# Hidden Regime Phase 3: Advanced Features Architecture

**Research and Design Document for Next-Generation Market Regime Detection**

This document outlines the technical architecture and implementation strategy for Phase 3 advanced features, transforming Hidden Regime from a traditional HMM system into a state-of-the-art Bayesian learning platform.

## Executive Summary

Phase 3 represents a significant evolution from deterministic HMMs to probabilistic, adaptive systems with uncertainty quantification. The key innovations include:

1. **Online Learning**: Real-time model adaptation with streaming data
2. **Bayesian Uncertainty Quantification**: MCMC sampling for parameter distributions
3. **Automatic Model Selection**: Data-driven optimization of model complexity
4. **Fat-Tailed Emission Models**: Crisis detection with Student-t distributions
5. **Multi-Asset Regime Correlation**: Cross-asset regime modeling

## Current State Analysis

### Phase 2 Capabilities ✅
- **Static HMMs**: Fixed parameters after training
- **Point Estimates**: Single values for all parameters
- **Batch Processing**: Complete retraining required for updates
- **Gaussian Emissions**: Normal distribution assumptions
- **Manual Model Selection**: User chooses number of states

### Phase 3 Transformation Goals 🎯
- **Adaptive HMMs**: Continuous learning from new data
- **Probabilistic Parameters**: Full uncertainty quantification
- **Streaming Processing**: Incremental updates without retraining
- **Flexible Emissions**: Student-t and mixture distributions
- **Automatic Optimization**: Data-driven model structure selection

## 1. Online Learning System

### 1.1 Theoretical Foundation

**Problem**: Traditional HMMs require complete retraining when new data arrives, making them impractical for real-time applications.

**Solution**: Implement recursive Bayesian updating for HMM parameters using sufficient statistics and exponential forgetting.

#### Mathematical Framework

```
θₜ = θₜ₋₁ + αₜ · ∇L(θₜ₋₁, xₜ)
```

Where:
- `θₜ`: Parameters at time t
- `αₜ`: Adaptive learning rate
- `L`: Log-likelihood function
- `xₜ`: New observation

#### Sufficient Statistics Approach

```python
class OnlineHMMStats:
    """Maintain sufficient statistics for online learning"""
    
    def __init__(self, n_states, forgetting_factor=0.99):
        self.n_states = n_states
        self.forgetting_factor = forgetting_factor
        
        # Sufficient statistics
        self.gamma_sum = np.zeros(n_states)  # Expected state counts
        self.xi_sum = np.zeros((n_states, n_states))  # Expected transition counts
        self.emission_stats = {
            'mean_num': np.zeros(n_states),
            'mean_den': np.zeros(n_states),
            'var_num': np.zeros(n_states),
            'var_den': np.zeros(n_states)
        }
        
    def update_stats(self, new_observation, state_probs, transition_probs):
        """Update sufficient statistics with new observation"""
        # Apply exponential forgetting
        self.gamma_sum *= self.forgetting_factor
        self.xi_sum *= self.forgetting_factor
        
        # Update with new data
        self.gamma_sum += state_probs
        self.xi_sum += transition_probs
        
        # Update emission statistics
        for state in range(self.n_states):
            weight = state_probs[state]
            self.emission_stats['mean_num'][state] *= self.forgetting_factor
            self.emission_stats['mean_den'][state] *= self.forgetting_factor
            self.emission_stats['mean_num'][state] += weight * new_observation
            self.emission_stats['mean_den'][state] += weight
            
    def estimate_parameters(self):
        """Estimate parameters from sufficient statistics"""
        # Transition matrix
        transition_matrix = self.xi_sum / self.xi_sum.sum(axis=1, keepdims=True)
        
        # Emission parameters
        means = self.emission_stats['mean_num'] / self.emission_stats['mean_den']
        # Variance estimation would include second moments
        
        return transition_matrix, means
```

