fit-hidden-markov-model

# Fit Hidden Markov Model

Fit a hidden Markov model (HMM) to sequential observation data using the Baum-Welch expectation-maximization algorithm, decode the most likely hidden state sequence via Viterbi, and select the optimal number of hidden states through information criteria.

## When to Use

- You observe a sequence of emissions but the underlying generative states are not directly observable
- You suspect your data is generated by a system that switches between a finite number of regimes
- You need to segment a time series into latent phases (e.g., market regimes, speech phonemes, biological sequence annotation)
- You want to compute the probability of an observed sequence under a generative model
- You need the most likely sequence of hidden states given observations (decoding)
- You are comparing models with different numbers of hidden states for the best complexity-fit trade-off

## Inputs

### Required

| Input | Type | Description |
|-------|------|-------------|
| `observations` | sequence/matrix | Observed data sequence (univariate or multivariate) |
| `n_hidden_states` | integer | Number of hidden states to fit (or a range for model selection) |
| `emission_type` | string | Distribution family for emissions: `"gaussian"`, `"discrete"`, `"poisson"`, `"multinomial"` |

### Optional

| Input | Type | Default | Description |
|-------|------|---------|-------------|
| `initial_params` | dict | random/heuristic | Initial transition matrix, emission parameters, and start probabilities |
| `n_restarts` | integer | 10 | Number of random restarts to mitigate local optima |
| `max_iterations` | integer | 500 | Maximum EM iterations per restart |
| `convergence_tol` | float | 1e-6 | Log-likelihood convergence threshold for EM |
| `state_range` | list of ints | `[n_hidden_states]` | Range of state counts for model selection |
| `covariance_type` | string | `"full"` | For Gaussian emissions: `"full"`, `"diagonal"`, `"spherical"` |
| `regularization` | float | 1e-6 | Small constant added to diagonal of covariance matrices to prevent singularity |

## Procedure

### Step 1: Define Hidden States and Observation Model

1.1. Specify the number of hidden states `K` (or a candidate range for model selection in Step 5).

1.2. Choose the emission distribution family based on the data type:
   - Continuous data: Gaussian (univariate or multivariate)
   - Count data: Poisson or negative binomial
   - Categorical data: discrete/multinomial

1.3. Define the model components:
   - **Transition matrix** `A` of size `K x K`: `A[i,j] = P(z_t = j | z_{t-1} = i)`
   - **Emission parameters** `theta_k` for each state `k`: distribution-specific (e.g., mean and covariance for Gaussian)
   - **Initial state distribution** `pi`: `pi[k] = P(z_1 = k)`

1.4. Verify observation data is properly formatted: no missing values in the sequence, consistent dimensionality, and sufficient length relative to the number of parameters.

**Got:** A clearly specified HMM architecture with `K` states, a chosen emission family, and clean observation data of length `T >> K^2`.

**If fail:** If data contains missing values, impute or remove affected segments. If `T` is too small relative to `K`, reduce `K` or acquire more data.

### Step 2: Initialize Parameters

2.1. Generate initial parameters for each of the `n_restarts` restarts:
   - **Transition matrix**: Random stochastic matrix (each row drawn from a Dirichlet distribution) or a slightly perturbed uniform matrix.
   - **Emission parameters**: Use K-means clustering on the observations to initialize means; compute cluster variances for Gaussian emissions.
   - **Initial distribution**: Uniform or proportional to cluster sizes from K-means.

2.2. For the first restart, use the K-means-informed initialization (generally the strongest start). For subsequent restarts, use random perturbations.

2.3. Verify all initial parameters are valid:
   - Transition matrix rows sum to 1 with all entries positive.
   - Emission parameters are in the valid domain (e.g., covariance matrices are positive definite).
   - Initial distribution sums to 1.

**Got:** `n_restarts` sets of valid initial parameters, with at least one data-driven initialization.

**If fail:** If K-means fails to converge, use purely random initialization with more restarts. If covariance matrices are singular, add the regularization constant to the diagonal.

