis-budget guarantees. 3.1
Problem formulation
Let yt ∈ R be the observation at time t and Yt = (y1 , . . . , yt ). We consider a K-state HMM with la-
tent regimes zt ∈ K = {1, . . . , K} and path Zt = (z1 , . . . , zt ) ∈ Kt . Regimes evolve according to a known transition matrix π, p(zt | Zt−1 ) = p(zt | zt−1 ) = πzt−1 ,zt .
(1)
Each regime k is equipped with a (possibly nonparametric) predictive model. For a given path Zt , we assume access to regime-specific predictive summaries {bt,k }K k=1 , where bt,k encodes the information in the subsequence Yk (Zt ) ≜ (yτ : 1 ≤ τ ≤ t, zτ = k). The corresponding regime-k one-step predictive density is denoted fbt,k (y) ≜ p(yt+1 = y | Yk (Zt )). Our primary object is the one-step-ahead posterior predictive, X X p(yt+1 | Yt ) = p(Zt | Yt ) p(zt+1 | Zt ) fbt,zt+1 (yt+1 ). Zt
zt+1
(2) Here, the posterior predictive is a mixture over paths, where each path contributes a transition-weighted mixture of regime-specific predictive densities determined by its summaries {bt,k }. Recursive evaluation of (2) requires maintaining the path posterior p(Zt | Yt ) and, for each path, the ability to evaluate the corresponding path-conditional predictives fbt,k (y). 3.2
Sequential belief update
Let Zt = Zt−1 ∪ {zt } denote an extension of a path with zt ∈ K. Recall that each path Zt carries regimespecific predictive summaries {bt,j }K j=1 , where bt,j encodes the information in Yj (Zt ) and determines the regime-j predictive density fbt,j (·). When zt = k, only the k-th summary is updated using the new observation yt , while all other summaries remain unchanged: ( Update(bt−1,k , yt ) j = k, bt,j = (3) bt−1,j j ̸= k. Here Update denotes a recursive update operator. In conjugate parametric models, bt,k corresponds to the regime-specific posterior, yielding a closed-form recursion (Appendix B). More generally, it may implement robust or generalised Bayesian updates under model misspecification (Duran-Martin et al., 2024; Altamirano et al., 2024), or predictive-only updates that do not maintain explicit parameter posteriors (Hahn et al., 2018; Duran-Martin et al., 2025a).
Gerardo Duran-Martin
3.3
4
Sequential path posterior update
At time t, the path posterior takes the form p(Zt | Yt ) = p(Zt−1 | Yt−1 ) p(zt | Zt−1 ) fbt−1,zt (yt ), with known initial condition p(Z0 | Y0 ) = p(z0 ). Thus, updating the path posterior at time t requires: (i) access to the path posterior at time t − 1 , (ii) the transition probability from zt−1 to zt given by (1), and (iii) the regime-specific posterior predictive evaluated at yt . The number of possible paths grows exponentially in t; for K regimes, there are K t paths at time t, making exact maintenance of p(Zt | Yt ) intractable.
Beam search as a predictive KL projection
Here, we show that the beam search used in the streaming hidden Markov model (Section 3.4) follows directly from a constrained predictive optimisation problem. In particular, we show that under a budget of S possible paths, the forward-KL-optimal one-stepahead approximation is equivalent to retaining the S posterior paths with the largest weight and renormalising them. Theorem 4.1. Fix t. Let wt (Zt ) ≜ p(Zt | Yt ) denote the posterior path weights.
3.4
Beam search
A classical heuristic to control complexity is to keep the S ≥ 1 paths with most mass at each time step. n oS (s) (s) Let Zt−1 , wt−1 denote the retained paths and s=1 normalised weights at time t−1, defining the empirical posterior S X
νt−1 (Zt−1 ) ≜
(s)
(s)
wt−1 δ(Zt−1 − Zt−1 ).
s=1
At time t, each retained path branches to K candidates. For k ∈ K, the unnormalised weight of the (s) (s,k) = Zt−1 ∪ {k} is extension Zt (s,k)
ŵt
(s)
(s)
= wt−1 p(k | Zt−1 ) fbt−1 ,k (yt ).
