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Adaptive, Robust and Scalable Bayesian Filtering for Online Learning

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Bayesian filtering can serve as a single principled framework for online learning, and a new weighted-likelihood filter makes it provably robust to outliers at the computational cost of a standard Kalman filter.

desk verdict A well-written thesis that compiles the author's already-published Bayesian filtering work; the flagship robustness theorem has a real but easily patchable proof gap around exactly-zero weights. read the letter →

arxiv 2505.07267 v1 pith:OZAQECUA submitted 2025-05-12 stat.ML cs.LG

Gerardo Duran-Martin This is my paper · ORCID
classification stat.MLcs.LG MSC 62F1562M2068T05
keywords BayesianfilteringonlinelearninggeneralisedBayesoutlierrobustnessposteriorinfluencefunctionchangepointdetectionKalmanfilterlow-rankapproximation
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 thesis makes the case that Bayesian filtering—recursively updating a posterior distribution over model parameters—is a single principled foundation for online learning tasks such as continual learning, prequential forecasting, and contextual bandits. It contributes three toolkits: the BONE framework, which unifies many existing adaptive methods under a five-component design space; the WoLF filter, which replaces the likelihood in a Kalman-style update with a weighted log-likelihood and is proven to bound the influence of arbitrarily large outliers; and scalable approximations (subspace EKF, PULSE, low-rank EKF) that make filtering practical for deep neural networks. The central theoretical result is Theorem 4.5, which shows the WoLF posterior has a bounded posterior influence function under weighting schemes like inverse multi-quadratic and Mahalanobis weights. If the claims hold, robust Bayesian online learning can be achieved at the computational cost of a standard Kalman filter rather than expensive variational iterations.

What carries the argument

The central object carrying the robustness argument is the weighted observation likelihood: the loss (4.2) that replaces the standard log-likelihood in the recursive posterior update (4.1). Concretely, the update equals the Kalman filter with the measurement precision $R_t^{-1}$ replaced by $W^2(y_t,\hat y_t) R_t^{-1}$, so a single scalar weight controls how much the current observation moves the posterior. The proof of Theorem 4.5 decomposes the posterior influence function—the KL divergence between the posterior under a contaminated observation and the uncontaminated posterior—into three terms ((T.1), (T.2), (T.3)) and bounds each using spectral-norm inequalities; the key condition is that the weight decays fast enough relative to the growth of the contaminated point. For non-stationarity, the auxiliary variable $\psi_t$ (e.g., a runlength) is the object that encodes regime information, and the BONE framework's five components parameterise the design space. For scalability, the machinery is low-rank structure: a projection matrix $A \in \mathbb{R}^{D\times d}$ for subspace filters, and a diagonal-plus-low-rank precision matrix for LoFi.

What would settle it

Take the one-dimensional linear Gaussian state-space model from Section 4.7.1, use the threshold weight (4.7) with a finite threshold $c$, and send the contaminated observation $y_t^c$ to infinity. Since the weight becomes exactly zero for $\|y_t^c-\hat y_t\|>c$, the theorem's bound on the log-determinant term (T.3) may fail; compute the KL divergence between the contaminated and uncontaminated posteriors numerically as $y_t^c$ grows, and check whether it remains bounded as Theorem 4.5 would require.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that generalised Bayesian inference can be imported into filtering without sacrificing closed-form updates: defining the loss as $\ell_t(\theta_t) = -W^2(y_t,\hat y_t)\log p(y_t|\theta_t)$ and using it in place of the log-likelihood yields the same Kalman precision update with $R_t^{-1}$ scaled by $W^2$. Theorem 4.5 proves that whenever the weight satisfies $\sup_y W(y,\hat y)<\infty$ and $\sup_y W(y,\hat y)^k \|y-\hat y\|<\infty$ for $k\ge 2$, the posterior influence function is bounded, so a single arbitrarily large contaminated observation has bounded effect. The thesis also claims that its BONE framework, with components (M.1) measurement model, (M.2) auxiliary variable, (M.3) conditional prior, (A.1) posterior algorithm, and (A.2) weighting, subsumes a wide range of prior online-learning methods and that a new instantiation, RL[1]-OUPR*, handles both gradual and abrupt change; and that the low-rank/subspace filters make neural-network-scale filtering practical. These are presented as a unified toolkit for Bayesian online learning rather than as separate tricks.

