Doubly Outlier-Robust Online Infinite Hidden Markov Model

Horace Yiu; Leandro Sánchez-Betancourt; Álvaro Cartea; Gerardo Duran-Martin

stat.MLApr 15, 2026

Doubly Outlier-Robust Online Infinite Hidden Markov Model

Authors: Horace Yiu, Leandro Sánchez-Betancourt, Álvaro Cartea, Gerardo Duran-Martin

Organizations: Mathematical Institute, University of Oxford · Oxford-Man Institute of Quantitative Finance

Abstract

We derive a robust update rule for the online infinite hidden Markov model (iHMM) for when the streaming data contains outliers and the model is misspecified. Leveraging recent advances in generalised Bayesian inference, we define robustness via the posterior influence function (PIF), and provide conditions under which the online iHMM has bounded PIF. Imposing robustness inevitably induces an adaptation lag for regime switching. Our method, which is called Batched Robust iHMM (BR-iHMM), balances adaptivity and robustness with two additional tunable parameters. Across limit order book data, hourly electricity demand, and a synthetic high-dimensional linear system, BR-iHMM reduces one-step-ahead forecasting error by up to 67% relative to competing online Bayesian methods. Together with theoretical guarantees of bounded PIF, our results highlight the practicality of our approach for both forecasting and interpretable online learning.

Figures & tables

Figure 1: Data ( blue dotted line ) generated according to ( 139 ) with Student- $t$ observation noise. The outliers are marked with x . The solid red line shows our BR-iHMM , and the solid orange line shows the standard online iHMM. Regions around the posterior predictive mean cover $\pm 2$ standard deviations as error bounds.

Panel of Figure 2 — Figure 2: Bayesian architecture of the BR-iHMM . Variables in a circle are updated during inference. Red variables are fixed hyperparameters. For instance, the priors of $\gamma$ and $\alpha$ are both uninformed $\Gamma(1,1)$ for all our experiments.

Model	Synthetic	Electricity	OFI
BR-iHMM	46.1 $\pm$ 0.00301	0.47 $\pm$ 0.04	0.616 $\pm$ 0.082
WoLF-iHMM	103.8 $\pm 0.01155$	0.63 $\pm$ 0.03	0.623 $\pm$ 0.089
iHMM	101.7 $\pm 0.02636$	0.57 $\pm$ 0.03	0.620 $\pm$ 0.080
offline-iHMM	2.9	0.32	0.552
BOCD	$123.12\pm 0.01365$	$0.80\pm 0.11$	0.733

Table 1: One-step ahead RMSE (Mean

\pm

stdev) over 100 runs. The most precise online model is bolded.

Figure 3: Synthetic linear data. Outliers are marked by red vertical dashes. Yellow and blue regions correspond to regimes 2 and 3. First subplot tracks rolling RMSE of mean predictions across the 100 runs. Bottom plot shows average MAP state. First 900 predictions are plotted. The full Figure E.11 is in the appendix.

Figure 4: Cumulative RMSE for OFI predictions. offline-iHMM is not plotted to allow zooming into the difference between the online models. Vertical dotted red lines indicate an outlier (defined as outside 1.5 $\times$ inter-quartile range).

Model	PPV	TPR	Delay
BR-iHMM	$\textbf{0.963}\pm\textbf{0.001}$	$\textbf{0.996}\pm\textbf{0.001}$	$3.4\pm 0.0$
WoLF-iHMM	$0.533\pm 0.002$	$0.344\pm 0.002$	$\textbf{0.0}\pm\textbf{0.0}$
iHMM	$0.551\pm 0.002$	$0.363\pm 0.008$	$0.2\pm 0.0$
offline-iHMM	$0.996$	$0.997$	$0.0$

Table 2: Values are averaged over 100 runs for online models. Higher PPV and TPR is better, lower DD is better.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description
$\left\lVert\cdot\right\rVert_{2}$	$\ell_{2}$ norm.
$\left\lVert\cdot\right\rVert_{\mathbf{R}_{t}}$	Mahalanobis norm with respect to $\mathbf{R}_{t}$ .
$\mathcal{N}({\bm{y}}\mid\bm{\mu},\bm{\Sigma})$	Multivariate Gaussian density evaluated at ${\bm{y}}$ with mean $\bm{\mu}$ and covariance $\bm{\Sigma}$ .
${\rm DP}(\gamma,H)$	Dirichlet process with base measure $H$ and $\gamma>0$
${\rm Dir}(\bm{\pi}\mid\bm{\beta})$	Dirichlet density evaluated at $\bm{\pi}$ with base prior $\bm{\beta}$ .
$SB(\gamma)$	Stick breaking distribution with concentration $\gamma$

Appendix

Table A.3: Table of notations.

