DS-HDP-HMM: Disentangled Sticky HMM

• DS-HDP-HMM is a Bayesian nonparametric model that infers an unbounded number of hidden states while separately modeling self-persistence from switching transitions.
• The model employs dedicated Gibbs sampling techniques that jointly sample persistence indicators and latent states to enhance inference efficiency and mitigate over-segmentation.
• Positioned between HDP-HMM and recurrent sticky variants, DS-HDP-HMM provides independent control over state durations and transition similarities, improving predictive performance.

Searching arXiv for recent and foundational papers on DS-HDP-HMM and closely related HDP-HMM variants. DS-HDP-HMM most commonly denotes the Disentangled Sticky Hierarchical Dirichlet Process Hidden Markov Model, a Bayesian nonparametric HMM that infers an unbounded number of latent states while separating the probability of staying in the current state from the distribution over alternative next states. In the contemporary HDP-HMM literature, this model is positioned between the original HDP-HMM and newer recurrent sticky variants, and it was proposed to remove a specific coupling in the sticky HDP-HMM between self-persistence and transition-row similarity (Zhou et al., 2020). The abbreviation is, however, overloaded: in small-variance asymptotics it is also used for a deterministic or small-variance asymptotic HDP-HMM objective derived from the infinite HMM (Huggins et al., 2014).

1. Terminological scope and model lineage

The DS-HDP-HMM emerged from a sequence of modifications to the HDP-HMM family. The original HDP-HMM places a hierarchical Dirichlet process prior on transition rows so that the number of hidden states need not be fixed in advance. Its sticky extension adds a self-transition bias to mitigate the well-known failure mode of state over-segmentation and rapid switching, a problem emphasized in speaker diarization and other segmentation tasks (0905.2592). The disentangled sticky model retains the HDP prior over shared transition structure but separates the decision to remain in the same state from the decision of where to go when a switch occurs (Zhou et al., 2020).

The conceptual motivation is that an HMM transition matrix contains at least three distinct prior characteristics: transition-row similarity, average self-persistence, and variability of the self-persistence prior across states. In the sticky HDP-HMM, the self-transition bias $\kappa$ affects both the mean self-persistence probability and the concentration of the transition prior, so these features cannot be varied independently. DS-HDP-HMM was proposed precisely to disentangle these roles, giving independent control to the HDP concentration parameter $\alpha$ and the Beta prior parameters $(\rho_1,\rho_2)$ (Zhou et al., 2020).

This historical progression is also the backdrop for later generalizations. The recurrent sticky HDP-HMM treats DS-HDP-HMM as the version in which self-persistence is disentangled but still stationary, and then replaces the state-specific constant $\kappa_j$ with a time-varying, context-dependent stickiness mechanism (Słupiński et al., 2024). In that sense, DS-HDP-HMM occupies a middle position: more expressive than the sticky HDP-HMM, but still a Markov model with stationary state-level persistence.

2. Probabilistic formulation

The DS-HDP-HMM inherits the HDP-HMM’s global state-sharing structure. Its global transition weights are drawn from a stick-breaking prior,

$\beta \sim \mathrm{GEM}(\gamma),$

with the construction

$$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$

and state-specific latent parameters $\theta_j$ drawn from a base prior $H$ (Zhou et al., 2020).

The distinctive transition mechanism is

$$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$

Here $\kappa_j$ is the self-persistence probability of state $\alpha$0, $\alpha$1 is a point mass on $\alpha$2, and $\alpha$3 is the non-self transition distribution. The latent chain and observations then satisfy

$\alpha$4

The model therefore makes the stay-versus-switch decision explicit: with probability $\alpha$5 the chain remains in $\alpha$6, and otherwise it transitions according to the HDP-coupled distribution $\alpha$7 (Zhou et al., 2020).

An equivalent augmentation introduces binary indicators $\alpha$8: $\alpha$9 This augmentation is central because it exposes the disentangled semantics directly in the latent-variable structure: $(\rho_1,\rho_2)$0 denotes state persistence, while $(\rho_1,\rho_2)$1 denotes a genuine switch (Zhou et al., 2020).

The special-case structure clarifies how DS-HDP-HMM sits within the broader family. If $(\rho_1,\rho_2)$2 and $(\rho_1,\rho_2)$3, then $(\rho_1,\rho_2)$4 and the model reduces to the HDP-HMM. The 2020 paper also states that the sticky HDP-HMM is a special case of DS-HDP-HMM when $(\rho_1,\rho_2)$5 in the corresponding parameterization (Zhou et al., 2020). This makes the disentangled model a strict generalization of both the nonsticky and sticky formulations at the level of transition priors.