Load-bearing premise

The proof of the robustness theorem assumes that no observation is ever assigned weight exactly zero; the threshold weight WoLF-TMD does assign weight exactly zero to sufficiently far-out observations, so the written proof has a gap there, although the result can be repaired by treating zero-weight observations as making the posterior equal to the prior.

Editorial extensions

If this is right

  • A single weighting function converts the standard Kalman filter into an outlier-robust filter with the same $O(D^3)$ update cost, so robust filtering no longer requires repeated variational inner iterations.
  • The BONE framework provides a common language for adaptive online learning methods, so existing algorithms like BOCD, runlength priors, and changepoint-probability models become interchangeable design choices.
  • The new hybrid rule RL[1]-OUPR* gives a single-hypothesis algorithm that tracks both gradual drift and abrupt changepoints, which the experiments show outperforms pure runlength and pure OU methods on several benchmarks.
  • The scalable variants—subspace EKF, PULSE, and LoFi—reduce memory and time to the point where Bayesian filtering can update a deep neural network online; LoFi maintains a diagonal-plus-low-rank precision matrix to achieve this.

Reading between the lines

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

  • The W=0 gap for WoLF-TMD suggests a general design principle for future robust filters: either choose weights bounded strictly away from zero, or handle the zero-weight branch explicitly so that the log-determinant bound remains valid.
  • The bounded-influence condition $\sup_y W(y,\hat y)^k \|y-\hat y\|<\infty$ for $k\ge 2$ can be used as a recipe for designing new weighting functions beyond the IMQ and threshold families; smooth compactly-supported weights would need a zero-branch treatment.
  • Combining WoLF with online changepoint detection (as done in Section 4.7.4) opens a testable hypothesis: outlier contamination and genuine regime change can be disentangled, so a robust filter should reduce false changepoint alarms on datasets with labelled outliers and labelled changepoints.
  • The BONE framework's design space suggests that adaptive methods' performance differences often reduce to the choice of auxiliary variable and prior-reset rule, which could allow a systematic empirical comparison rather than the current ad-hoc benchmarking.
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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

2 major / 4 minor

Summary. This PhD thesis develops Bayesian filtering methods for online machine learning, organized around three goals: adaptivity, robustness, and scalability. Chapter 3 introduces BONE, a framework that unifies existing online-learning algorithms as choices of measurement model, auxiliary variable, conditional prior, posterior approximation, and weighting scheme, and proposes a new method RL[1]-OUPR* for environments with both gradual drift and abrupt changes. Chapter 4 introduces WoLF, a generalized-Bayes filter that replaces the log-likelihood with a weighted version, yielding closed-form KF/EKF-style updates at the same O(D^3) cost as the Kalman filter. The chapter states Theorem 4.5, which claims that WoLF has a bounded posterior influence function under weight conditions satisfied by the proposed IMQ, MD, and TMD weights, and presents experiments on 2D tracking, MLP regression, heavy-tailed regression, and EWMA smoothing. Chapter 5 proposes subspace and low-rank approximations for scaling filtering to neural networks. The central theoretical risk is the proof of Theorem 4.5, which has a gap for exactly-zero weights and singular information matrices; the gap is local and repairable.

Significance. If Theorem 4.5 is correct, WoLF is a practically valuable contribution: outlier-robust Bayesian filtering with closed-form updates, computational cost matching the standard Kalman filter, and a bounded posterior influence function under explicit conditions on the weight function. The empirical comparisons against KF-B, KF-IW, OGD, and RBPF on multiple tasks support the practical relevance, and the experiments are reported with clear metrics and baselines. The BONE framework is a useful descriptive taxonomy that connects Bayesian online changepoint detection, continual learning, and bandit algorithms, although it is not itself a new algorithm. Strengths include explicit assumptions rather than data-fitted claims, reproducible pseudocode, and the fact that the robustness theorem is a parameter-free derivation from stated moment conditions rather than a circular argument.