\displaystyle\omega_{t}^{(i)}=\omega_{t-1}^{(i)}\times\frac{P({\bm{y}}_{t}\mid\hat{s}_{t-1}^{(i)},{\bm{x}}_{t},\bm{\Phi}_{t-1}^{(i)},\bm{\Psi}_{t-1}^{(i)})}{\sum_{j=1}^{N}P({\bm{y}}_{t}\mid\hat{s}_{t-1}^{(i)},{\bm{x}}_{t},\bm{\Phi}_{t-1}^{(j)},\bm{\Psi}_{t-1}^{(j)})}

Appendix

Algorithm 6 Online iHMM via Particle Learning ( iHMM )

\displaystyle\omega_{t+B}^{(i)}\propto\omega_{t}^{(i)}\,\frac{P({\bm{y}}_{t+1:t+B}\mid\hat{s}_{t}^{(i)},\bm{\Phi}_{t}^{(i)},\bm{\Psi}_{t}^{(i)},{\bm{x}}_{t+1:t+B})}{\sum_{j=1}^{N}P({\bm{y}}_{t+1:t+B}\mid\hat{s}_{t}^{(i)},\bm{\Phi}_{t}^{(j)},\bm{\Psi}_{t}^{(j)},{\bm{x}}_{t+1:t+B})}

Appendix

Algorithm 7 Batched Online iHMM with WoLF ( BR-iHMM )

Parameter	Model
$\tau_{\text{ESS}}$	all	BayesOpt
$c^{2}$	WoLF-iHMM , BR-iHMM	BayesOpt
$B$	BR-iHMM	BayesOpt

Appendix

Table C.4 : Summary of hyperparameters across iHMMs.

Model	Synthetic	Electricity	OFI
BR-iHMM	46.1 $\pm$ 0.00301	0.47 $\pm$ 0.04	0.616 $\pm$ 0.082
WoLF-iHMM	103.8 $\pm 0.01155$	0.63 $\pm$ 0.03	0.623 $\pm$ 0.089
iHMM	101.7 $\pm 0.02636$	0.57 $\pm$ 0.03	0.620 $\pm$ 0.080
offline-iHMM	2.9	0.32	0.552
BOCD	$123.12\pm 0.01365$	$0.80\pm 0.11$	0.733

Appendix

Table E.5 : Forecast performance in terms of one-step ahead RMSE (Mean

\pm

stdev) over 100 runs. The most precise online model is bolded. The BOCD implementation for the OFI experiment is from Tsaknaki et al. (2025) and has no randomisation. offline-iHMM is only ran once.

Dataset	BR-iHMM	WoLF-iHMM	iHMM
Synthetic	64.0 $\pm$ 0.27	66.1 $\pm$ 0.35	66.3 $\pm$ 0.20
Electricity	274.2 $\pm$ 0.5	351.1 $\pm$ 1.7	351.9 $\pm$ 1.4
LOB	95.2 $\pm$ 0.6	94.3 $\pm$ 1.8	96.1 $\pm$ 1.0

Appendix

Table E.6 : Average runtime in seconds across 100 runs across experiments (Mean

\pm

stdev). Fastest model is highlighted.

Figure E.5 : State-space PIF. T he non-batched PIF grows unbounded and is irrespective of the IMQ weights. The batched PIFs remains bounded even under large offsets in contrast. $c^{2}$ is the IMQ scale.

Figure E.6 : The left hand plot shows a lack of preference if state variance is high. In the right hand plot, the batch size is $B=6$ and an outlier ${\bm{y}}_{1}^{c}$ is in the batch.

Figure E.7 : Difference in log-likelihood for the unknown mean unknown variance case.

Figure E.8 : BR-iHMM (with known variance) vs classical iHMM (unknown-var) . The unknown variance iHMM has a Student-t posterior predictive and is still sensitive to outliers. Student-t remains non-robust and changes states, yet the BR-iHMM model is robust.

Dataset	Hyperparameter	BR-iHMM	WoLF-iHMM	iHMM
Well-log	$\tau_{{\rm ESS}}$	10	10	54
	$c^{2}$	2.124	1.657	–
	$B$	21	–	–
Synthetic	$\tau_{{\rm ESS}}$	2	1	0
	$c^{2}$	1.348	3.589	–
	$B$	2	–	–

Appendix

Table E.7 : Summary of fitted hyperparameters obtained from Bayesian optimisation. Same hyperparameters are used in subsequent runs throughout the same dataset. Extra hyperparameters are included where appropriate.

