Hidden Markov Models as Statistical Models

Latent state sequences, probabilistic emissions, and inference along ordered data

Statistics

Advanced Modeling

Hidden Markov Models

Bioinformatics

Hidden Markov model generation, forward-backward inference, estimation, diagnostics, and bioinformatics interpretation.

Author

Jiahong Sun

Published

November 4, 2026

Modified

November 4, 2026

1 A Sequence of Hidden States Generates a Sequence of Observations

A hidden Markov model (HMM) is a latent-variable model for ordered data. At each position or time $t$, an unobserved state $S_t\in\{1,\ldots,K\}$ evolves as a first-order Markov chain, and an observation $Y_t$ is emitted from a distribution determined by the current state:

\[ P(S_1=k)=\pi_k, \]

\[ P(S_t=k\mid S_{1:t-1})=P(S_t=k\mid S_{t-1})=a_{S_{t-1},k}, \]

\[ p(y_t\mid S_{1:t},Y_{1:t-1})=p(y_t\mid S_t)=b_{S_t}(y_t). \]

The parameters are an initial distribution $\boldsymbol\pi$, transition matrix $\mathbf A=(a_{jk})$, and emission distributions $b_k$. For a state path $s_{1:T}$ and observations $y_{1:T}$, the joint distribution is

\[ \boxed{ p(s_{1:T},y_{1:T}) =\pi_{s_1}b_{s_1}(y_1) \prod_{t=2}^T a_{s_{t-1},s_t}b_{s_t}(y_t). } \]

The observed-data likelihood sums over all $K^T$ possible paths:

\[ p(y_{1:T})=\sum_{s_{1:T}}p(s_{1:T},y_{1:T}). \]

Dynamic programming evaluates this sum without enumerating every path. An HMM is therefore a probability model for a sequence, not merely a segmentation algorithm. State calls inherit uncertainty and assumptions from that model.

2 What the Assumptions Mean

The standard HMM makes two conditional-independence claims:

the next state depends on the past only through the current state; and
observations are independent given their contemporaneous states.

Marginally, neighboring observations are dependent because their states are dependent. A state can summarize relevant history, but the first-order property is restrictive if duration, long memory, or unmodeled covariates matter.

For a homogeneous HMM, $\mathbf A$ is constant across positions. If the chain is ergodic, a stationary distribution $\boldsymbol\delta$ satisfies

\[ \boldsymbol\delta^{\mathsf T} =\boldsymbol\delta^{\mathsf T}\mathbf A, \qquad \sum_k\delta_k=1. \]

Starting with $\boldsymbol\pi=\boldsymbol\delta$ makes the latent process stationary. This can be convenient, but genomic boundaries, developmental time, or designed interventions may make stationarity implausible.

Under an ordinary HMM, the length $D_k$ of a run in state $k$ has a geometric distribution:

\[ P(D_k=d)=(a_{kk})^{d-1}(1-a_{kk}), \qquad E(D_k)=\frac{1}{1-a_{kk}}. \]

If observed domains or dwell times have non-geometric tails, a hidden semi-Markov model or explicit duration model may be preferable. Increasing $a_{kk}$ merely to obtain smoother segments changes the assumed duration distribution.

3 Three Distinct Inferential Questions

HMM analyses often blur three tasks that require different calculations.

3.1 Sequence likelihood

What is $p(y_{1:T}\mid\theta)$? This is used for parameter estimation, model comparison, and predictive scoring. It is computed by the forward algorithm.

Define

\[ \alpha_t(k)=p(y_{1:t},S_t=k). \]

Initialization and recursion give

\[ \alpha_1(k)=\pi_k b_k(y_1), \]

\[ \boxed{ \alpha_t(k)=b_k(y_t)\sum_{j=1}^K\alpha_{t-1}(j)a_{jk}. } \]

Then $p(y_{1:T})=\sum_k\alpha_T(k)$. The cost is $O(TK^2)$ rather than $O(K^T)$. Scaling or log-space calculations are essential because products of many probabilities underflow numerically.

3.2 State probabilities

What is $P(S_t=k\mid y_{1:T})$ or $P(S_{t-1}=j,S_t=k\mid y_{1:T})$? The backward quantity

\[ \beta_t(j)=p(y_{t+1:T}\mid S_t=j) \]

satisfies

\[ \beta_T(j)=1, \qquad \beta_t(j)=\sum_{k=1}^K a_{jk}b_k(y_{t+1})\beta_{t+1}(k). \]

Combining the passes yields the smoothed probability

\[ \boxed{ \gamma_t(k)=P(S_t=k\mid y_{1:T}) =\frac{\alpha_t(k)\beta_t(k)}{p(y_{1:T})}. } \]

$\gamma_t(k)$ preserves local classification uncertainty. Filtering instead conditions only on $y_{1:t}$ and answers an online prediction question.