### 1.2 Implementation Architecture

```python
class OnlineHMM:
    """Online Hidden Markov Model with streaming updates"""
    
    def __init__(self, config: OnlineHMMConfig):
        self.config = config
        self.stats = OnlineHMMStats(config.n_states, config.forgetting_factor)
        self.current_params = None
        self.parameter_history = []
        self.likelihood_history = []
        
    async def add_observation(self, observation: float, timestamp: datetime):
        """Add new observation and update model"""
        # Forward step: compute current state probabilities
        state_probs = self._forward_step(observation)
        
        # Backward step: compute transition probabilities (approximate)
        transition_probs = self._compute_transition_probs(observation, state_probs)
        
        # Update sufficient statistics
        self.stats.update_stats(observation, state_probs, transition_probs)
        
        # Update parameters (with smoothing)
        new_params = self.stats.estimate_parameters()
        self._smooth_parameter_update(new_params)
        
        # Store history
        self.parameter_history.append({
            'timestamp': timestamp,
            'parameters': self.current_params.copy(),
            'likelihood': self._compute_likelihood(observation)
        })
        
        return {
            'state_probabilities': state_probs,
            'most_likely_state': np.argmax(state_probs),
            'parameter_change': self._compute_parameter_change(),
            'model_confidence': self._compute_model_confidence()
        }
    
    def _smooth_parameter_update(self, new_params):
        """Smooth parameter updates to prevent instability"""
        if self.current_params is None:
            self.current_params = new_params
        else:
            # Exponential moving average
            alpha = self.config.parameter_smoothing
            for key in self.current_params:
                self.current_params[key] = (
                    alpha * new_params[key] + 
                    (1 - alpha) * self.current_params[key]
                )
    
    def detect_regime_change(self, threshold=0.8):
        """Detect significant regime changes"""
        if len(self.parameter_history) < 10:
            return False
            
        recent_likelihood = [h['likelihood'] for h in self.parameter_history[-10:]]
        likelihood_trend = np.polyfit(range(len(recent_likelihood)), recent_likelihood, 1)[0]
        
        return likelihood_trend < -threshold
```

### 1.3 Change Point Detection

```python
class RegimeChangeDetector:
    """Detect structural breaks in time series"""
    
    def __init__(self, window_size=50, min_regime_length=10):
        self.window_size = window_size
        self.min_regime_length = min_regime_length
        self.change_points = []
        
    def detect_change_point(self, data, method='cusum'):
        """Detect change points using various methods"""
        if method == 'cusum':
            return self._cusum_detection(data)
        elif method == 'likelihood_ratio':
            return self._likelihood_ratio_test(data)
        elif method == 'bayesian':
            return self._bayesian_change_point(data)
    
    def _cusum_detection(self, data):
        """CUSUM algorithm for change point detection"""
        n = len(data)
        if n < self.window_size:
            return []
            
        # Compute CUSUM statistics
        mean_est = np.mean(data)
        cusum_pos = np.zeros(n)
        cusum_neg = np.zeros(n)
        
        threshold = 5.0  # Adjustable threshold
        
        change_points = []
        for t in range(1, n):
            cusum_pos[t] = max(0, cusum_pos[t-1] + data[t] - mean_est)
            cusum_neg[t] = min(0, cusum_neg[t-1] + data[t] - mean_est)
            
            if abs(cusum_pos[t]) > threshold or abs(cusum_neg[t]) > threshold:
                if len(change_points) == 0 or t - change_points[-1] > self.min_regime_length:
                    change_points.append(t)
                    cusum_pos[t] = 0
                    cusum_neg[t] = 0
        
        return change_points
```

## 2. Bayesian Uncertainty Quantification

### 2.1 Theoretical Framework

**Objective**: Replace point estimates with full posterior distributions over HMM parameters.

#### Prior Specifications

```python
@dataclass
class BayesianPriors:
    """Prior distributions for Bayesian HMM"""
    
    # Dirichlet priors for transition matrix
    transition_concentration: float = 1.0
    
    # Normal-Inverse-Gamma priors for emissions
    emission_mean_prior: float = 0.0
    emission_mean_precision: float = 1.0
    emission_var_shape: float = 2.0
    emission_var_scale: float = 0.01
    
    # Dirichlet prior for initial state distribution
    initial_concentration: float = 1.0
```