For each path Zt , define the path-conditional predictive fZt (y) ≜
where fbt,k (y) = p(yt+1 = y | Yk (Zt )) is the regime-k predictive density determined by the path-specific summaries bt,k . Then the one-step-ahead posterior predictive is X p(y) ≡ p(yt+1 = y | Yt ) = wt (Zt ) fZt (y). Zt
Let A ⊆ Kt be any subset of S paths and define X X δ(A) ≜ wt (Zt ), WA ≜ wt (Zt ) = 1−δ(A).
(s,k)
and
defining the truncated posterior (s)
Zt ∈A
Zt ∈A /
We retain the S candidates with largest ŵt renormalise: (s) ŵ (s) wt = PS t (j) , j=1 ŵt
νt (Zt ) =
p(zt+1 = k | Zt ) fbt,k (y),
k=1
This yields S × K candidate paths.
S X
K X
(s)
wt δ(Zt − Zt ).
s=1
All non-retained paths are assigned zero mass. This procedure corresponds to beam search (beam pruning) applied to sequential probabilistic models (Graves, 2012). Remark. Together, the recursive belief updates (Section 3.2) and the sequential path-posterior recursion (Section 3.3) define a fully recursive streaming HMM. The resulting truncated posterior yields both one-step and multi-step predictive distributions, as illustrated in Section 5. Further details on sequential update strategies are provided in Duran-Martin et al. (2025b).
Consider predictive approximations supported on A, X q(y) = α(Zt ) fZt (y), Zt ∈A
with α(Zt ) ≥ 0 and
P
Zt ∈A α(Zt ) = 1.
Assume that the χ2 -divergence between the discarded and retained predictive mixtures is finite: Z (pAc (y))2 dy − 1 ≤ C < ∞, (A1) χ2 (pAc ∥ pA ) = pA (y) where pA (y) =
X wt (Zt ) Zt ∈A
pAc (y) =
WA
X wt (Zt ) Zt ∈A /
δ(A)
fZt (y), fZt (y).
Define the renormalised truncated predictive X wt (Zt ) qA (y) = fZt (y). WA Zt ∈A
Then:
(4)
A Predictive View on Streaming Hidden Markov Models
(i) Among all q supported on A, qA ∈ arg min KL(p∥q). q
(ii) Under (A1), KL(p∥qA ) ≤ log 1 + δ(A)C .
(5)
(iii) The bound is increasing in δ(A); hence the optimal support of size S consists of the S largest posterior weights wt (Zt ).
Figure 2: Representative multi-step forecasts.
See Appendix A for the proof. 5.2 Remarks. The discarded mass δ(A) quantifies the posterior weight removed by truncation; when δ(A) is small, the truncated predictive remains close to the full mixture, as controlled by Bound (5). Thus, theorem 4.1 identifies beam search as the forward-KL optimal projection onto S-supported predictive mixtures.
5
Experiments
5.1
GP-HMM
We consider a two-regime HMM with regime-specific Gaussian Process (GP) emissions. Data are generated from a process with regime-dependent linear drift and oscillatory structure:
1D Gaussian HMM
We next consider a three-regime Gaussian HMM with known variance and unknown regime means. We compare SHMM against online EM (Cappé, 2011) and a Rao–Blackwellised particle filter (RBPF) (Murphy and Russell, 2001) with prior as proposal. For matched computational budgets, SHMM and RBPF use S = 2 hypotheses. Figure 3 shows the regime-mean estimates. With S = 2, RBPF fails to consistently track regime switches. Online EM recovers mean levels but exhibits higher variance. SHMM remains stable and tracks regime changes accurately under the same hypothesis budget.