### Step 3: Run Baum-Welch EM for Parameter Estimation

3.1. **E-step (Forward-Backward algorithm):**
   - Compute forward probabilities `alpha[t,k]` = P(o_1,...,o_t, z_t=k | model) using the recursion:
     - `alpha[1,k] = pi[k] * b_k(o_1)`
     - `alpha[t,k] = sum_j(alpha[t-1,j] * A[j,k]) * b_k(o_t)`
   - Compute backward probabilities `beta[t,k]` = P(o_{t+1},...,o_T | z_t=k, model):
     - `beta[T,k] = 1`
     - `beta[t,k] = sum_j(A[k,j] * b_j(o_{t+1}) * beta[t+1,j])`
   - Compute state posterior `gamma[t,k]` = P(z_t=k | O, model):
     - `gamma[t,k] = alpha[t,k] * beta[t,k] / P(O | model)`
   - Compute transition posterior `xi[t,i,j]` = P(z_t=i, z_{t+1}=j | O, model).

3.2. **M-step (Parameter re-estimation):**
   - Update transition matrix: `A[i,j] = sum_t(xi[t,i,j]) / sum_t(gamma[t,i])`
   - Update emission parameters using weighted sufficient statistics:
     - Gaussian mean: `mu_k = sum_t(gamma[t,k] * o_t) / sum_t(gamma[t,k])`
     - Gaussian covariance: weighted scatter matrix plus regularization
     - Discrete: `b_k(v) = sum_t(gamma[t,k] * I(o_t=v)) / sum_t(gamma[t,k])`
   - Update initial distribution: `pi[k] = gamma[1,k]`

3.3. Compute log-likelihood: `log P(O | model) = log sum_k(alpha[T,k])`. Use the log-sum-exp trick to prevent underflow.

3.4. **Scaling:** Use scaled forward-backward variables to prevent numerical underflow for long sequences. Normalize `alpha` at each time step and accumulate log scaling factors.

3.5. Repeat E-step and M-step until log-likelihood change is below `convergence_tol` or `max_iterations` is reached.

3.6. Across all restarts, keep the parameter set with the highest final log-likelihood.

**Got:** Monotonically non-decreasing log-likelihood across iterations, converging within `max_iterations`. Final parameters are valid (stochastic matrices, positive-definite covariances).

**If fail:** If log-likelihood decreases, there is a bug in the E-step or M-step -- verify formulas. If convergence is very slow, try better initialization or increase `max_iterations`. If covariance becomes singular, increase regularization.

### Step 4: Apply Viterbi Decoding for Most Likely State Sequence

4.1. Initialize Viterbi variables:
   - `delta[1,k] = log(pi[k]) + log(b_k(o_1))`
   - `psi[1,k] = 0` (no predecessor)

4.2. Recurse forward for `t = 2,...,T`:
   - `delta[t,k] = max_j(delta[t-1,j] + log(A[j,k])) + log(b_k(o_t))`
   - `psi[t,k] = argmax_j(delta[t-1,j] + log(A[j,k]))`

4.3. Terminate:
   - `z*_T = argmax_k(delta[T,k])`
   - Best path log-probability: `max_k(delta[T,k])`

4.4. Backtrace for `t = T-1,...,1`:
   - `z*_t = psi[t+1, z*_{t+1}]`

4.5. Output the decoded state sequence `z* = (z*_1, ..., z*_T)` and its log-probability.

4.6. Compare the Viterbi path probability to the total sequence probability from the forward algorithm to assess how dominant the best path is.

**Got:** A single most-likely state sequence of length `T` with each entry in `{1,...,K}`. The Viterbi log-probability should be less than or equal to the total log-likelihood.

**If fail:** If the Viterbi path has log-probability of negative infinity, some transition or emission probability is zero where it should not be. Add floor values to prevent log(0).

### Step 5: Perform Model Selection (BIC/AIC Across Model Orders)

5.1. For each candidate number of hidden states `K` in `state_range`, fit the full HMM (Steps 2-4).

5.2. Compute the number of free parameters `p`:
   - Transition matrix: `K * (K - 1)` (each row is a simplex)
   - Emission parameters: depends on family (e.g., Gaussian with full covariance in `d` dimensions: `K * (d + d*(d+1)/2)`)
   - Initial distribution: `K - 1`

5.3. Compute information criteria:
   - `BIC = -2 * log_likelihood + p * log(T)`
   - `AIC = -2 * log_likelihood + 2 * p`
   - `AICc = AIC + 2*p*(p+1) / (T - p - 1)` (small-sample correction)

5.4. Select the model with the lowest BIC (preferred for consistency) or AIC (preferred for prediction). Report both.

5.5. Tabulate results: for each `K`, show log-likelihood, number of parameters, BIC, AIC, and convergence status.

5.6. If the optimal `K` is at the boundary of `state_range`, extend the range and re-fit.

**Got:** A clear minimum in BIC/AIC identifying the optimal number of hidden states. The selected model should have converged and have interpretable state meanings.

