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Build hierarchical models, MCMC (NUTS), variational inference, LOO\u002FWAIC comparison, posterior checks, for probabilistic programming and inference.","transaurus\u002Fstaging-davila7-claude-code-templates","transaurus","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F210882017?v=4","01M6JH1R6ZCRXGJCTDP5ZKZBE6","Page Turner Agent","01M6JH1RBBR2P6Y0XWDGM4B0G7",0,"MIT","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fscientific\u002Fpymc\u002FSKILL.md",[],[19,20,21,22,23,24,25,26,27,28],"pymc","pymc bayesian modeling","modeling","inference","programming","models","build","comparison","with","for","2026-11-10 10:12:30",false,"---\nname: pymc-bayesian-modeling\ndescription: \"Bayesian modeling with PyMC. Build hierarchical models, MCMC (NUTS), variational inference, LOO\u002FWAIC comparison, posterior checks, for probabilistic programming and inference.\"\n---\n\n# PyMC Bayesian Modeling\n\n## Overview\n\nPyMC is a Python library for Bayesian modeling and probabilistic programming. Build, fit, validate, and compare Bayesian models using PyMC's modern API (version 5.x+), including hierarchical models, MCMC sampling (NUTS), variational inference, and model comparison (LOO, WAIC).\n\n## When to Use This Skill\n\nThis skill should be used when:\n- Building Bayesian models (linear\u002Flogistic regression, hierarchical models, time series, etc.)\n- Performing MCMC sampling or variational inference\n- Conducting prior\u002Fposterior predictive checks\n- Diagnosing sampling issues (divergences, convergence, ESS)\n- Comparing multiple models using information criteria (LOO, WAIC)\n- Implementing uncertainty quantification through Bayesian methods\n- Working with hierarchical\u002Fmultilevel data structures\n- Handling missing data or measurement error in a principled way\n\n## Standard Bayesian Workflow\n\nFollow this workflow for building and validating Bayesian models:\n\n### 1. Data Preparation\n\n```python\nimport pymc as pm\nimport arviz as az\nimport numpy as np\n\n# Load and prepare data\nX = ...  # Predictors\ny = ...  # Outcomes\n\n# Standardize predictors for better sampling\nX_mean = X.mean(axis=0)\nX_std = X.std(axis=0)\nX_scaled = (X - X_mean) \u002F X_std\n```\n\n**Key practices:**\n- Standardize continuous predictors (improves sampling efficiency)\n- Center outcomes when possible\n- Handle missing data explicitly (treat as parameters)\n- Use named dimensions with `coords` for clarity\n\n### 2. Model Building\n\n```python\ncoords = {\n    'predictors': ['var1', 'var2', 'var3'],\n    'obs_id': np.arange(len(y))\n}\n\nwith pm.Model(coords=coords) as model:\n    # Priors\n    alpha = pm.Normal('alpha', mu=0, sigma=1)\n    beta = pm.Normal('beta', mu=0, sigma=1, dims='predictors')\n    sigma = pm.HalfNormal('sigma', sigma=1)\n\n    # Linear predictor\n    mu = alpha + pm.math.dot(X_scaled, beta)\n\n    # Likelihood\n    y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y, dims='obs_id')\n```\n\n**Key practices:**\n- Use weakly informative priors (not flat priors)\n- Use `HalfNormal` or `Exponential` for scale parameters\n- Use named dimensions (`dims`) instead of `shape` when possible\n- Use `pm.Data()` for values that will be updated for predictions\n\n### 3. Prior Predictive Check\n\n**Always validate priors before fitting:**\n\n```python\nwith model:\n    prior_pred = pm.sample_prior_predictive(samples=1000, random_seed=42)\n\n# Visualize\naz.plot_ppc(prior_pred, group='prior')\n```\n\n**Check:**\n- Do prior predictions span reasonable values?\n- Are extreme values plausible given domain knowledge?\n- If priors generate implausible data, adjust and re-check\n\n### 4. Fit Model\n\n```python\nwith model:\n    # Optional: Quick exploration with ADVI\n    # approx = pm.fit(n=20000)\n\n    # Full MCMC inference\n    idata = pm.sample(\n        draws=2000,\n        tune=1000,\n        chains=4,\n        target_accept=0.9,\n        random_seed=42,\n        idata_kwargs={'log_likelihood': True}  # For model comparison\n    )\n```\n\n**Key parameters:**\n- `draws=2000`: Number of samples per chain\n- `tune=1000`: Warmup samples (discarded)\n- `chains=4`: Run 4 chains for convergence checking\n- `target_accept=0.9`: Higher for difficult posteriors (0.95-0.99)\n- Include `log_likelihood=True` for model comparison\n\n### 5. Check Diagnostics\n\n**Use the diagnostic script:**\n\n```python\nfrom scripts.model_diagnostics import check_diagnostics\n\nresults = check_diagnostics(idata, var_names=['alpha', 'beta', 'sigma'])\n```\n\n**Check:**\n- **R-hat \u003C 1.01**: Chains have converged\n- **ESS > 400**: Sufficient effective samples\n- **No divergences**: NUTS sampled successfully\n- **Trace plots**: Chains should mix well (fuzzy caterpillar)\n\n**If issues arise:**\n- Divergences → Increase `target_accept=0.95`, use non-centered parameterization\n- Low ESS → Sample more draws, reparameterize to reduce correlation\n- High R-hat → Run longer, check for multimodality\n\n### 6. Posterior Predictive Check\n\n**Validate model fit:**\n\n```python\nwith model:\n    pm.sample_posterior_predictive(idata, extend_inferencedata=True, random_seed=42)\n\n# Visualize\naz.plot_ppc(idata)\n```\n\n**Check:**\n- Do posterior predictions capture observed data patterns?