3. Posterior inference and computational structure

The DS-HDP-HMM paper develops two Gibbs samplers: a direct assignment Gibbs sampler and a weak-limit Gibbs sampler (Zhou et al., 2020). The direct sampler marginalizes out the transition distributions $(\rho_1,\rho_2)$6 and the emission parameters $(\rho_1,\rho_2)$7, then samples the discrete latent variables directly. A key detail is that it samples the block $(\rho_1,\rho_2)$8 sequentially rather than resampling only $(\rho_1,\rho_2)$9. That modification is specific to the disentangled persistence mechanism, because $\kappa_j$0 and $\kappa_j$1 determine whether adjacent transitions are interpreted as persistence or switching.

Conditioned on the persistence indicators, each $\kappa_j$2 admits a Beta-binomial update,

$\kappa_j$3

The global weights $\kappa_j$4 are updated using Chinese restaurant franchise auxiliary counts $\kappa_j$5, giving

$\kappa_j$6

Hyperparameters $\kappa_j$7 and $\kappa_j$8 are updated with the same Gamma-conjugate machinery used in earlier HDP-HMM work, while $\kappa_j$9 are handled through the reparameterization

$\beta \sim \mathrm{GEM}(\gamma),$0

with a grid-based posterior calculation in the paper (Zhou et al., 2020).

The weak-limit sampler replaces the infinite HDP with a truncation level $\beta \sim \mathrm{GEM}(\gamma),$1,

$\beta \sim \mathrm{GEM}(\gamma),$2

and uses forward-backward sampling over the enlarged latent space to jointly sample $\beta \sim \mathrm{GEM}(\gamma),$3. In this finite approximation, the non-self transition distributions satisfy

$\beta \sim \mathrm{GEM}(\gamma),$4

The direct sampler has per-iteration complexity $\beta \sim \mathrm{GEM}(\gamma),$5, whereas the weak-limit sampler has per-iteration complexity $\beta \sim \mathrm{GEM}(\gamma),$6 but mixes better because it resamples the full latent sequence jointly (Zhou et al., 2020). The same weak-limit design is highlighted as especially useful for longer sequences and for dynamical emissions such as ARHMMs and SLDS.

4. Relation to neighboring HDP-state-space models

The DS-HDP-HMM is best understood relative to four closely related models:

Model	Persistence mechanism	Distinguishing feature
HDP-HMM	Markov self-transition probabilities only	Unbounded state count via HDP
Sticky HDP-HMM	Adds self-transition bias $\beta \sim \mathrm{GEM}(\gamma),$7	Reduces rapid switching but keeps coupled prior
DS-HDP-HMM	$\beta \sim \mathrm{GEM}(\gamma),$8	Separates self-persistence from non-self transitions
HDP-HSMM	Explicit duration variable $\beta \sim \mathrm{GEM}(\gamma),$9	Models non-geometric state durations
RS-HDP-HMM	Time-varying $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$0 from logistic regression	Makes stickiness context-dependent

The distinction between DS-HDP-HMM and sticky HDP-HMM is a matter of prior geometry. In the sticky HDP-HMM, the mean self-persistence is governed by the ratio

$$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$1

but the same combined concentration $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$2 also governs how transition rows concentrate around the global structure. DS-HDP-HMM removes this coupling by allocating $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$3 to the non-self transition distributions $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$4 and $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$5 to the self-persistence probabilities $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$6 (Słupiński et al., 2024).

A common misconception is to equate DS-HDP-HMM with explicit-duration semi-Markov models. The HDP-HSMM addresses a different limitation of the HDP-HMM: the implicit geometric duration law induced by Markov self-transitions. Its segment-level formulation introduces explicit duration variables $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$7, rules out self-transitions at segment boundaries, and thereby models state persistence through state-specific duration distributions rather than through a diagonal transition bias (Johnson et al., 2012). DS-HDP-HMM, by contrast, preserves the Markov structure; the 2020 paper emphasizes that this is important in many applications, including control and reinforcement learning (Zhou et al., 2020).

The practical distinction is visible in application work. In naturalistic car-following analysis, HDP-HSMM was preferred to HDP-HMM and sticky HDP-HMM because explicit durations yielded better predictive log-likelihood and a much smaller proportion of segments shorter than $$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$8 s—$$v_i \sim \mathrm{Beta}(1,\gamma), \qquad \beta_i = v_i \prod_{\ell=1}^{i-1}(1-v_\ell), \quad i=1,2,\dots$$9 for HDP-HSMM versus $\theta_j$0 for sticky HDP-HMM and $\theta_j$1 for HDP-HMM—showing that duration modeling and stickiness modeling solve different temporal pathologies (Wang et al., 2017).