### 2.2 MCMC Implementation

```python
class BayesianHMM:
    """Bayesian HMM with MCMC parameter estimation"""
    
    def __init__(self, n_states: int, priors: BayesianPriors):
        self.n_states = n_states
        self.priors = priors
        self.samples = {'transitions': [], 'emissions': [], 'states': []}
        
    def gibbs_sampler(self, data: np.ndarray, n_samples: int = 1000, burn_in: int = 200):
        """Gibbs sampling for Bayesian HMM"""
        
        # Initialize parameters randomly
        current_states = self._initialize_states(data)
        current_transitions = self._initialize_transitions()
        current_emissions = self._initialize_emissions()
        
        for iteration in range(n_samples + burn_in):
            # Sample states given parameters (Forward-Backward)
            current_states = self._sample_states(data, current_transitions, current_emissions)
            
            # Sample transitions given states (Dirichlet posterior)
            current_transitions = self._sample_transitions(current_states)
            
            # Sample emissions given states and data (Normal-Inverse-Gamma posterior)
            current_emissions = self._sample_emissions(data, current_states)
            
            # Store samples (after burn-in)
            if iteration >= burn_in:
                self.samples['states'].append(current_states.copy())
                self.samples['transitions'].append(current_transitions.copy())
                self.samples['emissions'].append(current_emissions.copy())
                
        return self.samples
    
    def _sample_transitions(self, states):
        """Sample transition matrix from Dirichlet posterior"""
        transitions = np.zeros((self.n_states, self.n_states))
        
        for i in range(self.n_states):
            # Count transitions from state i
            counts = np.zeros(self.n_states)
            for t in range(len(states) - 1):
                if states[t] == i:
                    counts[states[t + 1]] += 1
            
            # Dirichlet posterior
            posterior_alpha = counts + self.priors.transition_concentration
            transitions[i, :] = np.random.dirichlet(posterior_alpha)
            
        return transitions
    
    def _sample_emissions(self, data, states):
        """Sample emission parameters from Normal-Inverse-Gamma posterior"""
        emissions = np.zeros((self.n_states, 2))  # [mean, variance]
        
        for k in range(self.n_states):
            # Data for this state
            state_data = data[states == k]
            n_k = len(state_data)
            
            if n_k > 0:
                # Sufficient statistics
                sum_x = np.sum(state_data)
                sum_x2 = np.sum(state_data ** 2)
                
                # Posterior parameters for Normal-Inverse-Gamma
                mu_0 = self.priors.emission_mean_prior
                kappa_0 = self.priors.emission_mean_precision
                alpha_0 = self.priors.emission_var_shape
                beta_0 = self.priors.emission_var_scale
                
                # Updated parameters
                kappa_n = kappa_0 + n_k
                mu_n = (kappa_0 * mu_0 + sum_x) / kappa_n
                alpha_n = alpha_0 + n_k / 2
                beta_n = beta_0 + 0.5 * (sum_x2 + kappa_0 * mu_0**2 - kappa_n * mu_n**2)
                
                # Sample variance from Inverse-Gamma
                variance = 1 / np.random.gamma(alpha_n, 1 / beta_n)
                
                # Sample mean from Normal
                mean = np.random.normal(mu_n, np.sqrt(variance / kappa_n))
                
                emissions[k, :] = [mean, variance]
            else:
                # Use priors if no data for this state
                variance = 1 / np.random.gamma(alpha_0, 1 / beta_0)
                mean = np.random.normal(mu_0, np.sqrt(variance / kappa_0))
                emissions[k, :] = [mean, variance]
                
        return emissions
```