\n- Are systematic deviations evident (model misspecification)?\n- Consider alternative models if fit is poor\n\n### 7. Analyze Results\n\n```python\n# Summary statistics\nprint(az.summary(idata, var_names=['alpha', 'beta', 'sigma']))\n\n# Posterior distributions\naz.plot_posterior(idata, var_names=['alpha', 'beta', 'sigma'])\n\n# Coefficient estimates\naz.plot_forest(idata, var_names=['beta'], combined=True)\n```\n\n### 8. Make Predictions\n\n```python\nX_new = ...  # New predictor values\nX_new_scaled = (X_new - X_mean) \u002F X_std\n\nwith model:\n    pm.set_data({'X_scaled': X_new_scaled})\n    post_pred = pm.sample_posterior_predictive(\n        idata.posterior,\n        var_names=['y_obs'],\n        random_seed=42\n    )\n\n# Extract prediction intervals\ny_pred_mean = post_pred.posterior_predictive['y_obs'].mean(dim=['chain', 'draw'])\ny_pred_hdi = az.hdi(post_pred.posterior_predictive, var_names=['y_obs'])\n```\n\n## Common Model Patterns\n\n### Linear Regression\n\nFor continuous outcomes with linear relationships:\n\n```python\nwith pm.Model() as linear_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n    sigma = pm.HalfNormal('sigma', sigma=1)\n\n    mu = alpha + pm.math.dot(X, beta)\n    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)\n```\n\n**Use template:** `assets\u002Flinear_regression_template.py`\n\n### Logistic Regression\n\nFor binary outcomes:\n\n```python\nwith pm.Model() as logistic_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n\n    logit_p = alpha + pm.math.dot(X, beta)\n    y = pm.Bernoulli('y', logit_p=logit_p, observed=y_obs)\n```\n\n### Hierarchical Models\n\nFor grouped data (use non-centered parameterization):\n\n```python\nwith pm.Model(coords={'groups': group_names}) as hierarchical_model:\n    # Hyperpriors\n    mu_alpha = pm.Normal('mu_alpha', mu=0, sigma=10)\n    sigma_alpha = pm.HalfNormal('sigma_alpha', sigma=1)\n\n    # Group-level (non-centered)\n    alpha_offset = pm.Normal('alpha_offset', mu=0, sigma=1, dims='groups')\n    alpha = pm.Deterministic('alpha', mu_alpha + sigma_alpha * alpha_offset, dims='groups')\n\n    # Observation-level\n    mu = alpha[group_idx]\n    sigma = pm.HalfNormal('sigma', sigma=1)\n    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)\n```\n\n**Use template:** `assets\u002Fhierarchical_model_template.py`\n\n**Critical:** Always use non-centered parameterization for hierarchical models to avoid divergences.\n\n### Poisson Regression\n\nFor count data:\n\n```python\nwith pm.Model() as poisson_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n\n    log_lambda = alpha + pm.math.dot(X, beta)\n    y = pm.Poisson('y', mu=pm.math.exp(log_lambda), observed=y_obs)\n```\n\nFor overdispersed counts, use `NegativeBinomial` instead.\n\n### Time Series\n\nFor autoregressive processes:\n\n```python\nwith pm.Model() as ar_model:\n    sigma = pm.HalfNormal('sigma', sigma=1)\n    rho = pm.Normal('rho', mu=0, sigma=0.5, shape=ar_order)\n    init_dist = pm.Normal.dist(mu=0, sigma=sigma)\n\n    y = pm.AR('y', rho=rho, sigma=sigma, init_dist=init_dist, observed=y_obs)\n```\n\n## Model Comparison\n\n### Comparing Models\n\nUse LOO or WAIC for model comparison:\n\n```python\nfrom scripts.model_comparison import compare_models, check_loo_reliability\n\n# Fit models with log_likelihood\nmodels = {\n    'Model1': idata1,\n    'Model2': idata2,\n    'Model3': idata3\n}\n\n# Compare using LOO\ncomparison = compare_models(models, ic='loo')\n\n# Check reliability\ncheck_loo_reliability(models)\n```\n\n**Interpretation:**\n- **Δloo \u003C 2**: Models are similar, choose simpler model\n- **2 \u003C Δloo \u003C 4**: Weak evidence for better model\n- **4 \u003C Δloo \u003C 10**: Moderate evidence\n- **Δloo > 10**: Strong evidence for better model\n\n**Check Pareto-k values:**\n- k \u003C 0.7: LOO reliable\n- k > 0.7: Consider WAIC or k-fold CV\n\n### Model Averaging\n\nWhen models are similar, average predictions:\n\n```python\nfrom scripts.model_comparison import model_averaging\n\naveraged_pred, weights = model_averaging(models, var_name='y_obs')\n```\n\n## Distribution Selection Guide\n\n### For Priors\n\n**Scale