Relative to the recurrent sticky HDP-HMM, DS-HDP-HMM is the stationary case. RS-HDP-HMM replaces the constant $\theta_j$2 with

$\theta_j$3

so that a state may be sticky in one context and not in another (Słupiński et al., 2024). This suggests that DS-HDP-HMM is most appropriate when heterogeneity in state persistence is important but covariate-dependent variation in stickiness is not the main phenomenon.

5. Empirical behavior and application domains

The empirical evaluation of the disentangled sticky model in its original paper uses both synthetic and real datasets, with normalized Hamming distance for training state recovery and predictive negative log-likelihood on held-out data (Zhou et al., 2020). Two synthetic regimes are emphasized. In one, the true process has different $\theta_j$4 across states but similar switching distributions $\theta_j$5, corresponding to variable dwell times under otherwise similar transition structure. In the other, the true process has similar $\theta_j$6 but different $\theta_j$7. Across 10 replicated datasets, DS-HDP-HMM recovered the appropriate hyperparameter pattern and outperformed both sticky HDP-HMM and HDP-HMM; the paper notes that the advantage is especially clear under Gaussian emissions as well (Zhou et al., 2020).

The real-data applications illustrate the model’s ability to separate persistence structure from switching structure. On a public hippocampal spike-train dataset from a freely moving rat, the observation model is

$\theta_j$8

with gamma priors on firing rates. DS-HDP-HMM achieved the lowest test negative log-likelihood, and the inferred discrete states correlated with spatial location of the rat (Zhou et al., 2020). On mouse behavior video, the authors used a 9-dimensional latent representation from a convolutional autoencoder together with an ARHMM emission model,

$\theta_j$9

and reported a substantially better held-out predictive log-likelihood than both baselines. The inferred transition matrix had strongly variable diagonal entries, indicating state-dependent self-persistence (Zhou et al., 2020).

Later work situates these gains more finely. In the RS-HDP-HMM comparison, DS-HDP-HMM achieved weighted F1 $H$0 and accuracy $H$1 on NASCAR trajectories, weighted F1 $H$2 and accuracy $H$3 on dancing bee behavior, and on mouse behavior it inferred an average of $H$4 states with an average of $H$5 switches (Słupiński et al., 2024). RS-HDP-HMM outperformed DS-HDP-HMM in settings where position or context strongly modulated the probability of persistence, but those results do not negate the central contribution of DS-HDP-HMM: the disentangling of state persistence from transition similarity.

6. Alternative small-variance asymptotic usage of the acronym

A distinct line of work uses DS-HDP-HMM to denote the deterministic or small-variance asymptotic form of the HDP-HMM rather than the disentangled sticky model (Huggins et al., 2014). In that usage, one starts from an HDP-HMM with Gaussian emissions,

$H$6

$H$7

then lets $H$8 while scaling $H$9 appropriately (Huggins et al., 2014).

The combinatorial derivation integrates out $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$0 and $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$1 and yields the objective

$$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$2

where $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$3 is the number of distinct transitions out of state $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$4. The squared-error term is the emission fit, $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$5 penalizes the number of states, $$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$6 penalizes transition complexity, and the final term regularizes state means (Huggins et al., 2014).

The same note also presents a direct, non-collapsed asymptotic objective,

$$\kappa_j \overset{iid}{\sim} \mathrm{Beta}(\rho_1,\rho_2), \qquad \bar{\pi}_j \overset{iid}{\sim} \mathrm{DP}(\alpha \beta), \qquad \pi_j = \kappa_j \delta_j + (1-\kappa_j)\bar{\pi}_j .$$7

This second formulation preserves explicit transition-distribution structure and replaces the count-based transition penalty with a log-transition term and a KL regularizer tying each transition row to the global weights (Huggins et al., 2014).

The coexistence of these two meanings is a genuine terminological ambiguity. In recent sequential-modeling literature, and especially in papers that compare DS-HDP-HMM to sticky HDP-HMM and RS-HDP-HMM, the abbreviation refers to the Disentangled Sticky HDP-HMM (Zhou et al., 2020, Słupiński et al., 2024). In small-variance asymptotics, it refers to a deterministic limit of the HDP-HMM (Huggins et al., 2014). Distinguishing these usages is necessary for accurate reading of the literature.