Pymc

1. Validating Variational Approximation Quality

The Gauge-Transformer uses variational inference (VFE minimization). PyMC can validate whether the variational posterior is a good approximation.

import pymc as pm
import arviz as az
import numpy as np
import torch

def validate_vfe_approximation(observed_data, mu_q, sigma_q, prior_mu, prior_sigma):
    """Compare the Gauge-Transformer's variational posterior to MCMC ground truth.

    Args:
        observed_data: numpy array of observed values
        mu_q: variational mean from the Gauge-Transformer
        sigma_q: variational std from the Gauge-Transformer
        prior_mu: prior mean used in VFE
        prior_sigma: prior std used in VFE
    """
    # Build the same generative model that VFE approximates
    with pm.Model() as true_model:
        # Prior (matching the Gauge-Transformer's prior)
        theta = pm.Normal('theta', mu=prior_mu, sigma=prior_sigma)

        # Likelihood
        y = pm.Normal('y', mu=theta, sigma=1.0, observed=observed_data)

        # MCMC ground truth
        trace = pm.sample(4000, tune=2000, return_inferencedata=True, random_seed=42)

    # Compare posteriors
    mcmc_mean = float(trace.posterior['theta'].mean())
    mcmc_std = float(trace.posterior['theta'].std())

    print(f"MCMC posterior:  mean={mcmc_mean:.4f}, std={mcmc_std:.4f}")
    print(f"VFE posterior:   mean={mu_q:.4f}, std={sigma_q:.4f}")
    print(f"Mean difference: {abs(mcmc_mean - mu_q):.4f}")
    print(f"Std ratio:       {sigma_q / mcmc_std:.4f}")

    # KL divergence between variational and true posterior
    kl = np.log(mcmc_std / sigma_q) + (sigma_q**2 + (mu_q - mcmc_mean)**2) / (2 * mcmc_std**2) - 0.5
    print(f"KL(q || p_mcmc): {kl:.4f}")

    return trace, {'mcmc_mean': mcmc_mean, 'mcmc_std': mcmc_std, 'kl': kl}

2. Multivariate Gaussian Belief Validation

def validate_multivariate_belief(observed, mu_q, cov_q, prior_mu, prior_cov):
    """Validate multivariate Gaussian belief against MCMC.

    For small-scale problems where the Gauge-Transformer learns
    multivariate beliefs q(z) = N(mu_q, Sigma_q).
    """
    K = len(mu_q)

    with pm.Model() as model:
        # Multivariate prior
        theta = pm.MvNormal('theta', mu=prior_mu, cov=prior_cov, shape=K)

        # Likelihood
        y = pm.MvNormal('y', mu=theta, cov=np.eye(K), observed=observed)

        # Sample
        trace = pm.sample(4000, tune=2000, return_inferencedata=True, random_seed=42)

    # Extract MCMC posterior statistics
    mcmc_means = trace.posterior['theta'].mean(dim=('chain', 'draw')).values
    mcmc_cov = np.cov(trace.posterior['theta'].values.reshape(-1, K).T)

    # Compare
    print("MCMC posterior mean:", mcmc_means)
    print("VFE posterior mean: ", mu_q)
    print("Mean L2 error:      ", np.linalg.norm(mcmc_means - mu_q))

    return trace, mcmc_means, mcmc_cov

3. Bayesian Hyperparameter Sensitivity Analysis

def hyperparameter_sensitivity(perplexity_data, hyperparams):
    """Bayesian analysis of how hyperparameters affect perplexity.

    Args:
        perplexity_data: dict mapping hyperparameter configs to perplexity measurements
        hyperparams: list of hyperparameter names
    """
    import pandas as pd

    # Build dataframe
    rows = []
    for config, perplexities in perplexity_data.items():
        for ppl in perplexities:
            row = dict(zip(hyperparams, config))
            row['perplexity'] = ppl
            rows.append(row)
    df = pd.DataFrame(rows)

    with pm.Model() as sensitivity_model:
        # Intercept
        alpha = pm.Normal('alpha', mu=df['perplexity'].mean(), sigma=20)

        # Hyperparameter effects
        betas = {}
        for hp in hyperparams:
            values = df[hp].values.astype(float)
            betas[hp] = pm.Normal(f'beta_{hp}', mu=0, sigma=10)

        # Linear predictor
        mu = alpha
        for hp in hyperparams:
            mu = mu + betas[hp] * df[hp].values.astype(float)

        # Noise
        sigma = pm.HalfNormal('sigma', sigma=10)

        # Likelihood
        y = pm.Normal('perplexity', mu=mu, sigma=sigma, observed=df['perplexity'].values)

        # Sample
        trace = pm.sample(2000, tune=1000, return_inferencedata=True, random_seed=42)

    # Summarize effects
    summary = az.summary(trace, var_names=[f'beta_{hp}' for hp in hyperparams])
    print(summary)

    # Forest plot of effects
    az.plot_forest(trace, var_names=[f'beta_{hp}' for hp in hyperparams],
                   combined=True, figsize=(8, 4))

    return trace, summary

4. Hierarchical Model for Multi-Head Attention

def hierarchical_attention_model(attention_entropies):
    """Hierarchical Bayesian model for attention head behavior.

    Models per-head entropy as drawn from a group-level distribution,
    mirroring the hierarchical structure of VFE (h -> s -> p -> q).