parameters** (σ, τ):\n- `pm.HalfNormal('sigma', sigma=1)` - Default choice\n- `pm.Exponential('sigma', lam=1)` - Alternative\n- `pm.Gamma('sigma', alpha=2, beta=1)` - More informative\n\n**Unbounded parameters**:\n- `pm.Normal('theta', mu=0, sigma=1)` - For standardized data\n- `pm.StudentT('theta', nu=3, mu=0, sigma=1)` - Robust to outliers\n\n**Positive parameters**:\n- `pm.LogNormal('theta', mu=0, sigma=1)`\n- `pm.Gamma('theta', alpha=2, beta=1)`\n\n**Probabilities**:\n- `pm.Beta('p', alpha=2, beta=2)` - Weakly informative\n- `pm.Uniform('p', lower=0, upper=1)` - Non-informative (use sparingly)\n\n**Correlation matrices**:\n- `pm.LKJCorr('corr', n=n_vars, eta=2)` - eta=1 uniform, eta>1 prefers identity\n\n### For Likelihoods\n\n**Continuous outcomes**:\n- `pm.Normal('y', mu=mu, sigma=sigma)` - Default for continuous data\n- `pm.StudentT('y', nu=nu, mu=mu, sigma=sigma)` - Robust to outliers\n\n**Count data**:\n- `pm.Poisson('y', mu=lambda)` - Equidispersed counts\n- `pm.NegativeBinomial('y', mu=mu, alpha=alpha)` - Overdispersed counts\n- `pm.ZeroInflatedPoisson('y', psi=psi, mu=mu)` - Excess zeros\n\n**Binary outcomes**:\n- `pm.Bernoulli('y', p=p)` or `pm.Bernoulli('y', logit_p=logit_p)`\n\n**Categorical outcomes**:\n- `pm.Categorical('y', p=probs)`\n\n**See:** `references\u002Fdistributions.md` for comprehensive distribution reference\n\n## Sampling and Inference\n\n### MCMC with NUTS\n\nDefault and recommended for most models:\n\n```python\nidata = pm.sample(\n    draws=2000,\n    tune=1000,\n    chains=4,\n    target_accept=0.9,\n    random_seed=42\n)\n```\n\n**Adjust when needed:**\n- Divergences → `target_accept=0.95` or higher\n- Slow sampling → Use ADVI for initialization\n- Discrete parameters → Use `pm.Metropolis()` for discrete vars\n\n### Variational Inference\n\nFast approximation for exploration or initialization:\n\n```python\nwith model:\n    approx = pm.fit(n=20000, method='advi')\n\n    # Use for initialization\n    start = approx.sample(return_inferencedata=False)[0]\n    idata = pm.sample(start=start)\n```\n\n**Trade-offs:**\n- Much faster than MCMC\n- Approximate (may underestimate uncertainty)\n- Good for large models or quick exploration\n\n**See:** `references\u002Fsampling_inference.md` for detailed sampling guide\n\n## Diagnostic Scripts\n\n### Comprehensive Diagnostics\n\n```python\nfrom scripts.model_diagnostics import create_diagnostic_report\n\ncreate_diagnostic_report(\n    idata,\n    var_names=['alpha', 'beta', 'sigma'],\n    output_dir='diagnostics\u002F'\n)\n```\n\nCreates:\n- Trace plots\n- Rank plots (mixing check)\n- Autocorrelation plots\n- Energy plots\n- ESS evolution\n- Summary statistics CSV\n\n### Quick Diagnostic Check\n\n```python\nfrom scripts.model_diagnostics import check_diagnostics\n\nresults = check_diagnostics(idata)\n```\n\nChecks R-hat, ESS, divergences, and tree depth.\n\n## Common Issues and Solutions\n\n### Divergences\n\n**Symptom:** `idata.sample_stats.diverging.sum() > 0`\n\n**Solutions:**\n1. Increase `target_accept=0.95` or `0.99`\n2. Use non-centered parameterization (hierarchical models)\n3. Add stronger priors to constrain parameters\n4. Check for model misspecification\n\n### Low Effective Sample Size\n\n**Symptom:** `ESS \u003C 400`\n\n**Solutions:**\n1. Sample more draws: `draws=5000`\n2. Reparameterize to reduce posterior correlation\n3. Use QR decomposition for regression with correlated predictors\n\n### High R-hat\n\n**Symptom:** `R-hat > 1.01`\n\n**Solutions:**\n1. Run longer chains: `tune=2000, draws=5000`\n2. Check for multimodality\n3. Improve initialization with ADVI\n\n### Slow Sampling\n\n**Solutions:**\n1. Use ADVI initialization\n2. Reduce model complexity\n3. Increase parallelization: `cores=8, chains=8`\n4. Use variational inference if appropriate\n\n## Best Practices\n\n### Model Building\n\n1. **Always standardize predictors** for better sampling\n2. **Use weakly informative priors** (not flat)\n3. **Use named dimensions** (`dims`) for clarity\n4. **Non-centered parameterization** for hierarchical models\n5. **Check prior predictive** before fitting\n\n### Sampling\n\n1. **Run multiple chains** (at least 4) for convergence\n2. **Use `target_accept=0.9`** as baseline (higher if needed)\n3. **Include `log_likelihood=True`** for model comparison\n4. **Set random seed** for reproducibility\n\n### Validation\n\n1. **Check diagnostics** before interpretation (R-hat, ESS, divergences)\n2. **Posterior predictive check** for model validation\n3. **Compare multiple models** when appropriate\n4. **Report uncertainty** (HDI intervals, not just point estimates)\n\n### Workflow\n\n1. Start simple, add complexity gradually\n2. Prior predictive check → Fit → Diagnostics → Posterior predictive check\n3. Iterate on model specification based on checks\n4. Document assumptions and prior choices\n\n## Resources\n\nThis skill includes:\n\n### References (`references\u002F`)\n\n- **`distributions.md`**: Comprehensive catalog of PyMC distributions organized by category (continuous, discrete, multivariate, mixture, time series). Use when selecting priors or likelihoods.