### 2.3 Uncertainty-Aware Predictions

```python
class UncertaintyQuantifiedPredictor:
    """Make predictions with uncertainty quantification"""
    
    def __init__(self, bayesian_hmm: BayesianHMM):
        self.model = bayesian_hmm
        
    def predict_with_uncertainty(self, new_observation: float, n_samples: int = 100):
        """Predict next state with uncertainty bounds"""
        
        predictions = []
        state_probs_samples = []
        
        # Sample from posterior
        sample_indices = np.random.choice(len(self.model.samples['transitions']), n_samples)
        
        for idx in sample_indices:
            transitions = self.model.samples['transitions'][idx]
            emissions = self.model.samples['emissions'][idx]
            
            # Compute state probabilities for this parameter sample
            state_probs = self._forward_step(new_observation, transitions, emissions)
            state_probs_samples.append(state_probs)
            
            # Most likely state for this sample
            predictions.append(np.argmax(state_probs))
        
        # Aggregate results
        state_probs_samples = np.array(state_probs_samples)
        
        return {
            'mean_state_probs': np.mean(state_probs_samples, axis=0),
            'std_state_probs': np.std(state_probs_samples, axis=0),
            'credible_intervals': {
                'lower': np.percentile(state_probs_samples, 5, axis=0),
                'upper': np.percentile(state_probs_samples, 95, axis=0)
            },
            'mode_prediction': np.argmax(np.mean(state_probs_samples, axis=0)),
            'prediction_entropy': self._compute_entropy(np.mean(state_probs_samples, axis=0))
        }
    
    def _compute_entropy(self, probs):
        """Compute entropy as measure of uncertainty"""
        return -np.sum(probs * np.log(probs + 1e-10))
```

## 3. Automatic Model Selection

### 3.1 Reversible Jump MCMC

```python
class ReversibleJumpMCMC:
    """Automatic model selection using RJ-MCMC"""
    
    def __init__(self, min_states=2, max_states=6):
        self.min_states = min_states
        self.max_states = max_states
        self.model_samples = []
        
    def rjmcmc_sampler(self, data, n_iterations=1000):
        """RJ-MCMC sampler for model selection"""
        
        # Initialize with random model
        current_k = np.random.randint(self.min_states, self.max_states + 1)
        current_model = BayesianHMM(current_k, BayesianPriors())
        current_likelihood = self._compute_marginal_likelihood(current_model, data)
        
        accept_counts = {'birth': 0, 'death': 0, 'update': 0}
        
        for iteration in range(n_iterations):
            # Choose move type
            move_type = self._choose_move_type(current_k)
            
            if move_type == 'birth':
                proposed_model = self._birth_move(current_model, data)
                acceptance_prob = self._compute_birth_acceptance(current_model, proposed_model, data)
                
            elif move_type == 'death':
                proposed_model = self._death_move(current_model, data)
                acceptance_prob = self._compute_death_acceptance(current_model, proposed_model, data)
                
            else:  # update
                proposed_model = self._update_move(current_model, data)
                acceptance_prob = self._compute_update_acceptance(current_model, proposed_model, data)
            
            # Accept or reject
            if np.random.random() < acceptance_prob:
                current_model = proposed_model
                current_k = proposed_model.n_states
                current_likelihood = self._compute_marginal_likelihood(proposed_model, data)
                accept_counts[move_type] += 1
            
            # Store sample
            self.model_samples.append({
                'iteration': iteration,
                'n_states': current_k,
                'model': current_model,
                'likelihood': current_likelihood
            })
        
        return self.model_samples, accept_counts
    
    def _birth_move(self, current_model, data):
        """Add new state to model"""
        new_k = current_model.n_states + 1
        if new_k > self.max_states:
            return current_model
        
        # Create new model with additional state
        new_model = BayesianHMM(new_k, BayesianPriors())
        
        # Initialize new parameters by splitting existing state
        split_state = np.random.randint(current_model.n_states)
        new_model = self._split_state(current_model, split_state)
        
        return new_model
    
    def _death_move(self, current_model, data):
        """Remove state from model"""
        new_k = current_model.n_states - 1
        if new_k < self.min_states:
            return current_model
        
        # Choose state to remove
        remove_state = np.random.randint(current_model.n_states)
        new_model = self._merge_states(current_model, remove_state)
        
        return new_model
```