3.3 Most probable path

What single path maximizes $P(S_{1:T}=s_{1:T}\mid y_{1:T})$? The Viterbi algorithm replaces the forward sum by a maximum and records back-pointers. The Viterbi state at a position need not equal the state with largest $\gamma_t(k)$, because a jointly valid path and positionwise marginal modes are different targets. Report which decoding rule was used; a hard path should not erase posterior or conditional uncertainty.

4 Estimating the Model

With labeled states, transition and emission parameters can be estimated from ordinary counts or state-specific likelihoods. With hidden states, maximum likelihood commonly uses the Baum–Welch algorithm, the HMM specialization of expectation–maximization.

The E-step computes smoothed state and transition probabilities,

\[ \xi_t(j,k)=P(S_t=j,S_{t+1}=k\mid y_{1:T},\theta^{(m)}). \]

For an unconstrained transition matrix, the M-step update is

\[ \boxed{ a_{jk}^{(m+1)} =\frac{\sum_{t=1}^{T-1}\xi_t(j,k)} {\sum_{t=1}^{T-1}\gamma_t(j)}. } \]

Emission parameters are updated using $\gamma_t(k)$ as fractional weights. Each iteration does not decrease the observed likelihood, but the likelihood can be multimodal. Fit dispersed starts, compare recurring solutions, and inspect scientifically important outputs rather than trusting one convergence flag.

A Bayesian HMM places priors on rows of $\mathbf A$, emission parameters, and possibly $K$. Posterior sampling or other algorithms propagate parameter and state uncertainty. Symmetric state priors create label-permutation symmetry; summaries should be invariant to labels or use a scientifically defined labeling rule. Priors that favor persistence can be valuable, but they directly affect domain length and must be checked through prior and posterior predictive simulation.

5 Interpretation and Identification

State numbers have no intrinsic meaning. Permuting every state label and the corresponding rows, columns, and emission parameters leaves the observed distribution unchanged. State-specific biological names require prespecified emission features, external annotations, or a transparent post-fit rule.

Beyond label switching, weak identification occurs when states have similar emissions, transitions are rarely observed, sequences are short, or some states have negligible occupancy. The model may predict the observations adequately while individual state paths remain uncertain. Conversely, sharply separated emissions can produce confident calls even if the transition model is wrong.

The number of states $K$ is a modeling choice. Likelihood criteria, held-out sequence prediction, scientific interpretability, and stability can inform it, but none reveals a uniquely true biological state count. Adding states may split one heterogeneous regime, approximate a non-geometric duration, or absorb an unmodeled batch effect. Compare conclusions across plausible $K$ and emission families.

6 Covariates, Multiple Sequences, and Hierarchy

Transitions or emissions may depend on covariates:

\[ P(S_t=k\mid S_{t-1}=j,x_t)=a_{jk}(x_t), \]

\[ Y_t\mid S_t=k,x_t\sim b_k(\cdot\mid x_t). \]

A covariate-dependent transition describes changing switching propensity; an emission coefficient describes a within-state observation change. These are different scientific mechanisms and may be difficult to distinguish without strong design or measurements.

For multiple subjects, chromosomes, or specimens, fitting one shared HMM assumes exchangeability after modeled covariates. Independent sequences contribute replication; adjacent positions within one sequence usually do not. Hierarchical HMMs can allow specimen-specific transitions or emissions with partial pooling. Naively concatenating sequences creates artificial transitions across boundaries, while fitting each short sequence independently may estimate transitions poorly.

7 Diagnostics and Sensitivity

Assumption or risk	Evidence to inspect	Sensitivity or remedy
Emission family is adequate	within-state residuals, tails, zeros, cross-feature dependence	revise distribution or add justified emission dependence
First-order state memory is adequate	residual dependence, run-length distribution, transition patterns by lag	higher-order or semi-Markov model
Homogeneous transitions are adequate	changes by position, time, batch, or covariate	model transition covariates or stratification
State count is stable	predictive scores, occupancy, split/merged solutions	report results across plausible $K$
Parameters are identified	emission overlap, rare states, profile/posterior geometry	simplify, regularize, or collect informative measurements
Optimization is reliable	likelihood and parameters across starts	increase and diversify starts
State calls are certain	$\gamma_t(k)$, entropy, boundary uncertainty	report probabilities and uncertain regions
Replication is valid	sequence, donor, chromosome, and batch structure	validate or resample independent experimental units

Predictive checks should simulate complete sequences, not independent positions. Compare marginal emission distributions, transition counts, run lengths, boundary density, autocorrelation, state occupancy, and specimen-to-specimen variation. Cross-validation should hold out the unit to which prediction will generalize: positions for interpolation, whole sequence segments for local forecasting, or entire specimens for population generalization.