\n\n- **`sampling_inference.md`**: Detailed guide to sampling algorithms (NUTS, Metropolis, SMC), variational inference (ADVI, SVGD), and handling sampling issues. Use when encountering convergence problems or choosing inference methods.\n\n- **`workflows.md`**: Complete workflow examples and code patterns for common model types, data preparation, prior selection, and model validation. Use as a cookbook for standard Bayesian analyses.\n\n### Scripts (`scripts\u002F`)\n\n- **`model_diagnostics.py`**: Automated diagnostic checking and report generation. Functions: `check_diagnostics()` for quick checks, `create_diagnostic_report()` for comprehensive analysis with plots.\n\n- **`model_comparison.py`**: Model comparison utilities using LOO\u002FWAIC. Functions: `compare_models()`, `check_loo_reliability()`, `model_averaging()`.\n\n### Templates (`assets\u002F`)\n\n- **`linear_regression_template.py`**: Complete template for Bayesian linear regression with full workflow (data prep, prior checks, fitting, diagnostics, predictions).\n\n- **`hierarchical_model_template.py`**: Complete template for hierarchical\u002Fmultilevel models with non-centered parameterization and group-level analysis.\n\n## Quick Reference\n\n### Model Building\n```python\nwith pm.Model(coords={'var': names}) as model:\n    # Priors\n    param = pm.Normal('param', mu=0, sigma=1, dims='var')\n    # Likelihood\n    y = pm.Normal('y', mu=..., sigma=..., observed=data)\n```\n\n### Sampling\n```python\nidata = pm.sample(draws=2000, tune=1000, chains=4, target_accept=0.9)\n```\n\n### Diagnostics\n```python\nfrom scripts.model_diagnostics import check_diagnostics\ncheck_diagnostics(idata)\n```\n\n### Model Comparison\n```python\nfrom scripts.model_comparison import compare_models\ncompare_models({'m1': idata1, 'm2': idata2}, ic='loo')\n```\n\n### Predictions\n```python\nwith model:\n    pm.set_data({'X': X_new})\n    pred = pm.sample_posterior_predictive(idata.posterior)\n```\n\n## Additional Notes\n\n- PyMC integrates with ArviZ for visualization and diagnostics\n- Use `pm.model_to_graphviz(model)` to visualize model structure\n- Save results with `idata.to_netcdf('results.nc')`\n- Load with `az.from_netcdf('results.nc')`\n- For very large models, consider minibatch ADVI or data subsampling\n","# PyMC Bayesian Modeling\n\n## Overview\n\nPyMC is a Python library for Bayesian modeling and probabilistic programming. Build, fit, validate, and compare Bayesian models using PyMC's modern API (version 5.x+), including hierarchical models, MCMC sampling (NUTS), variational inference, and model comparison (LOO, WAIC).\n\n## When to Use This Skill\n\nThis skill should be used when:\n- Building Bayesian models (linear\u002Flogistic regression, hierarchical models, time series, etc.)\n- Performing MCMC sampling or variational inference\n- Conducting prior\u002Fposterior predictive checks\n- Diagnosing sampling issues (divergences, convergence, ESS)\n- Comparing multiple models using information criteria (LOO, WAIC)\n- Implementing uncertainty quantification through Bayesian methods\n- Working with hierarchical\u002Fmultilevel data structures\n- Handling missing data or measurement error in a principled way\n\n## Standard Bayesian Workflow\n\nFollow this workflow for building and validating Bayesian models:\n\n### 1. Data Preparation\n\n```python\nimport pymc as pm\nimport arviz as az\nimport numpy as np\n\n# Load and prepare data\nX = ...  # Predictors\ny = ...  # Outcomes\n\n# Standardize predictors for better sampling\nX_mean = X.mean(axis=0)\nX_std = X.std(axis=0)\nX_scaled = (X - X_mean) \u002F X_std\n```\n\n**Key practices:**\n- Standardize continuous predictors (improves sampling efficiency)\n- Center outcomes when possible\n- Handle missing data explicitly (treat as parameters)\n- Use named dimensions with `coords` for clarity\n\n### 2. Model Building\n\n```python\ncoords = {\n    'predictors': ['var1', 'var2', 'var3'],\n    'obs_id': np.arange(len(y))\n}\n\nwith pm.Model(coords=coords) as model:\n    # Priors\n    alpha = pm.Normal('alpha', mu=0, sigma=1)\n    beta = pm.Normal('beta', mu=0, sigma=1, dims='predictors')\n    sigma = pm.HalfNormal('sigma', sigma=1)\n\n    # Linear predictor\n    mu = alpha + pm.math.dot(X_scaled, beta)\n\n    # Likelihood\n    y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y, dims='obs_id')\n```\n\n**Key practices:**\n- Use weakly informative priors (not flat priors)\n- Use `HalfNormal` or `Exponential` for scale parameters\n- Use named dimensions (`dims`) instead of `shape` when possible\n- Use `pm.Data()` for values that will be updated for predictions\n\n### 3. Prior Predictive Check\n\n**Always validate priors before fitting:**\n\n```python\nwith model:\n    prior_pred = pm.sample_prior_predictive(samples=1000, random_seed=42)\n\n# Visualize\naz.plot_ppc(prior_pred, group='prior')\n```\n\n**Check:**\n- Do prior predictions span reasonable values?