### 3.2 Model Selection Criteria

```python
class ModelSelectionCriteria:
    """Various criteria for model selection"""
    
    @staticmethod
    def compute_bic(log_likelihood, n_params, n_data):
        """Bayesian Information Criterion"""
        return -2 * log_likelihood + n_params * np.log(n_data)
    
    @staticmethod
    def compute_aic(log_likelihood, n_params):
        """Akaike Information Criterion"""
        return -2 * log_likelihood + 2 * n_params
    
    @staticmethod
    def compute_dic(log_likelihood_samples):
        """Deviance Information Criterion"""
        mean_deviance = -2 * np.mean(log_likelihood_samples)
        deviance_at_mean = -2 * np.mean(log_likelihood_samples)  # Would need actual implementation
        p_dic = mean_deviance - deviance_at_mean
        return mean_deviance + p_dic
    
    def cross_validation_score(self, model_class, data, k_folds=5):
        """Cross-validation for model selection"""
        fold_size = len(data) // k_folds
        scores = []
        
        for fold in range(k_folds):
            # Split data
            start_idx = fold * fold_size
            end_idx = (fold + 1) * fold_size
            
            test_data = data[start_idx:end_idx]
            train_data = np.concatenate([data[:start_idx], data[end_idx:]])
            
            # Train model
            model = model_class()
            model.fit(train_data)
            
            # Evaluate on test set
            test_likelihood = model.score(test_data)
            scores.append(test_likelihood)
        
        return np.mean(scores), np.std(scores)
```

## 4. Fat-Tailed Emission Models

### 4.1 Student-t Distributions

```python
class StudentTHMM:
    """HMM with Student-t emission distributions"""
    
    def __init__(self, n_states):
        self.n_states = n_states
        self.emission_params = None  # [location, scale, degrees_of_freedom]
        
    def fit_student_t(self, data, states):
        """Fit Student-t parameters using EM algorithm"""
        self.emission_params = np.zeros((self.n_states, 3))
        
        for k in range(self.n_states):
            state_data = data[states == k]
            if len(state_data) > 3:  # Need at least 3 points for t-distribution
                # Initial estimates
                loc = np.mean(state_data)
                scale = np.std(state_data)
                df = 4.0  # Initial guess for degrees of freedom
                
                # EM iterations for Student-t
                for _ in range(20):
                    # E-step: compute weights
                    weights = self._compute_t_weights(state_data, loc, scale, df)
                    
                    # M-step: update parameters
                    loc = np.sum(weights * state_data) / np.sum(weights)
                    scale = np.sqrt(np.sum(weights * (state_data - loc)**2) / np.sum(weights))
                    df = self._update_degrees_of_freedom(state_data, loc, scale, weights)
                
                self.emission_params[k] = [loc, scale, df]
            else:
                # Fallback to robust estimates
                self.emission_params[k] = [np.median(state_data), 
                                         np.std(state_data), 4.0]
    
    def _compute_t_weights(self, data, loc, scale, df):
        """Compute weights for Student-t EM algorithm"""
        from scipy.stats import t
        
        standardized = (data - loc) / scale
        weights = (df + 1) / (df + standardized**2)
        return weights
    
    def _update_degrees_of_freedom(self, data, loc, scale, weights):
        """Update degrees of freedom parameter"""
        # Newton-Raphson iteration for degrees of freedom
        # This is a simplified version - full implementation would use numerical optimization
        
        from scipy.optimize import minimize_scalar
        
        def negative_log_likelihood(df):
            from scipy.stats import t
            if df <= 0:
                return np.inf
            return -np.sum(t.logpdf((data - loc) / scale, df) - np.log(scale))
        
        result = minimize_scalar(negative_log_likelihood, bounds=(0.1, 100), method='bounded')
        return max(result.x, 2.1)  # Ensure df > 2 for finite variance
    
    def detect_tail_events(self, data, threshold_quantile=0.05):
        """Detect extreme tail events using fitted Student-t models"""
        tail_events = []
        
        for t, observation in enumerate(data):
            # Find most likely state
            state_probs = self._compute_state_probabilities(observation)
            most_likely_state = np.argmax(state_probs)
            
            # Check if observation is in tail of distribution
            loc, scale, df = self.emission_params[most_likely_state]
            from scipy.stats import t
            
            # Compute percentile of observation
            standardized = (observation - loc) / scale
            percentile = t.cdf(standardized, df)
            
            # Check if in lower or upper tail
            if percentile < threshold_quantile or percentile > (1 - threshold_quantile):
                tail_events.append({
                    'time': t,
                    'observation': observation,
                    'state': most_likely_state,
                    'percentile': percentile,
                    'severity': min(percentile, 1 - percentile)
                })
        
        return tail_events
```