Information criteria based on a nominal parameter count can be unreliable near singularities or with poorly occupied states. Standard errors that condition on one decoded path ignore state uncertainty. Parametric bootstrap, profile methods, or a full posterior can better expose nonregular uncertainty, provided the simulation reproduces the sequence and sampling design.

8 Bioinformatics Example: Chromatin-state Annotation

Suppose several histone marks are binarized in consecutive genomic bins. A multivariate HMM can use a latent chromatin state $S_t$ and conditionally independent Bernoulli emissions:

\[ P(\mathbf Y_t=\mathbf y_t\mid S_t=k) =\prod_{m=1}^M q_{km}^{y_{tm}}(1-q_{km})^{1-y_{tm}}, \]

where $q_{km}$ is the state-specific probability of observing mark $m$. The transition matrix encourages spatial persistence, while the emission vector describes combinations of marks. This is the structure used by ChromHMM.

Genomic bins are ordered observational units, not independent biological replicates. For a treatment or population contrast, independently sampled donors, organisms, or specimens are usually the experimental units. Technical replicates, sequencing reads, and adjacent bins improve measurement or spatial resolution but do not create independent treatment replication. Batch and treatment must be separated by design whenever possible.

Statistically, a decoded state is the model-supported label for a bin under the chosen marks, bin size, $K$, transition assumptions, and training data. Biologically, names such as “active enhancer” require enrichment in external annotations, reproducibility across specimens, and functional validation. A state can absorb assay quality or batch instead of chromatin biology. Compare emissions, occupancies, and downstream conclusions across initializations, state counts, bin sizes, and replicate handling.

Profile HMMs answer a different sequence question: match, insertion, and deletion states encode variation around a protein or nucleic-acid family alignment. Their state topology is designed rather than interpreted as a generic segmentation. The shared HMM framework does not make chromatin annotation and homology search scientifically equivalent.

9 Connections and Take-home Messages

Latent-variable models provides the general joint-model and marginalization framework.
Finite mixture models use independent latent labels; an HMM connects labels through a Markov transition law.
Identifiability and label switching explains permutation symmetry and invariant targets.
Expectation–maximization derives the fitting principle behind Baum–Welch.
Posterior predictive checks develops generative criticism, here applied to whole sequences.
Cross-validation clarifies why the held-out unit must match the prediction target.

An HMM combines a Markov model for latent states with an observation model for emissions. Forward–backward probabilities, Viterbi paths, and sequence likelihoods answer different questions. Sound use requires explicit transition and emission assumptions, uncertainty-aware decoding, checks of duration and dependence, sensitivity to state count and initialization, and replication at the biological level relevant to the scientific claim.

10 References

Baum, L. E. and Petrie, T. “Statistical Inference for Probabilistic Functions of Finite State Markov Chains.” Annals of Mathematical Statistics 37 (1966): 1554–1563. Article.
Rabiner, L. R. “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.” Proceedings of the IEEE 77 (1989): 257–286. Article.
Zucchini, W., MacDonald, I. L., and Langrock, R. Hidden Markov Models for Time Series: An Introduction Using R, 2nd ed. Chapman & Hall/CRC, 2016. Publisher page.
Durbin, R., Eddy, S. R., Krogh, A., and Mitchison, G. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998. Publisher page.
Eddy, S. R. “Profile Hidden Markov Models.” Bioinformatics 14 (1998): 755–763. Article.
Ernst, J. and Kellis, M. “ChromHMM: Automating Chromatin-state Discovery and Characterization.” Nature Methods 9 (2012): 215–216. Article.