\n- Are extreme values plausible given domain knowledge?\n- If priors generate implausible data, adjust and re-check\n\n### 4. Fit Model\n\n```python\nwith model:\n    # Optional: Quick exploration with ADVI\n    # approx = pm.fit(n=20000)\n\n    # Full MCMC inference\n    idata = pm.sample(\n        draws=2000,\n        tune=1000,\n        chains=4,\n        target_accept=0.9,\n        random_seed=42,\n        idata_kwargs={'log_likelihood': True}  # For model comparison\n    )\n```\n\n**Key parameters:**\n- `draws=2000`: Number of samples per chain\n- `tune=1000`: Warmup samples (discarded)\n- `chains=4`: Run 4 chains for convergence checking\n- `target_accept=0.9`: Higher for difficult posteriors (0.95-0.99)\n- Include `log_likelihood=True` for model comparison\n\n### 5. Check Diagnostics\n\n**Use the diagnostic script:**\n\n```python\nfrom scripts.model_diagnostics import check_diagnostics\n\nresults = check_diagnostics(idata, var_names=['alpha', 'beta', 'sigma'])\n```\n\n**Check:**\n- **R-hat \u003C 1.01**: Chains have converged\n- **ESS > 400**: Sufficient effective samples\n- **No divergences**: NUTS sampled successfully\n- **Trace plots**: Chains should mix well (fuzzy caterpillar)\n\n**If issues arise:**\n- Divergences → Increase `target_accept=0.95`, use non-centered parameterization\n- Low ESS → Sample more draws, reparameterize to reduce correlation\n- High R-hat → Run longer, check for multimodality\n\n### 6. Posterior Predictive Check\n\n**Validate model fit:**\n\n```python\nwith model:\n    pm.sample_posterior_predictive(idata, extend_inferencedata=True, random_seed=42)\n\n# Visualize\naz.plot_ppc(idata)\n```\n\n**Check:**\n- Do posterior predictions capture observed data patterns?\n- Are systematic deviations evident (model misspecification)?\n- Consider alternative models if fit is poor\n\n### 7. Analyze Results\n\n```python\n# Summary statistics\nprint(az.summary(idata, var_names=['alpha', 'beta', 'sigma']))\n\n# Posterior distributions\naz.plot_posterior(idata, var_names=['alpha', 'beta', 'sigma'])\n\n# Coefficient estimates\naz.plot_forest(idata, var_names=['beta'], combined=True)\n```\n\n### 8. Make Predictions\n\n```python\nX_new = ...  # New predictor values\nX_new_scaled = (X_new - X_mean) \u002F X_std\n\nwith model:\n    pm.set_data({'X_scaled': X_new_scaled})\n    post_pred = pm.sample_posterior_predictive(\n        idata.posterior,\n        var_names=['y_obs'],\n        random_seed=42\n    )\n\n# Extract prediction intervals\ny_pred_mean = post_pred.posterior_predictive['y_obs'].mean(dim=['chain', 'draw'])\ny_pred_hdi = az.hdi(post_pred.posterior_predictive, var_names=['y_obs'])\n```\n\n## Common Model Patterns\n\n### Linear Regression\n\nFor continuous outcomes with linear relationships:\n\n```python\nwith pm.Model() as linear_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n    sigma = pm.HalfNormal('sigma', sigma=1)\n\n    mu = alpha + pm.math.dot(X, beta)\n    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)\n```\n\n**Use template:** `assets\u002Flinear_regression_template.py`\n\n### Logistic Regression\n\nFor binary outcomes:\n\n```python\nwith pm.Model() as logistic_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n\n    logit_p = alpha + pm.math.dot(X, beta)\n    y = pm.Bernoulli('y', logit_p=logit_p, observed=y_obs)\n```\n\n### Hierarchical Models\n\nFor grouped data (use non-centered parameterization):\n\n```python\nwith pm.Model(coords={'groups': group_names}) as hierarchical_model:\n    # Hyperpriors\n    mu_alpha = pm.Normal('mu_alpha', mu=0, sigma=10)\n    sigma_alpha = pm.HalfNormal('sigma_alpha', sigma=1)\n\n    # Group-level (non-centered)\n    alpha_offset = pm.Normal('alpha_offset', mu=0, sigma=1, dims='groups')\n    alpha = pm.Deterministic('alpha', mu_alpha + sigma_alpha * alpha_offset, dims='groups')\n\n    # Observation-level\n    mu = alpha[group_idx]\n    sigma = pm.HalfNormal('sigma', sigma=1)\n    y = pm.Normal('y', mu=mu, sigma=sigma, observed=y_obs)\n```\n\n**Use template:** `assets\u002Fhierarchical_model_template.py`\n\n**Critical:** Always use non-centered parameterization for hierarchical models to avoid divergences.