informative-priors-for-mmm

# Guide: Calculating Informative Priors for MMM This guide describes how to calculate tighter, more informative priors for PyMC-Marketing MMM based on data characteristics. **Important:** The default wide priors (`intercept: Normal(mu=0, sigma=2)`) often lead to better-calibrated posteriors. Only use informative priors when you have strong domain knowledge or specific requirements. Overly tight priors can lead to overconfident (underdispersed) posteriors. ## When to Use Informative Priors - You have domain knowledge about expected ROAS ranges, decay rates, or effect sizes - You have results from previous studies or models to inform your beliefs - Business constraints require certain parameter ranges - You're doing sensitivity analysis to understand prior impact ## Step 1: Calculate Scaled Statistics PyMC-Marketing uses MaxAbsScaler for target and channels (divides by max absolute value). ```python # Calculate how your data looks in scaled space y_scaled_max = 1.0 # By definition (MaxAbsScaler) y_scaled_mean = y.mean() / y.max() # Typically 0.6-0.8 y_scaled_min = y.min() / y.max() # Typically 0.4-0.6 print(f"Scaled y statistics:") print(f" min: {y_scaled_min:.3f}") print(f" mean: {y_scaled_mean:.3f}") print(f" max: {y_scaled_max:.3f}") ``` ## Step 2: Reason About Intercept Prior The intercept represents predicted y when all channels = 0 (no marketing spend). ```python # Intercept should be LESS than scaled mean (marketing contributes positively!) # Reasonable range in scaled space: 0.3 to 0.7 # If you believe ~20% of sales come from marketing: intercept_mu = y_scaled_mean * 0.8 # Sigma controls uncertainty - smaller = more confident # 0.2 = tight (use if you have strong beliefs) # 0.5 = moderate # 1.0+ = wide (closer to default) intercept_sigma = 0.2 print(f"Intercept prior: Normal(mu={intercept_mu:.2f}, sigma={intercept_sigma})") ``` ## Step 3: Reason About Channel Effect Priors `saturation_beta` represents the effect size per channel in scaled space. ```python # If 3 channels share ~20% of y, each contributes ~0.07 on average n_channels = len(channel_columns) expected_total_channel_contribution = 1 - intercept_mu # What's left after baseline expected_per_channel = expected_total_channel_contribution / n_channels # Prior sigma: allow 2-3x expected value for exploration beta_sigma = expected_per_channel * 2.5 print(f"Channel beta prior: HalfNormal(sigma={beta_sigma:.2f})") print(f" Expected per-channel contribution: ~{expected_per_channel:.2f}") ``` ## Step 4: Set model_config ```python from pymc_extras.prior import Prior model_config = { "intercept": Prior("Normal", mu=intercept_mu, sigma=intercept_sigma), "saturation_beta": Prior("HalfNormal", sigma=beta_sigma), # adstock_alpha: Beta(2,3) favors values 0.3-0.6, good default "adstock_alpha": Prior("Beta", alpha=2, beta=3), } mmm = MMM( date_column=date_column, channel_columns=channel_columns, adstock=GeometricAdstock(l_max=l_max), saturation=LogisticSaturation(), model_config=model_config, # ... other parameters ) ``` ## Step 5: Validate with Prior Predictive Checks After setting informative priors, always validate: ```python mmm.build_model(X=X, y=y) mmm.sample_prior_predictive(X=X, samples=1000, random_seed=None # Set for reproducibility) # Check prior predictive coverage from mmm_lib import check_prior_predictive_coverage coverage = check_prior_predictive_coverage(mmm, y, hdi_prob=0.94) # With informative priors, expect: # - HDI should be ~2-5x observed range (tighter than defaults) # - Some negative predictions OK, but <10% of samples # - Coverage should still be high (>80%) ``` ## Adjusting Based on Prior Predictive | Observation | Action | |-------------|--------| | HDI too wide (>10x observed range) | Decrease sigma values | | HDI too narrow (<2x observed range) | Increase sigma values | | Too many negative predictions (>10%) | Increase intercept mu or use HalfNormal | | Coverage too low (<50%) | Loosen priors (increase sigmas) | ## Example: Full Workflow ```python import pandas as pd import numpy as np from pymc_marketing.mmm.multidimensional import MMM from pymc_marketing.mmm import GeometricAdstock, LogisticSaturation from pymc_extras.prior import Prior # Load data df = pd.read_parquet('cleaned_data.parquet') y = df[target_column] channel_columns = ['tv', 'digital', 'radio'] # Step 1: Calculate scaled statistics y_scaled_mean = y.mean() / y.max() print(f"Scaled y mean: {y_scaled_mean:.3f}") # Step 2: Set intercept (assume 25% from marketing) intercept_mu = y_scaled_mean * 0.75 intercept_sigma = 0.3 # Step 3: Set channel effects n_channels = len(channel_columns) expected_per_channel = (1 - intercept_mu) / n_channels beta_sigma = expected_per_channel * 2 # Step 4: Create model config model_config = { "intercept": Prior("Normal", mu=intercept_mu, sigma=intercept_sigma), "saturation_beta": Prior("HalfNormal", sigma=beta_sigma), "adstock_alpha": Prior("Beta", alpha=2, beta=3), } print(f"Model config:") print(f" intercept: Normal(mu={intercept_mu:.2f}, sigma={intercept_sigma})") print(f" saturation_beta: HalfNormal(sigma={beta_sigma:.2f})") print(f" adstock_alpha: Beta(alpha=2, beta=3)") # Create MMM with informative priors mmm = MMM( date_column='date', channel_columns=channel_columns, adstock=GeometricAdstock(l_max=8), saturation=LogisticSaturation(), model_config=model_config, ) # Step 5: Validate X = df[['date'] + channel_columns] mmm.build_model(X=X, y=y) mmm.sample_prior_predictive(X=X, samples=1000, random_seed=None # Set for reproducibility) # Check and adjust if needed... ``` ## Warning: Posterior Calibration Informative priors that are too tight can cause **underdispersed posteriors** (overconfident uncertainty estimates). Signs of this: - 94% HDI coverage << 94% (e.g., 50-60%) - Posterior intervals that don't contain true values If you observe this, **loosen your priors** by increasing sigma values, or revert to defaults. ## Additional Prior Types ### Beta Distribution for Bounded Parameters For parameters that must be between 0 and 1 (like adstock decay): ```python # Beta(2, 3) - mode around 0.33, favors lower values # Beta(2, 2) - symmetric around 0.5 # Beta(3, 2) - mode around 0.67, favors higher values model_config = { "adstock_alpha": Prior("Beta", alpha=2, beta=3), } ``` ### HalfNormal for Positive Parameters For strictly positive parameters: ```python model_config = { "saturation_beta": Prior("HalfNormal", sigma=0.5), "sigma": Prior("HalfNormal", sigma=1.0), } ``` ### LogNormal for Positive Parameters with Heavy Tails When you expect occasional large values: ```python model_config = { "saturation_lam": Prior("LogNormal", mu=0, sigma=1), } ``` ## Domain Knowledge Integration ### From Historical Data If you have previous MMM results: ```python # Use posterior mean from previous model as prior mean # Use posterior std * 2 as prior std (to allow for model differences) previous_intercept_mean = 0.65 previous_intercept_std = 0.1 model_config = { "intercept": Prior("Normal", mu=previous_intercept_mean, sigma=previous_intercept_std * 2), } ``` ### From Industry Benchmarks If you have industry benchmarks for ROAS or effect sizes: ```python # Example: TV typically has ROAS of 2-5x # In scaled space, this might translate to beta of 0.05-0.15 model_config = { "saturation_beta": Prior("TruncatedNormal", mu=0.1, sigma=0.05, lower=0), } ``` ### From Business Constraints If business logic requires certain constraints: ```python # Example: Marketing cannot contribute more than 50% of sales # Set intercept prior to be at least 0.5 model_config = { "intercept": Prior("TruncatedNormal", mu=0.7, sigma=0.1, lower=0.5), } ``` ## When to Use This Skill - Setting up initial priors for a new MMM - Adjusting priors based on prior predictive checks - Incorporating domain knowledge into the model - Debugging overly wide or narrow posterior distributions - Sensitivity analysis to understand prior impact on results