(k) (k) yt = yt−1 + slope(k) ∆t + w1 sin w2 tlocal ∆t + εt , where k ∈ {0, 1} and εt ∼ N (0, σ 2 ). Slopes are ±0.15 and both regimes share (w1 , w2 ) = (0.3, 0.5). Each regime uses a GP predictive with a Gaussian plus periodic kernel (Williams and Rasmussen, 2006; Duvenaud, 2014). Figure 1 shows one-step-ahead predictions. Uncertainty increases near regime switches and contracts within stable segments. Figure 3: Estimated regime means. Table 1 reports predictive accuracy and runtime (mean ± std). SHMM achieves the lowest MAE and RMSE while remaining computationally competitive. Further results with varying S are shown in Appendix C.1.
Figure 1: GP-HMM one-step-ahead predictions. Figure 2 shows multi-step forecasts that are close to and far from a regime change.
Method Online EM SHMM RBPF
MAE 0.9 ± 0.01 0.8 ± 0.02 1.0 ± 0.04
RMSE 1.2 ± 0.02 1.1 ± 0.02 1.3 ± 0.06
Time (s) 0.04 ± 0.01 0.10 ± 0.03 0.26 ± 0.01
Table 1: Predictive performance and runtime.
Gerardo Duran-Martin
References Matias Altamirano, François-Xavier Briol, and Jeremias Knoblauch. Robust and scalable bayesian online changepoint detection. In International Conference on Machine Learning, pages 642–663. PMLR, 2023. Matias Altamirano, Francois-Xavier Briol, and Jeremias Knoblauch. Robust and conjugate Gaussian process regression. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, editors, Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 1155–1185. PMLR, 21–27 Jul 2024. URL https://proceedings.mlr.press/ v235/altamirano24a.html. Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, volume 4. Springer, 2006. Olivier Cappé. Online em algorithm for hidden markov models. Journal of Computational and Graphical Statistics, 20(3):728–749, 2011. Carlos M. Carvalho, Michael S. Johannes, Hedibert F. Lopes, and Nicholas G. Polson. Particle Learning and Smoothing. Statistical Science, 25(1):88 – 106, 2010. doi: 10.1214/10-STS325. URL https://doi. org/10.1214/10-STS325. Gerardo Duran-Martin, Matias Altamirano, Alex Shestopaloff, Leandro Sánchez-Betancourt, Jeremias Knoblauch, Matt Jones, Francois-Xavier Briol, and Kevin Patrick Murphy. Outlier-robust kalman filtering through generalised Bayes. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, editors, Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 12138–12171. PMLR, 21–27 Jul 2024. URL https://proceedings.mlr.press/ v235/duran-martin24a.html. Gerardo Duran-Martin, Leandro Sánchez-Betancourt, Álvaro Cartea, and Kevin Murphy. Martingale posterior neural networks for fast sequential decision making. arXiv preprint arXiv:2506.11898, 2025a. Gerardo Duran-Martin, Leandro Sánchez-Betancourt, Alex Shestopaloff, and Kevin Patrick Murphy. A unifying framework for generalised bayesian online learning in non-stationary environments. Transactions on Machine Learning Research, 2025b. ISSN 2835-8856. URL https://openreview.net/forum? id=osesw2V10u.
David Duvenaud. Automatic model construction with Gaussian processes. PhD thesis, 2014. Edwin Fong, Chris Holmes, and Stephen G Walker. Martingale posterior distributions. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(5):1357–1391, 2023. Roger Frigola, Yutian Chen, and Carl E Rasmussen. Variational gaussian process state-space models. Advances in neural information processing systems, 27, 2014. Joao Gama. Knowledge discovery from data streams. CRC Press, 2010. Roman Garnett, Michael A Osborne, and Stephen J Roberts. Sequential bayesian prediction in the presence of changepoints. In Proceedings of the 26th Annual International Conference on Machine Learning, pages 345–352, 2009. Alex Graves. Sequence transduction with recurrent neural networks. arXiv preprint arXiv:1211.3711, 2012. P Richard Hahn, Ryan Martin, and Stephen G Walker. On recursive bayesian predictive distributions. Journal of the American Statistical Association, 113 (523):1085–1093, 2018. James D Hamilton. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica: Journal of the econometric society, pages 357–384, 1989. Chris C Holmes and Stephen G Walker. Statistical inference with exchangeability and martingales. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 381 (2247), 2023. Yann McLatchie, Badr-Eddine Cherief-Abdellatif, David T Frazier, and Jeremias Knoblauch. Predictively oriented posteriors. arXiv preprint arXiv:2510.01915, 2025. Bhavya Mor, Sunita Garhwal, and Ajay Kumar. A systematic review of hidden markov models and their applications: B. mor et al. Archives of computational methods in engineering, 28(3):1429–1448, 2021. Kevin Murphy and Stuart Russell. Rao-blackwellised particle filtering for dynamic bayesian networks. In Sequential Monte Carlo methods in practice, pages 499–515. Springer, 2001. H. Ney, D. Mergel, A. Noll, and A. Paeseler. Data driven search organization for continuous speech recognition. IEEE Transactions on Signal Processing, 40(2):272–281, 1992. doi: 10.1109/78.124938.