### 4.2 Crisis Detection System

```python
class CrisisDetectionSystem:
    """Advanced crisis detection using fat-tailed models"""
    
    def __init__(self, student_t_hmm: StudentTHMM):
        self.model = student_t_hmm
        self.crisis_threshold = 0.01  # 1% tail probability
        self.alert_history = []
        
    def real_time_crisis_monitoring(self, new_observation):
        """Monitor for crisis conditions in real-time"""
        
        # Get state probabilities
        state_probs = self.model._compute_state_probabilities(new_observation)
        most_likely_state = np.argmax(state_probs)
        
        # Analyze tail behavior for current state
        loc, scale, df = self.model.emission_params[most_likely_state]
        
        from scipy.stats import t
        standardized = (new_observation - loc) / scale
        tail_prob = min(t.cdf(standardized, df), 1 - t.cdf(standardized, df))
        
        # Crisis indicators
        crisis_indicators = {
            'extreme_return': abs(new_observation) > 0.05,  # 5% daily return
            'tail_event': tail_prob < self.crisis_threshold,
            'low_degrees_freedom': df < 3.0,  # Heavy tails
            'high_volatility_state': scale > 0.03,  # 3% volatility
        }
        
        # Overall crisis score
        crisis_score = sum(crisis_indicators.values()) / len(crisis_indicators)
        
        # Generate alert if needed
        alert = None
        if crisis_score > 0.6:  # 60% of indicators triggered
            alert = {
                'timestamp': pd.Timestamp.now(),
                'level': 'HIGH' if crisis_score > 0.8 else 'MEDIUM',
                'crisis_score': crisis_score,
                'indicators': crisis_indicators,
                'tail_probability': tail_prob,
                'degrees_of_freedom': df,
                'observation': new_observation
            }
            
            self.alert_history.append(alert)
        
        return {
            'crisis_score': crisis_score,
            'indicators': crisis_indicators,
            'alert': alert,
            'state_analysis': {
                'most_likely_state': most_likely_state,
                'state_probability': state_probs[most_likely_state],
                'tail_probability': tail_prob
            }
        }
```

## 5. Multi-Asset Regime Correlation

### 5.1 Cross-Asset HMM

```python
class MultiAssetHMM:
    """HMM for multiple correlated assets"""
    
    def __init__(self, n_assets, n_states, correlation_structure='full'):
        self.n_assets = n_assets
        self.n_states = n_states
        self.correlation_structure = correlation_structure
        
        # Parameters
        self.transition_matrix = None
        self.emission_means = None  # [n_states, n_assets]
        self.emission_covs = None   # [n_states, n_assets, n_assets]
        
    def fit_multivariate(self, data):
        """Fit multivariate HMM to multiple asset returns"""
        n_obs, n_assets = data.shape
        
        # Initialize parameters
        self._initialize_multivariate_parameters(data)
        
        # EM algorithm for multivariate case
        log_likelihoods = []
        
        for iteration in range(100):  # Max iterations
            # E-step: compute state probabilities
            alpha = self._multivariate_forward(data)
            beta = self._multivariate_backward(data)
            
            gamma = alpha * beta
            gamma = gamma / gamma.sum(axis=1, keepdims=True)
            
            xi = self._compute_multivariate_xi(data, alpha, beta)
            
            # M-step: update parameters
            self._update_multivariate_parameters(data, gamma, xi)
            
            # Check convergence
            log_likelihood = self._compute_multivariate_log_likelihood(data)
            log_likelihoods.append(log_likelihood)
            
            if len(log_likelihoods) > 1:
                if abs(log_likelihoods[-1] - log_likelihoods[-2]) < 1e-6:
                    break
        
        return log_likelihoods
    
    def _multivariate_forward(self, data):
        """Forward algorithm for multivariate emissions"""
        n_obs = len(data)
        alpha = np.zeros((n_obs, self.n_states))
        
        # Initialize
        alpha[0] = self.initial_probs * self._multivariate_emission_prob(data[0])
        
        # Forward pass
        for t in range(1, n_obs):
            emission_probs = self._multivariate_emission_prob(data[t])
            alpha[t] = emission_probs * (alpha[t-1] @ self.transition_matrix)
            
            # Normalize to prevent underflow
            alpha[t] /= alpha[t].sum()
        
        return alpha
    
    def _multivariate_emission_prob(self, observation):
        """Compute emission probabilities for multivariate observation"""
        probs = np.zeros(self.n_states)
        
        for k in range(self.n_states):
            mean = self.emission_means[k]
            cov = self.emission_covs[k]
            
            # Multivariate normal probability
            from scipy.stats import multivariate_normal
            probs[k] = multivariate_normal.pdf(observation, mean, cov)
        
        return probs + 1e-10  # Prevent zeros
    
    def analyze_regime_correlations(self):
        """Analyze correlations between assets in different regimes"""
        correlations = {}
        
        for k in range(self.n_states):
            cov_matrix = self.emission_covs[k]
            # Convert covariance to correlation
            std_devs = np.sqrt(np.diag(cov_matrix))
            corr_matrix = cov_matrix / np.outer(std_devs, std_devs)
            
            correlations[f'regime_{k}'] = {
                'correlation_matrix': corr_matrix,
                'mean_correlation': np.mean(corr_matrix[np.triu_indices_from(corr_matrix, k=1)]),
                'max_correlation': np.max(corr_matrix[np.triu_indices_from(corr_matrix, k=1)]),
                'min_correlation': np.min(corr_matrix[np.triu_indices_from(corr_matrix, k=1)])
            }
        
        return correlations
```

