Explanation: Core MMM Methodology

Version: 2.5.1

Marketing Mix Modelling (MMM) quantifies the impact of marketing activities on sales or other KPIs. AMMM provides a Bayesian framework for building flexible, interpretable MMM models.

Core Principles

Bayesian Approach: AMMM uses PyMC to treat model parameters as probability distributions, providing uncertainty quantification rather than point estimates. This yields credible intervals (HDIs) and enables incorporation of prior knowledge.

Flexibility: Customizable media transformations (adstock, saturation), control variables, and prior distributions allow models to reflect business reality.

Interpretability: Model outputs include channel-specific coefficients, response curves, ROI estimates, and contribution decomposition.

Actionability: Results directly inform budget optimization and scenario planning.

Bayesian Framework

Key Concepts

Parameters as Distributions: Model parameters (channel effectiveness, adstock rates, saturation points) are probability distributions reflecting uncertainty about their true values.

Prior Distributions: Express initial beliefs about parameters before observing data. Common choices:

• HalfNormal: For positive-only parameters (effectiveness coefficients, error terms)
• Beta: For parameters between 0-1 (adstock retention rates)
• Gamma: For positive parameters with specific shapes (saturation parameters)
• Normal: For unconstrained parameters

Likelihood Function: Quantifies how probable the observed data is given parameter values.

Posterior Distributions: Updated beliefs about parameters after observing data, obtained via Bayes’ Theorem:

P(parameter | Data) ∝ P(Data | parameter) × P(parameter)

MCMC Sampling: PyMC uses NUTS (No-U-Turn Sampler) to draw samples from posterior distributions, which are typically too complex for analytical solutions.

Benefits

• Credible intervals (HDIs) for all parameters
• Full posterior distributions for detailed inspection
• Direct probability statements about parameters
• Formal incorporation of prior knowledge

Core Model Equation

The typical AMMM model structure is a Bayesian regression with transformed media inputs and optional control variables:

y_t = α + Σ_m β_m · saturation(adstock(x_{m,t})) + Σ_c γ_c · z_{c,t} + ε_t

It’s also common to define a shorthand baseline term:

baseline_t := α + Σ_c γ_c · z_{c,t}
y_t = baseline_t + Σ_m β_m · saturation(adstock(x_{m,t})) + ε_t

Components

Target Variable (y_t): The outcome being modelled (sales, revenue, conversions).

Intercept (α): Fundamental base level.

Control Variables (z_c,t): Baseline drivers not attributable to paid media (and any engineered time features). Coefficients γ_c are estimated in the Bayesian model. Common examples:

• External factors (promotions, competitor activity, macro indicators)
• Prophet-derived time components when enabled (trend/seasonality/holidays)

Media Inputs (x_m,t): Raw marketing effort (spend, impressions, GRPs).

Media Transformations:

1. Adstock (Carry-over Effects): Models lagged advertising impact

   • Geometric adstock: adstocked_t = (1-θ) × input_t + θ × adstocked_{t-1}
   • Parameter θ (retention rate): 0-1, typically has Beta prior
   • ad_stock_max_lag: Maximum periods for carry-over calculation

2. Saturation (Diminishing Returns): Models non-linear response

   • Michaelis-Menten: Hyperbolic response curve
   • Logistic: S-shaped response curve
   • Parameters control shape and steepness, typically Gamma or HalfNormal priors

Channel Effectiveness (β_m): Scales transformed media input. Represents marginal impact on target. Typically HalfNormal prior (positive effect).

Error Term (ε_t): Random variation not explained by model. Typically Normal(0, σ_error) with HalfNormal prior on σ.