major comments (2)
  1. [§4.6.2, Proposition 4.18 (bound for T.3)] The proof of Theorem 4.5 is incomplete as written. Equation (4.39) gives the valid bound log|bar Sigma_t| <= -log(|Sigma_pred^{-1}| + bar(w_t)^{2D}|H'R^{-1}H|), whose right-hand side is bounded above by -log|Sigma_pred^{-1}| < infinity because Sigma_pred^{-1} is positive definite. Instead of stopping there, the proof passes to the 'minimum' bound (4.40) and then to (4.41), which invokes -2D log(bar(w_t)). For the threshold weight WoLF-TMD in (4.7), bar(w_t) can be exactly zero, and -log(0) = +infinity; the assertion 'sup bar(w_t) < infinity implies sup log bar(w_t) < infinity' is false when the weight attains zero. The same branch fails whenever H'R^{-1}H is singular, which occurs whenever the observation dimension is less than the state dimension, including the 2D tracking experiment of Section 4.7.1 with D=4 and o=2. The theorem is repairable by stopping at (4.39), but as written the proof does not establish the claimed robustness for the recommended WoLF-TMD weight or for common singular information matrices.
  2. [§4.6.2, Lemma 4.15 and Eq. (4.29)] The derivation of the bound for the mean-difference term (T.2) contains a division-by-zero issue when weights can be zero. In the chain leading to (4.29), the proof introduces an expression with w_t^{-2} in the denominator; for W = 0, this is undefined. The desired bound can be obtained directly from Lemma 4.14 without this step, because the denominator in the expression before the w^{-2} substitution is at least sigma_min(Sigma_pred^{-1})^2 > 0, so the proof should be rewritten to avoid the undefined intermediate expression. As written, this is a technical error in a load-bearing bound, though it is also locally repairable.
minor comments (4)
  1. [§4.5, Theorem 4.5 statement] The theorem states the weight condition as 'sup_y W(y, ŷ) < inf' and 'sup_y W(y, ŷ)^k ||y|| < inf'; presumably these should be '< infinity'. This formulation should be corrected.
  2. [§4.4, Eq. (4.7)] The text says the weighting functions satisfy W : R^o x R^o -> R_{++}, but the threshold weight WoLF-TMD in (4.7) takes the value 0. The codomain should be R_{+} or the statement should be adjusted.
  3. [§4.6.2, proof of Theorem 4.5] In the paragraph before Lemma 4.13, the goal is stated as showing that 'qLG is outlier robust'; this should read q^{W-LG} to avoid confusion with the non-robust linear Gaussian filter of Theorem 4.4.
  4. [§3.4, Algorithm 10 (RL[1]-OUPR*)] There are minor notation inconsistencies: the changepoint probability is denoted 'pi' in the pseudocode but 'kappa' in the text, and the weight 'nu_t(r(1))' is missing a subscript in some places. These should be harmonized for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the robustness theorem is proved from explicit weight-moment assumptions, BONE is a descriptive taxonomy, and the new methods are validated against external baselines; the only issue is a repairable proof gap in Prop 4.18, which is a correctness concern, not circularity.

full rationale

The central derivation in Chapter 4 is not circular. Theorem 4.5 states that a weighted-observation posterior has bounded posterior influence when the weight function satisfies sup_y W(y,yhat) < infinity and sup_y W(y,yhat)^k ||y|| < infinity for k >= 2. The proof then bounds the three KL terms (T.1), (T.2), and (T.3) using the Kalman precision update and these moment conditions. The assumptions are not the same as the conclusion: for unweighted KF, W = 1 violates the moment condition, and Theorem 4.4 separately proves an unbounded PIF, so the theorem has genuine content. The proposed weights (IMQ, MD, TMD) are inputs whose robustness is established by the theorem, not fitted values dressed up as predictions. The BONE framework in Chapter 3 is explicitly a unifying representation of existing methods through modelling and algorithmic choices; it does not derive a numerical prediction from its own definitions, and the new RL[1]-OUPR* method is a specific combination of an OU prior and prior reset that is evaluated empirically against external baselines on forecasting, classification, and bandit tasks. Chapter 5 presents previously published scalable-filter methods and compares them on Fashion MNIST; the self-citations are to prior papers by the same group, but they are not used as the sole justification for the thesis's new claims. The empirical sections tune hyperparameters on warmup data and then compare on held-out or sequential data, so there is no fitted parameter being renamed as a prediction. One technical issue is present in the written proof of Proposition 4.18: the min-branch (4.40)-(4.41) can be infinite for the threshold weight W = 0 (WoLF-TMD) or when H'R^{-1}H is singular. This is a correctness gap in the proof, not a circularity, and it is repairable by stopping at the finite bound in (4.39). Overall, no load-bearing self-citation, no ansatz smuggled in by citation, and no derivation that reduces to its inputs by construction.