\n\n### Poisson Regression\n\nFor count data:\n\n```python\nwith pm.Model() as poisson_model:\n    alpha = pm.Normal('alpha', mu=0, sigma=10)\n    beta = pm.Normal('beta', mu=0, sigma=10, shape=n_predictors)\n\n    log_lambda = alpha + pm.math.dot(X, beta)\n    y = pm.Poisson('y', mu=pm.math.exp(log_lambda), observed=y_obs)\n```\n\nFor overdispersed counts, use `NegativeBinomial` instead.\n\n### Time Series\n\nFor autoregressive processes:\n\n```python\nwith pm.Model() as ar_model:\n    sigma = pm.HalfNormal('sigma', sigma=1)\n    rho = pm.Normal('rho', mu=0, sigma=0.5, shape=ar_order)\n    init_dist = pm.Normal.dist(mu=0, sigma=sigma)\n\n    y = pm.AR('y', rho=rho, sigma=sigma, init_dist=init_dist, observed=y_obs)\n```\n\n## Model Comparison\n\n### Comparing Models\n\nUse LOO or WAIC for model comparison:\n\n```python\nfrom scripts.model_comparison import compare_models, check_loo_reliability\n\n# Fit models with log_likelihood\nmodels = {\n    'Model1': idata1,\n    'Model2': idata2,\n    'Model3': idata3\n}\n\n# Compare using LOO\ncomparison = compare_models(models, ic='loo')\n\n# Check reliability\ncheck_loo_reliability(models)\n```\n\n**Interpretation:**\n- **Δloo \u003C 2**: Models are similar, choose simpler model\n- **2 \u003C Δloo \u003C 4**: Weak evidence for better model\n- **4 \u003C Δloo \u003C 10**: Moderate evidence\n- **Δloo > 10**: Strong evidence for better model\n\n**Check Pareto-k values:**\n- k \u003C 0.7: LOO reliable\n- k > 0.7: Consider WAIC or k-fold CV\n\n### Model Averaging\n\nWhen models are similar, average predictions:\n\n```python\nfrom scripts.model_comparison import model_averaging\n\naveraged_pred, weights = model_averaging(models, var_name='y_obs')\n```\n\n## Distribution Selection Guide\n\n### For Priors\n\n**Scale parameters** (σ, τ):\n- `pm.HalfNormal('sigma', sigma=1)` - Default choice\n- `pm.Exponential('sigma', lam=1)` - Alternative\n- `pm.Gamma('sigma', alpha=2, beta=1)` - More informative\n\n**Unbounded parameters**:\n- `pm.Normal('theta', mu=0, sigma=1)` - For standardized data\n- `pm.StudentT('theta', nu=3, mu=0, sigma=1)` - Robust to outliers\n\n**Positive parameters**:\n- `pm.LogNormal('theta', mu=0, sigma=1)`\n- `pm.Gamma('theta', alpha=2, beta=1)`\n\n**Probabilities**:\n- `pm.Beta('p', alpha=2, beta=2)` - Weakly informative\n- `pm.Uniform('p', lower=0, upper=1)` - Non-informative (use sparingly)\n\n**Correlation matrices**:\n- `pm.LKJCorr('corr', n=n_vars, eta=2)` - eta=1 uniform, eta>1 prefers identity\n\n### For Likelihoods\n\n**Continuous outcomes**:\n- `pm.Normal('y', mu=mu, sigma=sigma)` - Default for continuous data\n- `pm.StudentT('y', nu=nu, mu=mu, sigma=sigma)` - Robust to outliers\n\n**Count data**:\n- `pm.Poisson('y', mu=lambda)` - Equidispersed counts\n- `pm.NegativeBinomial('y', mu=mu, alpha=alpha)` - Overdispersed counts\n- `pm.ZeroInflatedPoisson('y', psi=psi, mu=mu)` - Excess zeros\n\n**Binary outcomes**:\n- `pm.Bernoulli('y', p=p)` or `pm.Bernoulli('y', logit_p=logit_p)`\n\n**Categorical outcomes**:\n- `pm.Categorical('y', p=probs)`\n\n**See:** `references\u002Fdistributions.md` for comprehensive distribution reference\n\n## Sampling and Inference\n\n### MCMC with NUTS\n\nDefault and recommended for most models:\n\n```python\nidata = pm.sample(\n    draws=2000,\n    tune=1000,\n    chains=4,\n    target_accept=0.9,\n    random_seed=42\n)\n```\n\n**Adjust when needed:**\n- Divergences → `target_accept=0.95` or higher\n- Slow sampling → Use ADVI for initialization\n- Discrete parameters → Use `pm.Metropolis()` for discrete vars\n\n### Variational Inference\n\nFast approximation for exploration or initialization:\n\n```python\nwith model:\n    approx = pm.fit(n=20000, method='advi')\n\n    # Use for initialization\n    start = approx.sample(return_inferencedata=False)[0]\n    idata = pm.sample(start=start)\n```\n\n**Trade-offs:**\n- Much faster than MCMC\n- Approximate (may underestimate uncertainty)\n- Good for large models or quick exploration\n\n**See:** `references\u002Fsampling_inference.md` for detailed sampling guide\n\n## Diagnostic Scripts\n\n### Comprehensive Diagnostics\n\n```python\nfrom scripts.model_diagnostics import create_diagnostic_report\n\ncreate_diagnostic_report(\n    idata,\n    var_names=['alpha', 'beta', 'sigma'],\n    output_dir='diagnostics\u002F'\n)\n```\n\nCreates:\n- Trace plots\n- Rank plots (mixing check)\n- Autocorrelation plots\n- Energy plots\n- ESS evolution\n- Summary statistics CSV\n\n### Quick Diagnostic Check\n\n```python\nfrom scripts.model_diagnostics import check_diagnostics\n\nresults = check_diagnostics(idata)\n```\n\nChecks R-hat, ESS, divergences, and tree depth.