A Predictive View on Streaming Hidden Markov Models
Lawrence R Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2):257–286, 2002. Yunus Saatçi, Ryan D Turner, and Carl E Rasmussen. Gaussian process change point models. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 927–934, 2010. Andrew Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE transactions on Information Theory, 13(2): 260–269, 2003. Christopher KI Williams and Carl Edward Rasmussen. Gaussian processes for machine learning, volume 2. MIT press Cambridge, MA, 2006.
Gerardo Duran-Martin
A Predictive View on Streaming Hidden Markov Models: Supplementary Materials
A
Proofs
Proof. Fix a set A. Define pA (y) ≜
X wt (Zt ) WA
Zt ∈A
fZt (y),
pAc (y) ≜
X wt (Zt ) Zt ∈A /
δ(A)
fZt (y),
so that the posterior predicive is p(y) = WA pA (y) + δ(A) pAc (y).
(6)
By definition, pA = qA in (4). (i) Optimal weights for fixed support. Let q be any predictive supported on A, i.e. q(y) = Then Z KL(p∥q) =
P
Zt ∈A α(Zt )fZt (y) with α(Zt ) ≥ 0 and
p(y) log
P
A α(Zt ) = 1.
p(y) dy. q(y)
Insert pA : log hence
Z KL(p∥q) =
p(y) pA (y) p(y) = log + log , q(y) pA (y) q(y) p(y) p(y) log dy + pA (y)
Z p(y) log
pA (y) dy. q(y)
The first term does not depend on q. Using the decomposition (6) in the second term, Z Z Z pA (y) pA (y) pA (y) p(y) log dy = WA pA (y) log dy + δ(A) pAc (y) log dy. q(y) q(y) q(y)
(7)
(8)
The first term equals WA KL(pA ∥q) ≥ 0, with equality if and only if q = pA . Therefore q = pA = qA minimises KL(p∥q) among all q supported on A. (ii) Predictive bound. Since qA = pA , using (6),
so
p(y) pAc (y) = WA + δ(A) , pA (y) pA (y) Z pAc (y) KL(p∥pA ) = p(y) log WA + δ(A) dy. pA (y)
Assume that the χ2 -divergence between pAc and pA is finite, i.e. Z (pAc (y))2 2 χ (pAc ∥ pA ) = dy − 1 ≤ C < ∞. pA (y) By Jensen’s inequality applied to (9), Z KL(p∥pA ) ≤ log
h pAc (y) i p(y) WA + δ(A) dy . pA (y)
(9)
A Predictive View on Streaming Hidden Markov Models
Since
R
p(y) dy = 1, this becomes Z pAc (y) dy . KL(p∥pA ) ≤ log WA + δ(A) p(y) pA (y)
Using the mixture decomposition (6), Z Z Z pAc (y) pAc (y) pAc (y) p(y) dy = WA pA (y) dy + δ(A) pAc (y) dy. pA (y) pA (y) pA (y) The first term equals WA . The second term can be bounded using the χ2 condition: Z Z (pAc (y))2 pAc (y) dy = dy ≤ 1 + C. pAc (y) pA (y) pA (y) Therefore
Z p(y)
pAc (y) dy ≤ WA + δ(A)(1 + C). pA (y)
Substituting into the Jensen bound yields KL(p∥pA ) ≤ log WA + δ(A) WA + δ(A)(1 + C) . Using WA = 1 − δ(A) and simplifying gives KL(p∥qA ) ≤ log 1 + δ(A)C . (iii) Choice of support. The upper bound depends on A only through δ(A). Since C ≥ 0, the function log(1 + δ(A)C) is increasing in δ ∈ [0, 1). Therefore minimising the bound is equivalent to minimising δ(A), i.e. maximising X WA = wt (Zt ). Zt ∈A
This is achieved by selecting the S largest posterior weights wt (Zt ).