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

The central claims rest on standard Gaussian and subspace assumptions plus the explicit moment conditions on W. No new physical entities are introduced. The main free parameters are the hazard rate kappa, threshold epsilon, weight cutoff c, state noise Qt, and subspace dimension d, all set by hand or warmup tuning.

free parameters (5)
  • hazard rate kappa = tuned during warmup per experiment
    Prior probability of a changepoint in RL-PR and RL[1]-OUPR*; in Section 3.5.1 hyperparameters including kappa are tuned on the first 300 observations; sensitivity is shown in Figures 3.7 and 3.9.
  • restart threshold epsilon = chosen per experiment
    Threshold in RL[1]-OUPR* deciding between hard reset and OU soft update, equations (3.19)-(3.20); no default value is provided and it must be set by the user.
  • soft threshold c for weights = c=4 in Section 4.7.4, c=0.05 in Section 4.7.5
    Controls how quickly WoLF down-weights large residuals in the IMQ or Mahalanobis weighting functions; chosen by hand per experiment and not learned.
  • dynamics covariance Qt = e.g., Qt=10^-4 I in Section 4.7.3; tuned in Section 3.5.1
    State noise covariance in the SSM; central to the adaptivity vs stability tradeoff; selected manually or by warmup tuning.
  • subspace dimension d = not specified in available text
    Dimensionality of the affine subspace in Chapter 5 (Section 5.1); a free design choice inherited from the lottery-ticket and SVD warmup.
assumptions (5)
  • domain assumption State and measurement noise are Gaussian with known covariances
    Used throughout: Kalman filter, EKF, and WoLF update equations (Sections 2.5, 4.2). The thesis acknowledges this and relaxes it via generalized Bayes, but the closed-form updates rely on Gaussianity.
  • domain assumption Posterior density over model parameters is approximated as Gaussian
    All recursive methods maintain a Gaussian posterior (Sections 2.4, 3.1.5, 5.3). This is an approximation for non-linear models and may miss multi-modality.
  • domain assumption Weight function W satisfies sup_y W(y,yhat) < inf and sup_y W(y,yhat)^k ||y - yhat|| < inf for k>=2
    Explicit condition in Theorem 4.5 for outlier robustness; it holds for the proposed IMQ, Mahalanobis, and threshold weights, but is a non-trivial restriction on admissible weight functions.
  • domain assumption The lottery-ticket or subspace hypothesis: neural network parameters live in a low-dimensional affine subspace
    Used in Chapter 5.1 to justify subspace EKF; the thesis cites prior evidence (Li et al. 2018, Larsen et al. 2022) but does not prove it for the online setting.
  • ad hoc to paper The proof of Theorem 4.5 implicitly assumes W > 0 for the log-determinant bound
    Equation (4.41) takes sup log(W) < inf from sup W < inf, which fails when W=0; this is unflagged and excludes WoLF-TMD as written.