\n\n## Common Issues and Solutions\n\n### Divergences\n\n**Symptom:** `idata.sample_stats.diverging.sum() > 0`\n\n**Solutions:**\n1. Increase `target_accept=0.95` or `0.99`\n2. Use non-centered parameterization (hierarchical models)\n3. Add stronger priors to constrain parameters\n4. Check for model misspecification\n\n### Low Effective Sample Size\n\n**Symptom:** `ESS \u003C 400`\n\n**Solutions:**\n1. Sample more draws: `draws=5000`\n2. Reparameterize to reduce posterior correlation\n3. Use QR decomposition for regression with correlated predictors\n\n### High R-hat\n\n**Symptom:** `R-hat > 1.01`\n\n**Solutions:**\n1. Run longer chains: `tune=2000, draws=5000`\n2. Check for multimodality\n3. Improve initialization with ADVI\n\n### Slow Sampling\n\n**Solutions:**\n1. Use ADVI initialization\n2. Reduce model complexity\n3. Increase parallelization: `cores=8, chains=8`\n4. Use variational inference if appropriate\n\n## Best Practices\n\n### Model Building\n\n1. **Always standardize predictors** for better sampling\n2. **Use weakly informative priors** (not flat)\n3. **Use named dimensions** (`dims`) for clarity\n4. **Non-centered parameterization** for hierarchical models\n5. **Check prior predictive** before fitting\n\n### Sampling\n\n1. **Run multiple chains** (at least 4) for convergence\n2. **Use `target_accept=0.9`** as baseline (higher if needed)\n3. **Include `log_likelihood=True`** for model comparison\n4. **Set random seed** for reproducibility\n\n### Validation\n\n1. **Check diagnostics** before interpretation (R-hat, ESS, divergences)\n2. **Posterior predictive check** for model validation\n3. **Compare multiple models** when appropriate\n4. **Report uncertainty** (HDI intervals, not just point estimates)\n\n### Workflow\n\n1. Start simple, add complexity gradually\n2. Prior predictive check → Fit → Diagnostics → Posterior predictive check\n3. Iterate on model specification based on checks\n4. Document assumptions and prior choices\n\n## Resources\n\nThis skill includes:\n\n### References (`references\u002F`)\n\n- **`distributions.md`**: Comprehensive catalog of PyMC distributions organized by category (continuous, discrete, multivariate, mixture, time series). Use when selecting priors or likelihoods.\n\n- **`sampling_inference.md`**: Detailed guide to sampling algorithms (NUTS, Metropolis, SMC), variational inference (ADVI, SVGD), and handling sampling issues. Use when encountering convergence problems or choosing inference methods.\n\n- **`workflows.md`**: Complete workflow examples and code patterns for common model types, data preparation, prior selection, and model validation. Use as a cookbook for standard Bayesian analyses.\n\n### Scripts (`scripts\u002F`)\n\n- **`model_diagnostics.py`**: Automated diagnostic checking and report generation. Functions: `check_diagnostics()` for quick checks, `create_diagnostic_report()` for comprehensive analysis with plots.\n\n- **`model_comparison.py`**: Model comparison utilities using LOO\u002FWAIC. Functions: `compare_models()`, `check_loo_reliability()`, `model_averaging()`.\n\n### Templates (`assets\u002F`)\n\n- **`linear_regression_template.py`**: Complete template for Bayesian linear regression with full workflow (data prep, prior checks, fitting, diagnostics, predictions).\n\n- **`hierarchical_model_template.py`**: Complete template for hierarchical\u002Fmultilevel models with non-centered parameterization and group-level analysis.\n\n## Quick Reference\n\n### Model Building\n```python\nwith pm.Model(coords={'var': names}) as model:\n    # Priors\n    param = pm.Normal('param', mu=0, sigma=1, dims='var')\n    # Likelihood\n    y = pm.Normal('y', mu=..., sigma=..., observed=data)\n```\n\n### Sampling\n```python\nidata = pm.sample(draws=2000, tune=1000, chains=4, target_accept=0.9)\n```\n\n### Diagnostics\n```python\nfrom scripts.model_diagnostics import check_diagnostics\ncheck_diagnostics(idata)\n```\n\n### Model Comparison\n```python\nfrom scripts.model_comparison import compare_models\ncompare_models({'m1': idata1, 'm2': idata2}, ic='loo')\n```\n\n### Predictions\n```python\nwith model:\n    pm.set_data({'X': X_new})\n    pred = pm.sample_posterior_predictive(idata.posterior)\n```\n\n## Additional Notes\n\n- PyMC integrates with ArviZ for visualization and diagnostics\n- Use `pm.model_to_graphviz(model)` to visualize model structure\n- Save results with `idata.to_netcdf('results.nc')`\n- Load with `az.from_netcdf('results.nc')`\n- For very large models, consider minibatch ADVI or data subsampling\n",{"name":5,"description":7},"main","34fa6acf794b9a6310d298cbb8fb8c9420a5edcf","cli-tool\u002Fcomponents\u002Fskills\u002Fscientific\u002Fpymc",2,"https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates","https:\u002F\u002Fgithub.com\u002Ftransaurus","Automated works of Page Turner","2026-03-28 