### 5.2 Dynamic Correlation Modeling

```python
class DynamicCorrelationModel:
    """Model time-varying correlations between assets"""
    
    def __init__(self, n_assets):
        self.n_assets = n_assets
        self.correlation_history = []
        
    def dcc_garch_model(self, returns):
        """Dynamic Conditional Correlation GARCH model"""
        
        # Step 1: Fit univariate GARCH models
        garch_models = []
        standardized_returns = np.zeros_like(returns)
        
        for asset in range(self.n_assets):
            asset_returns = returns[:, asset]
            
            # Fit GARCH(1,1) - simplified version
            garch_params = self._fit_garch(asset_returns)
            garch_models.append(garch_params)
            
            # Standardize returns
            volatilities = self._predict_garch_volatility(asset_returns, garch_params)
            standardized_returns[:, asset] = asset_returns / volatilities
        
        # Step 2: Model dynamic correlations
        correlation_matrices = self._fit_dcc_model(standardized_returns)
        
        return garch_models, correlation_matrices
    
    def _fit_garch(self, returns):
        """Fit GARCH(1,1) model to single asset"""
        # Simplified GARCH implementation
        # In practice, would use arch library or similar
        
        from scipy.optimize import minimize
        
        def garch_likelihood(params):
            omega, alpha, beta = params
            
            if omega <= 0 or alpha < 0 or beta < 0 or alpha + beta >= 1:
                return 1e10
            
            n = len(returns)
            sigma2 = np.zeros(n)
            sigma2[0] = np.var(returns)
            
            for t in range(1, n):
                sigma2[t] = omega + alpha * returns[t-1]**2 + beta * sigma2[t-1]
            
            # Log-likelihood
            log_likelihood = -0.5 * np.sum(
                np.log(2 * np.pi * sigma2) + returns**2 / sigma2
            )
            
            return -log_likelihood
        
        # Initial guess
        initial_params = [np.var(returns) * 0.01, 0.1, 0.8]
        
        result = minimize(garch_likelihood, initial_params, method='L-BFGS-B',
                         bounds=[(1e-6, None), (0, 1), (0, 1)])
        
        return result.x
    
    def regime_based_correlation_forecast(self, current_regime, horizon=10):
        """Forecast correlations based on current regime"""
        
        if not self.correlation_history:
            return None
        
        # Get historical correlations for current regime
        regime_correlations = [
            corr for corr, regime in self.correlation_history 
            if regime == current_regime
        ]
        
        if len(regime_correlations) < 5:
            return None
        
        # Simple forecast: average of recent correlations in this regime
        recent_correlations = regime_correlations[-10:]  # Last 10 observations
        forecast_correlation = np.mean(recent_correlations, axis=0)
        
        return forecast_correlation
```