--- title: "Hidden Markov Models as Statistical Models" subtitle: "Latent state sequences, probabilistic emissions, and inference along ordered data" description: "Hidden Markov model generation, forward-backward inference, estimation, diagnostics, and bioinformatics interpretation." author: "Jiahong Sun" date: "2026-11-04" date-modified: "2026-11-04" categories: [Statistics, Advanced Modeling, Hidden Markov Models, Bioinformatics] toc: true number-sections: true draft: false --- # A Sequence of Hidden States Generates a Sequence of Observations A hidden Markov model (HMM) is a latent-variable model for ordered data. At each position or time $t$, an unobserved state $S_t\in\{1,\ldots,K\}$ evolves as a first-order Markov chain, and an observation $Y_t$ is emitted from a distribution determined by the current state: $$ P(S_1=k)=\pi_k, $$ $$ P(S_t=k\mid S_{1:t-1})=P(S_t=k\mid S_{t-1})=a_{S_{t-1},k}, $$ $$ p(y_t\mid S_{1:t},Y_{1:t-1})=p(y_t\mid S_t)=b_{S_t}(y_t). $$ The parameters are an initial distribution $\boldsymbol\pi$, transition matrix $\mathbf A=(a_{jk})$, and emission distributions $b_k$. For a state path $s_{1:T}$ and observations $y_{1:T}$, the joint distribution is $$ \boxed{ p(s_{1:T},y_{1:T}) =\pi_{s_1}b_{s_1}(y_1) \prod_{t=2}^T a_{s_{t-1},s_t}b_{s_t}(y_t). } $$ The observed-data likelihood sums over all $K^T$ possible paths: $$ p(y_{1:T})=\sum_{s_{1:T}}p(s_{1:T},y_{1:T}). $$ Dynamic programming evaluates this sum without enumerating every path. An HMM is therefore a probability model for a sequence, not merely a segmentation algorithm. State calls inherit uncertainty and assumptions from that model. # What the Assumptions Mean The standard HMM makes two conditional-independence claims: 1. the next state depends on the past only through the current state; and 2. observations are independent given their contemporaneous states. Marginally, neighboring observations are dependent because their states are dependent. A state can summarize relevant history, but the first-order property is restrictive if duration, long memory, or unmodeled covariates matter. For a homogeneous HMM, $\mathbf A$ is constant across positions. If the chain is ergodic, a stationary distribution $\boldsymbol\delta$ satisfies $$ \boldsymbol\delta^{\mathsf T} =\boldsymbol\delta^{\mathsf T}\mathbf A, \qquad \sum_k\delta_k=1. $$ Starting with $\boldsymbol\pi=\boldsymbol\delta$ makes the latent process stationary. This can be convenient, but genomic boundaries, developmental time, or designed interventions may make stationarity implausible. Under an ordinary HMM, the length $D_k$ of a run in state $k$ has a geometric distribution: $$ P(D_k=d)=(a_{kk})^{d-1}(1-a_{kk}), \qquad E(D_k)=\frac{1}{1-a_{kk}}. $$ If observed domains or dwell times have non-geometric tails, a hidden semi-Markov model or explicit duration model may be preferable. Increasing $a_{kk}$ merely to obtain smoother segments changes the assumed duration distribution. # Three Distinct Inferential Questions HMM analyses often blur three tasks that require different calculations. ## Sequence likelihood What is $p(y_{1:T}\mid\theta)$? This is used for parameter estimation, model comparison, and predictive scoring. It is computed by the forward algorithm. Define $$ \alpha_t(k)=p(y_{1:t},S_t=k). $$ Initialization and recursion give $$ \alpha_1(k)=\pi_k b_k(y_1), $$ $$ \boxed{ \alpha_t(k)=b_k(y_t)\sum_{j=1}^K\alpha_{t-1}(j)a_{jk}. } $$ Then $p(y_{1:T})=\sum_k\alpha_T(k)$. The cost is $O(TK^2)$ rather than $O(K^T)$. Scaling or log-space calculations are essential because products of many probabilities underflow numerically. ## State probabilities What is $P(S_t=k\mid y_{1:T})$ or $P(S_{t-1}=j,S_t=k\mid y_{1:T})$? The backward quantity $$ \beta_t(j)=p(y_{t+1:T}\mid S_t=j) $$ satisfies $$ \beta_T(j)=1, \qquad \beta_t(j)=\sum_{k=1}^K a_{jk}b_k(y_{t+1})\beta_{t+1}(k). $$ Combining the passes yields the smoothed probability $$ \boxed{ \gamma_t(k)=P(S_t=k\mid y_{1:T}) =\frac{\alpha_t(k)\beta_t(k)}{p(y_{1:T})}. } $$ $\gamma_t(k)$ preserves local classification uncertainty. Filtering instead conditions only on $y_{1:t}$ and answers an online prediction question. ## Most probable path What single path maximizes $P(S_{1:T}=s_{1:T}\mid y_{1:T})$? The Viterbi algorithm replaces the forward sum by a maximum and records back-pointers. The Viterbi state at a position need not equal the state with largest $\gamma_t(k)$, because a jointly valid path and positionwise marginal modes are different targets. Report which decoding rule was used; a hard path should not erase posterior or conditional uncertainty. # Estimating the Model With labeled states, transition and emission parameters can be estimated from ordinary counts or state-specific likelihoods. With