10:12:59",{"source":8,"skillName":5,"commands":43,"prompt":53,"zipAvailable":54},[44,47,50],{"tool":45,"command":46},"npx","npx skills 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Key insight: Zapier optimizes for simplicity and integrations (7000+ apps), Make optimizes for power","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fworkflow-automation\u002Fzapier-make-patterns\u002FSKILL.md",[95],{"categoryNo":96,"categoryName":97,"taxonomyType":98,"taxonomyTypeDesc":99},"01JBSEEDCAT000000000000023","Skill与智能体开发","SCENARIO","使用场景",[101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,28,119],"zapier make patterns","patterns","make","zapier","writing","workflow","automation","code","integrations","business","use","power","complexity","platform","apps","how to","No-Code","which","spawner",{"skillNo":121,"name":122,"nameZh":6,"description":123,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":124,"categories":125,"keywords":126,"publishedAt":29},"01M6JMR8YK6SWPCTG1MCQTMN9S","yeet","Use only when the user explicitly asks to stage, commit, push, and open a GitHub pull request in one flow using the GitHub CLI (`gh`).","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fworkflow-automation\u002Fyeet\u002FSKILL.md",[],[122,127,128,129,130,131,132,133,134,135,136,137,111,138,139,140],"cli","github","commit","pull request","openai","pull","push","flow","open","gh","user","only","stage","github cli",{"skillNo":142,"name":143,"nameZh":6,"description":144,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":145,"categories":146,"keywords":148,"publishedAt":29},"01M6JMQNFE64ZASQK90HARRXW8","workflow-automation","Workflow automation is the infrastructure that makes AI agents reliable. Without durable execution, a network hiccup during a 10-step payment flow means lost money and angry customers. With it, workflows resume exactly where they left off.  This skill covers the platforms (n8n, Temporal, Inngest) and patterns (sequential, parallel, orchestrator-worker) that turn brittle scripts into production-grade automation.  Key insight: The platforms make different tradeoffs. n8n optimizes for accessibility","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fworkflow-automation\u002Fworkflow-automation\u002FSKILL.md",[147],{"categoryNo":96,"categoryName":97,"taxonomyType":98,"taxonomyTypeDesc":99},[106,107,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,27,28,119],"Workflow Automation","ai agents","accessibility","workflows","apache","execution","inngest","parallel","orchestrator","resume","n8n","payment","worker","scripts","temporal",{"skillNo":165,"name":166,"nameZh":6,"description":167,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":168,"categories":169,"keywords":170,"publishedAt":29},"01M6JMQDYJ0XETERPHGDR3EVGN","trigger-dev","Trigger.dev expert for background jobs, AI workflows, and reliable async execution with excellent developer experience and TypeScript-first design. Use when: trigger.dev, trigger dev, background task, ai background job, long running task.","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fworkflow-automation\u002Ftrigger-dev\u002FSKILL.md",[],[171,172,173,174,175,152,176,153,154,177,111,178,179,180,181,182,183,27,28,119],"dev","trigger dev","trigger","design","typescript","async execution","first","async","developer experience","expert","task","background jobs","long",{"skillNo":185,"name":186,"nameZh":6,"description":187,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":188,"categories":189,"keywords":193,"publishedAt":29},"01M6JMQ5J6KK3GSX9FJCWWK6G6","planning-with-files","Transforms workflow to use Manus-style persistent markdown files for planning, progress tracking, and knowledge storage. 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Use when building new workflows, designing workflow structure, choosing workflow patterns, planning workflow architecture, or asking about webhook processing, HTTP API integration, database operations, AI agent workflows, or scheduled tasks.","https:\u002F\u002Fgithub.com\u002Ftransaurus\u002Fstaging-davila7-claude-code-templates\u002Fblob\u002Fmain\u002Fcli-tool\u002Fcomponents\u002Fskills\u002Fworkflow-automation\u002Fn8n\u002Fn8n-workflow-patterns\u002FSKILL.md",[216,219],{"categoryNo":217,"categoryName":218,"taxonomyType":98,"taxonomyTypeDesc":99},"01JBSEEDCAT000000000000010","后端与接口",{"categoryNo":96,"categoryName":97,"taxonomyType":98,"taxonomyTypeDesc":99},[106,221,159,222,102,223,224,152,225,199,195,226,227,228,229,230,231,232,233,234],"n8n workflow patterns","workflow patterns","n8n workflow","ai agent","architecture","database","api","planning workflow","structure","database operations","webhook","processing","http api","scheduled"]