**If fail:** If no clear minimum exists (monotonically decreasing BIC), the model may be misspecified -- consider a different emission family. If all models have poor log-likelihood, the data may not follow an HMM structure.

### Step 6: Validate with Held-Out Data and Posterior Decoding

6.1. Split data into training and validation sets (e.g., 80/20 or use multiple sequences if available).

6.2. Fit the model on training data. Compute log-likelihood on held-out data using the forward algorithm (do not re-fit parameters).

6.3. **Posterior decoding** (alternative to Viterbi):
   - For each time step, assign the state with highest posterior probability: `z^_t = argmax_k(gamma[t,k])`
   - This maximizes the expected number of correctly decoded states (vs. Viterbi which maximizes the joint path probability).

6.4. Compare Viterbi and posterior decoding:
   - Compute agreement rate between the two decoded sequences.
   - Regions of disagreement indicate ambiguous state assignments.

6.5. Assess state interpretability:
   - Examine emission parameters for each state (means, variances, discrete distributions).
   - Verify states correspond to meaningful regimes in the domain context.
   - Check that state dwell times (implied by diagonal of `A`) are reasonable.

6.6. Compute held-out log-likelihood per observation and compare across model orders to confirm the training-set model selection.

**Got:** Held-out log-likelihood is reasonably close to training log-likelihood (no severe overfitting). Viterbi and posterior decoding agree on 90%+ of time steps. States have distinct, interpretable emission distributions.

**If fail:** If held-out likelihood is much worse than training, the model is overfitting -- reduce `K` or increase regularization. If states are not interpretable, try different initializations or a different emission family.

## Validation

- Log-likelihood is monotonically non-decreasing across Baum-Welch iterations for each restart
- The transition matrix is row-stochastic (rows sum to 1, all entries non-negative)
- Emission parameters are in the valid domain (positive-definite covariances, valid probability distributions)
- The Viterbi path log-probability does not exceed the total sequence log-probability
- BIC/AIC curves show a clear minimum at the selected model order
- Held-out log-likelihood confirms the model generalizes beyond the training set
- Forward and backward probability computations agree: `P(O) = sum_k(alpha[T,k]) = sum_k(pi[k] * b_k(o_1) * beta[1,k])`

## Pitfalls

- **Local optima in EM**: The Baum-Welch algorithm converges to a local maximum, not necessarily the global one. Always use multiple random restarts and pick the best.
- **Numerical underflow**: Forward-backward probabilities shrink exponentially with sequence length. Use log-space computation or scaled variables to prevent underflow to zero.
- **Overfitting with too many states**: Each additional hidden state adds `O(K + d^2)` parameters. Use BIC (not just likelihood) for model selection and validate on held-out data.
- **Label switching**: Hidden states are identifiable only up to permutation. When comparing models across restarts, match states by emission parameters, not by index.
- **Degenerate states**: A state may collapse to explain a single observation (Gaussian with near-zero variance). Regularization on covariance matrices prevents this.
- **Confusing Viterbi and posterior decoding**: Viterbi gives the single best joint path; posterior decoding gives the best marginal state at each time step. They answer different questions and can disagree significantly.
- **Ignoring state dwell times**: The geometric dwell-time distribution implicit in standard HMMs may be a poor fit for data with long regime durations. Consider hidden semi-Markov models if dwell times are non-geometric.

## Related Skills

- [Model Markov Chain](../model-markov-chain/SKILL.md) -- prerequisite for understanding the transition structure that underlies the hidden layer
- [Simulate Stochastic Process](../simulate-stochastic-process/SKILL.md) -- can be used to generate synthetic HMM data for testing and to simulate from a fitted model for posterior predictive checks