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Cite this review

Pith. "Pith review of Adaptive, Robust and Scalable Bayesian Filtering for Online Learning." pith.science (2026). https://pith.science/paper/OZAQECUA

@misc{pith2026250507267,
  author       = {Pith},
  title        = {Pith review of: Adaptive, Robust and Scalable Bayesian Filtering for Online Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZAQECUA}},
  note         = {Machine review of arXiv:2505.07267}
}
read the original abstract

In this thesis, we introduce Bayesian filtering as a principled framework for tackling diverse sequential machine learning problems, including online (continual) learning, prequential (one-step-ahead) forecasting, and contextual bandits. To this end, this thesis addresses key challenges in applying Bayesian filtering to these problems: adaptivity to non-stationary environments, robustness to model misspecification and outliers, and scalability to the high-dimensional parameter space of deep neural networks. We develop novel tools within the Bayesian filtering framework to address each of these challenges, including: (i) a modular framework that enables the development adaptive approaches for online learning; (ii) a novel, provably robust filter with similar computational cost to standard filters, that employs Generalised Bayes; and (iii) a set of tools for sequentially updating model parameters using approximate second-order optimisation methods that exploit the overparametrisation of high-dimensional parametric models such as neural networks. Theoretical analysis and empirical results demonstrate the improved performance of our methods in dynamic, high-dimensional, and misspecified models.

Figures

Figures reproduced from arXiv: 2505.07267 by the authors.

