Marketing Mix Model v2 — Adstock + Diminishing Returns with PyMC

What Changed from v1

Version 1 modelled only saturation (diminishing returns via the Hill function). Version 2 adds geometric adstock before saturation, creating the standard two-stage media transformation pipeline used in production MMMs:

raw spend → adstock(λ) → Hill(K, S) → β·saturated → Σ + α → sales

Why Adstock Before Saturation?

Advertising effects don't vanish the moment you stop spending. A TV ad seen on Monday still influences purchase decisions on Tuesday and Wednesday — this is the carryover or adstock effect. By applying adstock before the Hill function, we model a realistic pipeline:

1. Adstock accumulates and decays spend over time → "effective exposure"
2. Hill saturation maps effective exposure to a bounded sales response

The ordering matters. Applying saturation first would compress spend into [0, 1] before accumulation, losing the important distinction between "one big day" and "sustained moderate spend over a week." The adstock-then-saturate ordering is the standard in Meta Robyn, Google Meridian, and the academic MMM literature (Jin et al., 2017).

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Model Specification

$$\text{sales} = \alpha + \sum_{c} \beta_c \cdot \text{hill}\bigl(\text{adstock}(x_c;; \lambda_c);; K_c, S_c\bigr) + \varepsilon$$

Geometric Adstock:

$$\text{adstock}_t = x_t + \lambda_c \cdot \text{adstock}_{t-1}$$

where $\lambda_c \in [0, 1)$ is the decay (retention) rate. The half-life of the carryover effect is $t_{1/2} = -\ln 2 / \ln \lambda$.

After adstock, the series is re-normalised to [0, 1] before entering the Hill function.

Hill Saturation Function:

$$\text{hill}(x;; K, S) = \frac{x^S}{K^S + x^S}$$

Parameters

Parameter	Description
$\lambda_c$ (decay rate)	Carryover retention per day for channel $c$. λ=0 means no carryover; λ=0.7 means ~2-day half-life
$K_c$ (half-saturation)	The effective-exposure level at which channel $c$ reaches 50% of its maximum effect
$S_c$ (Hill exponent)	Controls curvature — $S \approx 1$ gives Michaelis–Menten; $S > 1$ creates an S-shape
$\beta_c$ (coefficient)	Maximum possible sales lift (in z-scored units) from channel $c$ at full saturation
$\alpha$ (intercept)	Baseline sales level when all channels are at zero spend
$\sigma$ (noise)	Observation noise standard deviation

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Prior Choices & Justification

Parameter	Prior	Rationale
$\lambda$	Beta(2, 5)	Mode ≈ 0.17, mean ≈ 0.29. Most advertising carryover decays within a few days. This prior centres on fast decay while allowing moderate carryover (λ ≈ 0.5–0.7 for TV) if the data supports it. Chosen over Beta(1,1)/Uniform because very high λ values (>0.9) imply month-long half-lives, which are implausible for direct sales response.
$K$	Beta(2, 2)	Same as v1. Symmetric prior on [0, 1], mildly informative.
$S$	Gamma(3, 1)	Same as v1. Centres the Hill exponent around 2–3.
$\beta_c$	HalfNormal(1)	Same as v1. Enforces positivity and mild regularisation.
$\alpha$	Normal(0, 0.5)	Weakly informative in z-space.
$\sigma$	HalfNormal(0.5)	Weakly informative.

Note on Identifiability

Adding adstock introduces potential identifiability tension between λ (how much carry-over) and K (where saturation kicks in). A channel with high λ and low K can produce similar fits to one with low λ and high K. The informative priors on both parameters help regularise this, and the diagnostic checks (r̂, ESS, divergences) should be examined carefully. If identifiability issues arise, consider:

1. Fixing λ for one channel as a reference
2. Using stronger priors informed by experimental data (e.g., geo-lift tests)
3. Reparameterising λ via logit-normal for better sampling geometry

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Sampling Configuration

Setting	v1	v2	Rationale for change
Tune	1,500	2,000	Adstock scan creates longer dependency chains
Draws	2,000	2,000	Unchanged
Chains	4	4	Unchanged
target_accept	0.95	0.97	Higher to handle scan-induced posterior geometry

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Outputs

The script generates five plots in ./results/:

File	Description
adstock_decay.png	Impulse response curves and posterior distributions of λ per channel
prior_vs_posterior_curves_v2.png	Prior vs posterior Hill saturation curves (same as v1 but with adstock-adjusted K values)
posterior_params_v2.png	KDE plots of all four parameter types: K, S, β, λ
posterior_predictive_v2.png	Observed sales vs model posterior predictive (90% interval)
channel_decomposition_v2.png	Stacked area chart showing each channel's contribution to sales over time

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What's Still Missing (v3 Roadmap)

• Trend & seasonality — linear/spline trend + Fourier terms or day-of-week effects
• Control variables — pricing, promotions, holidays, competitor activity
• Weibull adstock — more flexible than geometric; can model delayed peak effects
• Out-of-sample validation — time-based train/test split with MAPE and coverage metrics
• Budget optimiser — marginal ROI–based reallocation given the fitted model
• Experimental calibration — using geo-lift test results as informative priors on β