hidden states, maximum likelihood commonly uses the Baum--Welch algorithm, the HMM specialization of [expectation--maximization](expectation-maximization.qmd). The E-step computes smoothed state and transition probabilities, $$ \xi_t(j,k)=P(S_t=j,S_{t+1}=k\mid y_{1:T},\theta^{(m)}). $$ For an unconstrained transition matrix, the M-step update is $$ \boxed{ a_{jk}^{(m+1)} =\frac{\sum_{t=1}^{T-1}\xi_t(j,k)} {\sum_{t=1}^{T-1}\gamma_t(j)}. } $$ Emission parameters are updated using $\gamma_t(k)$ as fractional weights. Each iteration does not decrease the observed likelihood, but the likelihood can be multimodal. Fit dispersed starts, compare recurring solutions, and inspect scientifically important outputs rather than trusting one convergence flag. A Bayesian HMM places priors on rows of $\mathbf A$, emission parameters, and possibly $K$. Posterior sampling or other algorithms propagate parameter and state uncertainty. Symmetric state priors create label-permutation symmetry; summaries should be invariant to labels or use a scientifically defined labeling rule. Priors that favor persistence can be valuable, but they directly affect domain length and must be checked through prior and posterior predictive simulation. # Interpretation and Identification State numbers have no intrinsic meaning. Permuting every state label and the corresponding rows, columns, and emission parameters leaves the observed distribution unchanged. State-specific biological names require prespecified emission features, external annotations, or a transparent post-fit rule. Beyond label switching, weak identification occurs when states have similar emissions, transitions are rarely observed, sequences are short, or some states have negligible occupancy. The model may predict the observations adequately while individual state paths remain uncertain. Conversely, sharply separated emissions can produce confident calls even if the transition model is wrong. The number of states $K$ is a modeling choice. Likelihood criteria, held-out sequence prediction, scientific interpretability, and stability can inform it, but none reveals a uniquely true biological state count. Adding states may split one heterogeneous regime, approximate a non-geometric duration, or absorb an unmodeled batch effect. Compare conclusions across plausible $K$ and emission families. # Covariates, Multiple Sequences, and Hierarchy Transitions or emissions may depend on covariates: $$ P(S_t=k\mid S_{t-1}=j,x_t)=a_{jk}(x_t), $$ $$ Y_t\mid S_t=k,x_t\sim b_k(\cdot\mid x_t). $$ A covariate-dependent transition describes changing switching propensity; an emission coefficient describes a within-state observation change. These are different scientific mechanisms and may be difficult to distinguish without strong design or measurements. For multiple subjects, chromosomes, or specimens, fitting one shared HMM assumes exchangeability after modeled covariates. Independent sequences contribute replication; adjacent positions within one sequence usually do not. Hierarchical HMMs can allow specimen-specific transitions or emissions with partial pooling. Naively concatenating sequences creates artificial transitions across boundaries, while fitting each short sequence independently may estimate transitions poorly. # Diagnostics and Sensitivity | Assumption or risk | Evidence to inspect | Sensitivity or remedy | |---|---|---| | Emission family is adequate | within-state residuals, tails, zeros, cross-feature dependence | revise distribution or add justified emission dependence | | First-order state memory is adequate | residual dependence, run-length distribution, transition patterns by lag | higher-order or semi-Markov model | | Homogeneous transitions are adequate | changes by position, time, batch, or covariate | model transition covariates or stratification | | State count is stable | predictive scores, occupancy, split/merged solutions | report results across plausible $K$ | | Parameters are identified | emission overlap, rare states, profile/posterior geometry | simplify, regularize, or collect informative measurements | | Optimization is reliable | likelihood and parameters across starts | increase and diversify starts | | State calls are certain | $\gamma_t(k)$, entropy, boundary uncertainty | report probabilities and uncertain regions | | Replication is valid | sequence, donor, chromosome, and batch structure | validate or resample independent experimental units | Predictive checks should simulate complete sequences, not independent positions. Compare marginal emission distributions, transition counts, run lengths, boundary density, autocorrelation, state occupancy, and specimen-to-specimen variation. Cross-validation should hold out the unit