B
Parametric updates
As a convenient special case, consider a regime-parameterised emission family with parameters θk , collected as ΘK = (θ1 , . . . , θK ). Conditional on (Zt , ΘK ), observations are independent with regime-specific emissions p(yt | zt , ΘK ) = p(yt | θzt ),
p(Yt | Zt , ΘK ) =
t Y
p(yτ | θzτ ).
τ =1
We assume independent priors across regimes, p(ΘK ) =
K Y
p(θk ).
k=1
Posterior factorisation.
Conditioned on a path Zt , the parameter posterior factorises across regimes: p(ΘK | Zt , Yt ) =
K Y k=1
p(θk | Yk (Zt )),
(10)
Gerardo Duran-Martin
where Yk (Zt ) = (yτ : 1 ≤ τ ≤ t, zτ = k) denotes the observations assigned to regime k under Zt . Equivalently, p(ΘK | Zt , Yt ) =
K Y
p(θk )
k=1
t Y
! 1{zτ =k}
p(yτ | θk )
.
τ =1
Thus each retained path maintains regime-specific posteriors that are updated only when the corresponding regime is active. Sequential update.
If Zt = Zt−1 ∪ {k}, only the k-th posterior changes:
p(θk | Yk (Zt )) ∝ p(θk | Yk (Zt−1 )) p(yt | θk ),
p(θj | Yj (Zt )) = p(θj | Yj (Zt−1 )) for j ̸= k.
Under conjugate priors this recursion is closed form and reduces to a constant-time sufficient-statistics update. Given the regime-k posterior under Zt , the one-step-ahead predictive is Z p(yt+1 | zt+1 = k, Yt , Zt ) = p(yt+1 | θk ) p(θk | Yk (Zt )) dθk ,
Path-conditional predictive.
which is available in closed form for exponential-family models with conjugate priors and for linear-Gaussian emissions.
C
Further experimental results
C.1
1D Gaussian HMM
Figure 4 compares one-step-ahead predictions across methods for S = 2. SHMM closely tracks regime changes and stabilises rapidly after transitions. RBPF exhibits delayed adaptation and occasionally interpolates between regimes when particle diversity collapses. Online EM produces smoother trajectories but with increased variance near switching points. Figure 5 examines predictive accuracy and runtime as the particle count S varies. SHMM attains low MAE and RMSE with small S and exhibits only marginal gains beyond S = 5, indicating that most posterior mass is captured by a small hypothesis set. In contrast, RBPF requires larger S to approach comparable accuracy, reflecting the variance induced by stochastic resampling. Runtime increases approximately linearly with S for both methods, though SHMM remains consistently faster across the evaluated range.
A Predictive View on Streaming Hidden Markov Models
Figure 4: One-step-ahead predictions for the two-regime Gaussian HMM with S = 2. Grey denotes observations, black the true regime mean, and red the predictive mean. SHMM adapts rapidly to regime changes, while RBPF and Online EM exhibit delayed or smoother transitions.
Figure 5: Predictive accuracy (MAE, RMSE) and runtime as a function of particle count S for SHMM and RBPF. SHMM attains low error with small S, while RBPF requires larger S for comparable accuracy. Runtime increases approximately linearly with S for both methods.