\Sigma_{0}

is the initial variance for newly initiated states, for an

m

-dimensional state vector, the initial covariance matrix is then set to

\Sigma_{0}\mathbf{I}_{m}

where

\mathbf{I}_{m}

is the

m

-dimensional identity matrix. For the Well-log and Synthetic experiment,

\Sigma_{0}

is set to

1

\rho^{*}

is the autoregressive coefficient in the OFI DGP ( 141 ).

p

denotes the degree of feature engineering of the weather covariates, see Section E.11 . MAX_STATES = 30 for all three online iHMM models.

Figure E.9 : Comparison of iHMM filtered mean and maximum a-posteriori (MAP) state inference with different degrees of robustness on the well-log data. Top subplot overlays the well-log data with the filtered values of iHMM , iHMM (unknown-var) , WoLF-iHMM , BR-iHMM , DSM-BOCD , and offline-iHMM . Bottom subplot shows the MAP state inference.

Model	PPV	TPR	Delay
BR-iHMM	$\textbf{0.4739}\pm\textbf{0.0636}$	$\textbf{0.5797}\pm\textbf{0.0885}$	$78.06\pm 46.95$
WoLF-iHMM	$0.3384\pm 0.588$	$0.3384\pm 0.0588$	$63.20\pm 16.50$
iHMM	$0.4086\pm 0.0635$	$0.4970\pm 0.0924$	$\textbf{19.19}\pm\textbf{11.38}$

Appendix

Table E.8 : Regime classification performances on synthetic data. PPV, TPR, and delays are averaged over 100 runs for online models. Higher PPV and TPR is better, lower detection delay is better.

Figure E.10 : Lookahead MAE and detection delay for different batch sizes. The blue graph, corresponding to the axis on the left, plots the detection delay of BR-iHMM as we increase the batch size. The orange graph, corresponding to the axis on the right, plots the lookahead MAE for different batch sizes.

Figure E.11 : Synthetic linear data. Outlier observations are marked by red vertical dashes. Yellow and blue regions correspond to regimes 2 and 3. First subplot tracks rolling RMSE of mean predictions across the 100 runs. Second subplot compares the L2 distance between average latent state (per model) versus the true data generating state $\bm{\mu}_{t}$ . The bottom plot shows average MAP state. Each plot overlays the results for all four models. All data points are plotted.

Figure E.12 : RMSE and detection delay for different batch sizes. The blue graph, corresponding to the axis on the left, plots the detection delay of BR-iHMM as we increase the batch size. The orange graph, corresponding to the axis on the right, plots the RMSE for different batch sizes.

Model	PPV	TPR	Delay
BR-iHMM	$\textbf{0.963}\pm\textbf{0.001}$	$\textbf{0.996}\pm\textbf{0.001}$	$3.4\pm 0.0$
WoLF-iHMM	$0.533\pm 0.002$	$0.344\pm 0.002$	$\textbf{0.0}\pm\textbf{0.0}$
iHMM	$0.551\pm 0.002$	$0.363\pm 0.008$	$0.2\pm 0.0$
offline-iHMM	$0.996$	$0.997$	$0.0$

Appendix

Table E.9 : Values are averaged over 100 runs for online models. Higher PPV and TPR is better, lower DD is better.

Figure E.13 : IID synthetic data with Gaussian noise. Outliers are marked with a red cross . Top panel compares predicted value, lower panel compares MAP states.

Figure E.14 : IID synthetic data with t-noise. Outliers are marked with a red cross . Top panel compares predicted value, lower panel compares MAP states.

Figure E.15 : Hourly electricity demand. Top panel plots the rolling RMSE with a window of 100 for each model. Bottom panel plots MAP state. Each plot overlays the results for all three models. The MAP state of iHMM did not change, WoLF-iHMM detected a small change towards the end.

Figure E.16 : Electricity data. Standardised data for all 31912 data points standardised per dimension. Electricity demand is measured in Load (kW) . Data shows clear periodicity.

Figure E.17 : Electricity demand forecasts produced by iHMM , WoLF-iHMM , and BR-iHMM respectively. MAP states are plotted on the right hand axis for each model in purple . The true standardised electricity demands are plotted in blue in the background.

Figure F.18 : Top panel plots the observations. Middle panel plots the actual variance, along with variance estimates between a naive one-state GARCH and an iHMM of GARCH models. Bottom panel plots inferred regime probabilities for the iHMM GARCH model.

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