Prior Specification

Priors guide model estimation and reflect domain knowledge:

Channel Effectiveness (β_m):

• HalfNormal: For positive effects
• Inform based on past studies, domain expertise, or lift tests

Adstock (θ_m):

• Beta distribution (0-1 constraint)
• Digital channels: Lower values (shorter memory)
• Traditional media: Higher values (longer memory)

Saturation (λ_m):

• Gamma or HalfNormal
• Inform using lift test results when available
• Otherwise use weakly informative priors

Best Practices:

• Start with weakly informative priors
• Visualize priors before fitting
• Test sensitivity to prior choices
• Document reasoning

Model Fitting

MCMC Process:

1. Initialize parameter values
2. Run multiple independent chains (typically 4)
3. Tuning phase: Sampler adapts (typically 2000 iterations)
4. Sampling phase: Collect posterior samples (typically 2000+ draws)

Key Parameters:

• draws: Posterior samples per chain
• tune: Tuning iterations
• chains: Number of independent chains
• target_accept: Acceptance rate (0.8-0.99)

Convergence Diagnostics:

• R-hat: Should be ≈1.0 (< 1.01 ideal)

  • Compares within-chain vs between-chain variance
  • Values > 1.05 indicate non-convergence

• Effective Sample Size (ESS): Should be > 100-400

  • Accounts for autocorrelation
  • Low ESS indicates inefficient sampling

• Trace Plots: Should show “fuzzy caterpillar”

  • Horizontal band (stationarity)
  • Good mixing between chains
  • No trends or patterns

• Divergences: Should be zero or minimal

  • Indicate sampler instability
  • Fix by increasing target_accept

Model Outputs

Channel Coefficients (β_m):

• Magnitude indicates effectiveness
• HDI indicates uncertainty
• If HDI excludes zero, strong evidence of effect

Response Curves:

• Visualize diminishing returns
• Identify optimal spend levels
• Compare channel dynamics

ROI Metrics:

• Overall ROI: Total contribution / total spend
• Marginal ROI (mROI): Return on next dollar spent
• Use mROI for budget allocation decisions

Contribution Analysis:

• Decomposes target into components
• Shows baseline vs marketing impact
• Tracks contribution over time

Budget Optimization:

• Allocates budget to maximize returns
• Uses response curves and mROI
• Supports constraints on channel spend

Scenario Planning:

• Tests “what-if” scenarios
• Predicts outcomes under different budgets
• Provides uncertainty intervals

Prophet Integration

AMMM can use Prophet to generate deterministic time features (trend/seasonality/holidays) that are then included as control variables in the Bayesian MMM.

In the current pipeline this is a two-stage (“hybrid”) setup:

1. Fit a Prophet model to the target time series to obtain components like trend, yearly, weekly, holidays.
2. Add those components as columns in the training data and include them in extra_features_cols, so they enter the PyMC model via the control term Σ_c γ_c · z_{c,t}.

The Bayesian MMM (channel effects, adstock, saturation, control coefficients) is then estimated conditional on those Prophet-derived features.

Configuration

Configuration:

• yearly_seasonality: Annual patterns (all data frequencies)
• weekly_seasonality: Day-of-week patterns (for daily or sub-daily data)
• daily_seasonality: Hour-of-day patterns (for sub-daily/intra-day data)
• trend: Long-term trends
• include_holidays: Holiday effects

Data Frequency Guidance:

• Weekly data: Use yearly_seasonality=True, set weekly_seasonality=False and daily_seasonality=False
• Daily data: Use yearly_seasonality=True and weekly_seasonality=True, set daily_seasonality=False
• Intra-day data: Use all three seasonality parameters as needed

Prophet components are automatically added to the model as control variables when enabled.

Convergence Best Practices

If convergence fails:

1. Increase tune and draws
2. Increase target_accept (0.95-0.99)
3. Use more informative priors
4. Simplify model (fewer channels/features)
5. Check data quality

For large datasets:

• Use more chains and draws
• Increase computational resources
• Monitor memory usage

For complex models:

• Start simple, add complexity gradually
• Validate at each step
• Document changes

References

• PyMC Documentation
• ArviZ Diagnostics
• Prophet

See also:

• Configuration Guide
• Prior Calibration
• Lift Test Calibration