Explanation: Core MMM Methodology

This document explains the core concepts and techniques used in the Abacus MMM library.

(Placeholder: This section needs content explaining the Bayesian approach, the specific model structure (linear model with transformations), the concept and implementation of geometric adstock, and the concept and implementation of logistic/tanh saturation.)

Bayesian Approach

Abacus uses a Bayesian framework (via PyMC) for modelling. This means:

  • Parameters (like channel effectiveness, adstock decay) are treated as probability distributions, not single point estimates.

  • Prior beliefs about parameters can be incorporated using prior distributions (see Custom Priors (custom_priors) in the Configuration Guide).

  • Results are posterior distributions, representing updated beliefs after observing the data. This provides inherent uncertainty quantification (e.g., HDIs).

Core Model Structure

The underlying model is typically a linear regression model on the transformed media data:

\[y_{t} = \alpha + \sum_{m=1}^{M}\beta_{m}f(x_{m, t}) + \sum_{c=1}^{C}\gamma_{c}z_{c, t} + \varepsilon_{t}\]

Where:

  • y_t: Target variable at time t.

  • alpha: Intercept (baseline).

  • beta_m: Coefficient representing the effectiveness of channel m.

  • f(x_{m, t}): Transformed media variable for channel m at time t. This transformation usually involves applying both adstock and saturation.

  • gamma_c: Coefficient for control variable c.

  • z_{c, t}: Value of control variable c at time t.

  • epsilon_t: Error term, typically assumed to be Normally distributed.

Media Transformations

The function f() combines two key concepts:

1. Adstock (Carry-over Effect)

  • Concept: The effect of advertising lingers beyond the initial exposure period. Past spending continues to influence the target variable.

  • Implementation (Geometric Adstock): Abacus uses a geometric decay model. The adstocked value at time t is a weighted sum of current and past spend, where weights decay geometrically based on a decay rate parameter alpha. A higher alpha means effects decay faster (shorter memory). The adstock_max_lag parameter truncates this effect.

    # Conceptual formula (see core.transformers.geometric_adstock)
adstock[t] = x[t] + alpha * adstock[t-1]

  • Parameter: alpha (typically Beta distributed, 0 to 1).

2. Saturation (Diminishing Returns)

  • Concept: The effectiveness of advertising diminishes as spend increases. The first pound/dollar spent typically has more impact than the millionth.

  • Implementation (Logistic/Michaelis-Menten): Abacus applies a saturation function to the adstocked media values.

    • Logistic: Uses a sigmoid-like curve. Controlled by parameter lam. Higher lam means saturation occurs more quickly. (See core.transformers.logistic_saturation).

    • Michaelis-Menten: Uses a curve common in enzyme kinetics, also representing diminishing returns. Controlled by parameters beta (representing the maximum effect, similar to Vmax) and lam (controlling the shape/steepness, similar to Km). (See core.transformers.michaelis_menten_saturation).

  • Parameters: lam (typically Gamma distributed for Logistic), or beta and lam (for Michaelis-Menten).

The combination f(x) = saturation(adstock(x)) captures both the delayed and the diminishing effects of media spend.