Carryover and saturation

Media enters the model through two transforms applied in a fixed order: carryover first, then diminishing returns. The order is a modelling claim, not a convention. Saturating each week in isolation and then accumulating would understate the concavity a large flight actually runs into, because the flight’s own weeks would never be summed before the curve bent.

carried[g,t,m] = sum_l w[l,m] * spend_scaled[g,t-l,m] / sum_l w[l,m]
delta[g,t,m]   = hill(carried[g,t,m]) - hill(0)

Weights are divided by their own sum. Without that normalization, raising the decay parameter inflates the whole transformed series, the coefficient shrinks to compensate, and the two become a ridge that NUTS explores badly. Worse for this engine specifically: an ROI prior would stop meaning the same thing at different decay rates, and the entire elicitation story in chapter 03 would be built on a moving unit.

geometric          w[l] = alpha ** l
delayed_geometric  w[l] = alpha ** ((l - theta) ** 2)
weibull_cdf        w[l] = exp(-((l / scale) ** shape))

Geometric decay is monotone from the first period, which cannot express a brand response that builds for a few weeks before fading. delayed_geometric peaks at lag theta; the Weibull survival form holds the response up before dropping it when shape > 1 and decays fast with a long tail when shape < 1.

source · engine/mmm/transforms.py

Half-saturation ec is expressed in scaled media units, so ec = 1means the channel is at 50% saturation at its median historical spend. That is what makes the prior in chapter 01 interpretable without knowing the client’s budget. With slope = 1this reduces to the Michaelis–Menten form.

hill(u) = (u + eps) ** s / ((u + eps) ** s + ec ** s)

Subtracting hill(0) matters more than it looks. The epsilon lifts the curve off the floor by around 1e-12, so without the subtraction a channel with zero spend would contribute a tiny positive amount in every geo and period — and those amounts, summed over a five-year panel, are what stop a decomposition from reconciling.