    Args:
        attention_entropies: (n_layers, n_heads, n_samples) array
    """
    n_layers, n_heads, n_samples = attention_entropies.shape

    with pm.Model() as hierarchical_model:
        # Hyperpriors (group-level)
        mu_global = pm.Normal('mu_global', mu=0, sigma=5)
        sigma_global = pm.HalfNormal('sigma_global', sigma=2)

        # Layer-level
        mu_layer = pm.Normal('mu_layer', mu=mu_global, sigma=sigma_global, shape=n_layers)
        sigma_layer = pm.HalfNormal('sigma_layer', sigma=1, shape=n_layers)

        # Head-level
        mu_head = pm.Normal('mu_head',
                           mu=mu_layer[:, None],
                           sigma=sigma_layer[:, None],
                           shape=(n_layers, n_heads))

        # Observation noise
        sigma_obs = pm.HalfNormal('sigma_obs', sigma=1)

        # Likelihood
        for l in range(n_layers):
            for h in range(n_heads):
                pm.Normal(f'entropy_L{l}_H{h}',
                         mu=mu_head[l, h],
                         sigma=sigma_obs,
                         observed=attention_entropies[l, h])

        trace = pm.sample(2000, tune=1000, return_inferencedata=True, random_seed=42)

    return trace

5. Model Comparison (Gauge-Transformer vs. Baselines)

def bayesian_model_comparison(results_gauge, results_standard, metric='perplexity'):
    """Bayesian comparison of Gauge-Transformer vs standard transformer.

    Args:
        results_gauge: array of metric values from Gauge-Transformer runs
        results_standard: array of metric values from standard transformer runs
    """
    with pm.Model() as comparison:
        # Priors for each model's performance
        mu_gauge = pm.Normal('mu_gauge', mu=np.mean(results_gauge), sigma=20)
        mu_standard = pm.Normal('mu_standard', mu=np.mean(results_standard), sigma=20)

        sigma_gauge = pm.HalfNormal('sigma_gauge', sigma=10)
        sigma_standard = pm.HalfNormal('sigma_standard', sigma=10)

        # Likelihoods
        pm.Normal('y_gauge', mu=mu_gauge, sigma=sigma_gauge, observed=results_gauge)
        pm.Normal('y_standard', mu=mu_standard, sigma=sigma_standard, observed=results_standard)

        # Derived: difference and effect size
        diff = pm.Deterministic('difference', mu_gauge - mu_standard)
        pooled_sigma = pm.math.sqrt((sigma_gauge**2 + sigma_standard**2) / 2)
        effect_size = pm.Deterministic('effect_size', diff / pooled_sigma)

        trace = pm.sample(4000, tune=2000, return_inferencedata=True, random_seed=42)

    # Summarize
    print(az.summary(trace, var_names=['difference', 'effect_size']))

    # Probability that gauge model is better (lower perplexity)
    prob_better = float(np.mean(trace.posterior['difference'].values < 0))
    print(f"\nP(Gauge-Transformer has lower {metric}) = {prob_better:.3f}")

    # Plot
    az.plot_posterior(trace, var_names=['difference'], ref_val=0)

    return trace, prob_better

Diagnostics and Convergence

Essential Convergence Checks

def check_convergence(trace, var_names=None):
    """Run standard MCMC convergence diagnostics."""
    summary = az.summary(trace, var_names=var_names)

    # Check R-hat (should be < 1.01)
    rhat_issues = summary[summary['r_hat'] > 1.01]
    if len(rhat_issues) > 0:
        print("WARNING: R-hat > 1.01 for:")
        print(rhat_issues[['r_hat']])

    # Check ESS (should be > 400)
    ess_issues = summary[summary['ess_bulk'] < 400]
    if len(ess_issues) > 0:
        print("WARNING: Low ESS for:")
        print(ess_issues[['ess_bulk', 'ess_tail']])

    # Trace plots
    az.plot_trace(trace, var_names=var_names)

    # Rank plots (better than trace plots for convergence)
    az.plot_rank(trace, var_names=var_names)

    return summary

Posterior Predictive Checks

def posterior_predictive_check(trace, model):
    """Generate and visualize posterior predictive samples."""
    with model:
        ppc = pm.sample_posterior_predictive(trace, random_seed=42)

    az.plot_ppc(az.from_pymc3(posterior_predictive=ppc, model=model))
    return ppc

ArviZ Integration

PyMC integrates with ArviZ for visualization and diagnostics:

import arviz as az

# Summary statistics
az.summary(trace)

# Posterior plots
az.plot_posterior(trace, var_names=['theta'])

# Forest plots (compare parameters)
az.plot_forest(trace, var_names=['mu_layer'])

# Pair plots (check correlations between parameters)
az.plot_pair(trace, var_names=['mu_gauge', 'mu_standard'])

# LOO cross-validation (model comparison)
az.loo(trace)

# WAIC
az.waic(trace)

Best Practices

1. Always check convergence — R-hat < 1.01, ESS > 400
2. Use weakly informative priors — not flat, but not too narrow
3. Run posterior predictive checks — verify the model generates plausible data
4. Report full posterior summaries — mean, SD, credible intervals
5. Use ArviZ for all visualization — consistent, publication-quality plots
6. Start simple — begin with simple models, add complexity incrementally
7. Compare models with LOO — use az.compare() for formal model comparison
8. Set random seeds for reproducibility
9. Scale data if parameters are on very different scales
10. Use pm.find_MAP() as a sanity check before full MCMC

Relevance to the Gauge-Transformer

The Gauge-Transformer's core framework is Bayesian:

• Beliefs q(z) = N(μ_q, Σ_q) are variational posteriors
• VFE = KL[q || p] - E_q[log p(x|z)] is the variational objective
• Hierarchical structure (h → s → p → q) maps to hierarchical Bayesian models
• Gauge invariance of KL divergence has direct Bayesian interpretation

PyMC provides the gold-standard MCMC inference to validate that the Gauge-Transformer's fast variational inference produces accurate posteriors, especially on small-scale problems where full MCMC is tractable.

Dependencies

pip install pymc arviz