## 6. Implementation Timeline

### Phase 3.1: Online Learning Foundation (Months 1-2)
- [ ] Implement `OnlineHMM` with sufficient statistics
- [ ] Add change point detection algorithms
- [ ] Create streaming data interfaces
- [ ] Develop parameter smoothing mechanisms
- [ ] Build comprehensive testing suite

### Phase 3.2: Bayesian Infrastructure (Months 3-4)  
- [ ] Implement `BayesianHMM` with MCMC sampling
- [ ] Add prior specification framework
- [ ] Create uncertainty quantification methods
- [ ] Develop diagnostic tools for MCMC
- [ ] Build uncertainty-aware prediction API

### Phase 3.3: Model Selection & Fat Tails (Months 5-6)
- [ ] Implement Reversible Jump MCMC
- [ ] Add Student-t emission models
- [ ] Create crisis detection system
- [ ] Develop model selection criteria
- [ ] Build tail risk analysis tools

### Phase 3.4: Multi-Asset Integration (Months 7-8)
- [ ] Implement `MultiAssetHMM` 
- [ ] Add dynamic correlation modeling
- [ ] Create cross-asset regime analysis
- [ ] Develop portfolio optimization tools
- [ ] Build comprehensive visualization system

## 7. Technical Challenges & Solutions

### Challenge 1: Computational Complexity
**Problem**: MCMC sampling and online learning significantly increase computational requirements.

**Solution**: 
- Implement GPU acceleration using JAX/PyTorch
- Use variational inference as faster alternative to MCMC
- Implement hierarchical sampling for efficiency
- Add parallel processing for multi-asset models

### Challenge 2: Numerical Stability
**Problem**: Bayesian methods and online learning can suffer from numerical issues.

**Solution**:
- Implement log-space computations throughout
- Add adaptive regularization mechanisms  
- Use robust initialization strategies
- Implement numerical stability checks

### Challenge 3: Model Validation
**Problem**: Complex Bayesian models are harder to validate than simple HMMs.

**Solution**:
- Develop comprehensive diagnostic tools
- Implement cross-validation for complex models
- Add synthetic data testing frameworks
- Create interpretability tools for Bayesian outputs

### Challenge 4: Real-Time Requirements
**Problem**: Online learning must operate within strict latency constraints.

**Solution**:
- Implement asynchronous processing architecture
- Use approximate inference methods when needed
- Cache intermediate computations
- Optimize critical paths for low latency

## 8. Success Metrics

### Performance Metrics
- **Prediction Accuracy**: 15-20% improvement over Phase 2 static models
- **Uncertainty Calibration**: 90%+ of predictions within confidence intervals
- **Crisis Detection**: <24 hour detection time for major market events
- **Real-Time Latency**: <100ms for online parameter updates

### Business Metrics  
- **Risk-Adjusted Returns**: 25-40% improvement in Sharpe ratios
- **Downside Protection**: 50-70% reduction in maximum drawdowns
- **Model Stability**: 80%+ reduction in parameter volatility
- **Cross-Asset Insights**: Identify regime correlations not visible in single-asset models

## Conclusion

Phase 3 represents a fundamental transformation of Hidden Regime from a traditional econometric tool to a cutting-edge Bayesian learning system. The architecture provides:

1. **Real-Time Adaptation**: Models that continuously learn and adapt to market changes
2. **Uncertainty Quantification**: Full probabilistic understanding of model confidence
3. **Automatic Optimization**: Data-driven selection of optimal model complexity
4. **Crisis Detection**: Early warning systems for market stress periods
5. **Multi-Asset Intelligence**: Understanding of cross-asset regime dynamics

This technical foundation enables the eventual Model Context Protocol integration in Phase 4, creating a comprehensive AI-powered financial analysis ecosystem.

The implementation will be challenging but achievable, building incrementally on the solid Phase 2 foundation while introducing state-of-the-art Bayesian machine learning techniques to financial regime detection.