A Predictive View on Streaming Hidden Markov Models

A hidden Markov model (HMM) combines three roles: inference of a hidden-state belief from observations, propagation through a Markov transition, and emission back to observation space. We show that full, time-indexed Predictive Information Bottleneck VJEPA (PIB-VJEPA) exposes the same computational structure: a stochastic context encoder plays the role of an amortized filtering distribution, a probabilistic predictor defines latent-state dynamics, and a decoder, inverse target encoder, or induced implicit conditional supplies the emission direction. We distinguish 4 progressively stronger levels of correspondence and give sufficient conditions for exact sequence-level HMM equivalence. To make the connection concrete, we introduce Markov-Chain JEPA (MCJEPA), which replaces the latent predictor by a learned transition matrix; in the finite time-homogeneous case, matrix powers guarantee exact multi-horizon Chapman--Kolmogorov consistency. Conditioned discrete-state transitions, continuous-state Markov kernels, and continuous-time dynamics extend this construction, while deterministic temporal JEPA appears as a degenerate Dirac-kernel special case. We further interpret predictive information-bottleneck learning as seeking a compact predictive state: compression promotes minimality, while residual predictability tests sufficiency. Controlled experiments support transition composition, the filtering interpretation, predictive Markovization in a known synthetic process, and the distinction between JEPA latent prediction and HMM-style sequence learning. Together, these results give temporal JEPA a principled state-space interpretation.

Accurate posterior prediction need not require accurate approximation of Bayesian updates. We prove that an unbounded gap between the update maps can coexist with vanishing predictive KL for every fixed finite $K\ge2$ in a stationary symmetric Gaussian HMM. Exact Bayesian mixing and an explicit deterministic radial filter act on the same $K-1$ belief coordinates. As $q\to0^+$, their separation in centered logits in the worst case grows at least linearly in the natural confidence scale $L_K(q)$, while their categorical $D_{\mathrm{KL}}(\mathrm{exact}\|\mathrm{radial})$ vanishes at the same explicit witness. Along stationary HMM trajectories, the expected terminal KL between filtered posteriors also converges to zero at $H(q)=\lceil-\log(q)/c\rceil+1$. Typical blocks without switches drive both filters into a common confidence cone, where softmax curvature suppresses their disagreement; a single Gaussian maximal event controls adaptive noise. A sweep with equally spaced Gaussians over $K\in\{2,4,8\}$ illustrates the opposing trends, and binary controls at long horizons compare saturating and nonsaturating recurrences. The result isolates two missing links between internal update gaps and predictive cost: the contribution of separating states to expected loss and decoder sensitivity. Thus even an unbounded internal update gap does not by itself certify predictive failure. The construction is fixed in $K$ and does not provide a universal criterion for when compression is harmless or characterize when internal gaps must incur task loss.

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.