Figure 1.1
Figure 1.1. One-dimensional projection of a noisy-two dimensional dynamical system. In the top panel, [PITH_FULL_IMAGE:figures/full_fig_p017_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. (Left panel) Mean estimate of the parameters. The solid lines correspond to the recursive￾Bayes estimate of the mean. The dashed lines correspond to the offline estimate of the mean using Ridge regression. (Right panel) RMSE on a held-out test set. estimate µt tend to, and eventually match, the Ridge estimate of model parameters. As a consequence, the RMSE on the held-out set for the recursive-Bayes estimate match t… view at source ↗
Figure 2.2
Figure 2.2. Decision boundaries for the logistic regression model as a function of the number of processed [PITH_FULL_IMAGE:figures/full_fig_p030_2_2.png] view at source ↗
Figures from the paper (36 more)
Figure 2.3
Figure 2.3. Figure 2.3: The solid lines show the posterior mean estimate of model parameters and two standard [PITH_FULL_IMAGE:figures/full_fig_p030_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: Decision boundaries for the classification problem using a neural network trained using the [PITH_FULL_IMAGE:figures/full_fig_p031_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Cumulative prequential accuracy for the non-linear classification problem trained using the [PITH_FULL_IMAGE:figures/full_fig_p031_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Sample run of the piecewise regression process. Each box titled [PITH_FULL_IMAGE:figures/full_fig_p036_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: (Left panel) Rolling prequential RMSE of the linear model trained using various levels of Qt. (Right panel) Total prequential RMSE of the linear model varying levels of prediction error. When Qt is close to zero, the model exhibits limited adaptability, retaining mor…
Figure 2.8
Figure 2.8. Figure 2.8: Rolling prequential accuracy for the non-linear classification problem, trained using R-VGA [PITH_FULL_IMAGE:figures/full_fig_p040_2_8.png]
Figure 3.1
Figure 3.1. Figure 3.1: Two-levelled hierarchical state-space model (SSM) with known dynamics, motivating our [PITH_FULL_IMAGE:figures/full_fig_p043_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: The top panel shows the target variable (electricity consumption) from March 1 2020 to March 12 2020. The bottom panel shows the twelve-hour rolling relative absolute error of predictions for the same time window. The dotted black line corresponds to March 7 2020, wh…
Figure 3.3
Figure 3.3. Figure 3.3: One day ahead electricity forecasting results for Figure [PITH_FULL_IMAGE:figures/full_fig_p069_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: One day ahead electricity forecasting results for [PITH_FULL_IMAGE:figures/full_fig_p069_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Distribution of the 5-day mean absolute error (MAE) for each of the competing methods on [PITH_FULL_IMAGE:figures/full_fig_p070_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Misclassification rate of various methods on the online classification with periodic drift task. [PITH_FULL_IMAGE:figures/full_fig_p070_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: Accuracy of predictions for RL[1]-PR as a function of the number of hypothesis and the prior probability of a changepoint κ. The black dotted line is the performance of RL[1]-OUPR* reported in [PITH_FULL_IMAGE:figures/full_fig_p071_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: Misclassification rate of various methods on the online classification with drift and jumps [PITH_FULL_IMAGE:figures/full_fig_p071_3_8.png]
Figure 3.9
Figure 3.9. Figure 3.9: Accuracy of predictions for RL[K]-PR as a function of the number of hypotheses (K) and the probability of a changepoint κ. The black dotted line is the performance of RL[1]-OUPR* reported in [PITH_FULL_IMAGE:figures/full_fig_p072_3_9.png]
Figure 3.10
Figure 3.10. Figure 3.10: Regret of competing methods on the contextual bandits task. Confidence bands are com [PITH_FULL_IMAGE:figures/full_fig_p073_3_10.png]
Figure 3.11
Figure 3.11. Figure 3.11: The left panel shows a sample run of the piecewise polynomial regression with dependence across segments. The x-axis is for the features, the (left) y-axis is for measurements together with the estimations made by RL[1]-PR, RL-MMPR, and RL[1]-OUPR*, the (right) y-ax…
Figure 3.12
Figure 3.12. Figure 3.12: Count of changepoints over an experiment for 100 trials. The orange line shows the true [PITH_FULL_IMAGE:figures/full_fig_p074_3_12.png]
Figure 4.1
Figure 4.1. Figure 4.1: Weighted likelihood (unnormalised) for a standard Gaussian. [PITH_FULL_IMAGE:figures/full_fig_p078_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: First state component of the SSM (4.42). The grey dots are measurements sampled from (4.42) and the red crosses are measurements sampled from an outlier measurement process. The dotted blue line shows the KF posterior mean estimate and the solid orange line shows our…
Figure 4.3
Figure 4.3. Figure 4.3: The left panel shows a sample path using the Student variant and the right panel shows a [PITH_FULL_IMAGE:figures/full_fig_p089_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Distribution (across 500 2d tracking trials) of RMSE for first component of the state vector, [PITH_FULL_IMAGE:figures/full_fig_p090_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: RMedSE versus time per step (relative to the [PITH_FULL_IMAGE:figures/full_fig_p092_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Results with sorted data. Left panel shows a run of each filter on the 1d regression, with [PITH_FULL_IMAGE:figures/full_fig_p093_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Results with unsorted inputs. The left panel shows a run of each filter with the underlying [PITH_FULL_IMAGE:figures/full_fig_p094_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: The figure shows a run of each filter with the underlying data-generating function in solid [PITH_FULL_IMAGE:figures/full_fig_p094_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Sample run of the heavy-tailed-regression process. Each box corresponds to the samples [PITH_FULL_IMAGE:figures/full_fig_p096_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: shows the distribution of the RMSE for all methods after 30 trials [PITH_FULL_IMAGE:figures/full_fig_p096_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Segmentation of the non-stationary linear regression problem. The left panel shows the seg [PITH_FULL_IMAGE:figures/full_fig_p096_4_11.png]
Figure 4.12
Figure 4.12. Figure 4.12: Log-returns of DJI from 2019 to 2024. The outliers at the beginning of 2021 and at the [PITH_FULL_IMAGE:figures/full_fig_p100_4_12.png]
Figure 4.13
Figure 4.13. Figure 4.13: EWMA and WolF-EWMA estimates over DJI log-returns from 2019 to 2024. [PITH_FULL_IMAGE:figures/full_fig_p101_4_13.png]
Figure 4.14
Figure 4.14. Figure 4.14: Smoothing factors for EWMA and WolF-EWMA for the DJI log-returns from 2019 to 2024. [PITH_FULL_IMAGE:figures/full_fig_p101_4_14.png]
Figure 5.1
Figure 5.1. Figure 5.1: Comparison of the prequential accuracy on the Fashion MNIST dataset for the subspace [PITH_FULL_IMAGE:figures/full_fig_p119_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Comparison of the prequential accuracy on the Fashion MNIST dataset for the PULSE [PITH_FULL_IMAGE:figures/full_fig_p120_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Comparison of the prequential accuracy on the Fashion MNIST dataset. The [PITH_FULL_IMAGE:figures/full_fig_p120_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Comparison of the prequential misclassification rate on the Fashion MNIST dataset on the [PITH_FULL_IMAGE:figures/full_fig_p121_5_4.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.