to which prediction will generalize: positions for interpolation, whole sequence segments for local forecasting, or entire specimens for population generalization. Information criteria based on a nominal parameter count can be unreliable near singularities or with poorly occupied states. Standard errors that condition on one decoded path ignore state uncertainty. Parametric bootstrap, profile methods, or a full posterior can better expose nonregular uncertainty, provided the simulation reproduces the sequence and sampling design. # Bioinformatics Example: Chromatin-state Annotation Suppose several histone marks are binarized in consecutive genomic bins. A multivariate HMM can use a latent chromatin state $S_t$ and conditionally independent Bernoulli emissions: $$ P(\mathbf Y_t=\mathbf y_t\mid S_t=k) =\prod_{m=1}^M q_{km}^{y_{tm}}(1-q_{km})^{1-y_{tm}}, $$ where $q_{km}$ is the state-specific probability of observing mark $m$. The transition matrix encourages spatial persistence, while the emission vector describes combinations of marks. This is the structure used by ChromHMM. Genomic bins are ordered observational units, not independent biological replicates. For a treatment or population contrast, independently sampled donors, organisms, or specimens are usually the experimental units. Technical replicates, sequencing reads, and adjacent bins improve measurement or spatial resolution but do not create independent treatment replication. Batch and treatment must be separated by design whenever possible. Statistically, a decoded state is the model-supported label for a bin under the chosen marks, bin size, $K$, transition assumptions, and training data. Biologically, names such as “active enhancer” require enrichment in external annotations, reproducibility across specimens, and functional validation. A state can absorb assay quality or batch instead of chromatin biology. Compare emissions, occupancies, and downstream conclusions across initializations, state counts, bin sizes, and replicate handling. Profile HMMs answer a different sequence question: match, insertion, and deletion states encode variation around a protein or nucleic-acid family alignment. Their state topology is designed rather than interpreted as a generic segmentation. The shared HMM framework does not make chromatin annotation and homology search scientifically equivalent. # Connections and Take-home Messages - [Latent-variable models](latent-variable-models.qmd) provides the general joint-model and marginalization framework. - [Finite mixture models](finite-mixture-models.qmd) use independent latent labels; an HMM connects labels through a Markov transition law. - [Identifiability and label switching](identifiability-and-label-switching.qmd) explains permutation symmetry and invariant targets. - [Expectation--maximization](expectation-maximization.qmd) derives the fitting principle behind Baum--Welch. - [Posterior predictive checks](posterior-predictive-checks.qmd) develops generative criticism, here applied to whole sequences. - [Cross-validation](../regression-models/cross-validation.qmd) clarifies why the held-out unit must match the prediction target. An HMM combines a Markov model for latent states with an observation model for emissions. Forward--backward probabilities, Viterbi paths, and sequence likelihoods answer different questions. Sound use requires explicit transition and emission assumptions, uncertainty-aware decoding, checks of duration and dependence, sensitivity to state count and initialization, and replication at the biological level relevant to the scientific claim. # References - Baum, L. E. and Petrie, T. “Statistical Inference for Probabilistic Functions of Finite State Markov Chains.” *Annals of Mathematical Statistics* 37 (1966): 1554--1563. [Article](https://doi.org/10.1214/aoms/1177699147). - Rabiner, L. R. “A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.” *Proceedings of the IEEE* 77 (1989): 257--286. [Article](https://doi.org/10.1109/5.18626). - Zucchini, W., MacDonald, I. L., and Langrock, R. *Hidden Markov Models for Time Series: An Introduction Using R*, 2nd ed. Chapman & Hall/CRC, 2016. [Publisher page](https://doi.org/10.1201/b20790). - Durbin, R., Eddy, S. R., Krogh, A., and Mitchison, G. *Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids*. Cambridge University Press, 1998. [Publisher page](https://doi.org/10.1017/CBO9780511790492). - Eddy, S. R. “Profile Hidden Markov Models.” *Bioinformatics* 14 (1998): 755--763. [Article](https://doi.org/10.1093/bioinformatics/14.9.755). - Ernst, J. and Kellis, M. “ChromHMM: Automating Chromatin-state Discovery and Characterization.” *Nature Methods* 9 (2012): 215--216. [Article](https://doi.org/10.1038/nmeth.1906).