Modelling the Marketing Funnel

[TeemuSailynoja]

Context

Model the causal relationships of the demand generating (upper funnel) and demand capturing (lower funnel) channels in an MMM.
Often, the lower funnel spend is increased by demand generated in the upper funnel channels, and throttled by budget limits set for the lower funnel channels.

@lfiaschi requested a feature like this is #1840. This has been implemented in the past using the MuEffect class, but it required quite heavy modifiimplemecations to the plotting functions as wells as tot the MultidimensionalBudgetOptimizerWrapper.
With @cetagostini 's PR introducing the MaskedPrior class (#1945), this implementation gets simpler, but there are still a couple of hurdles.

Hurdles

1. The BudgetOptimizer doesn't expect observed RVs in the graph for the target variable, so the marketing spend on the demand capturing channels would need to be replaced with for example a deterministic computed from the posterior.

2. Often, the demand marketing spend on the demand generation channels includes a maximum budget per month or week, this would then require adding a censored likelihood into the model, which offers an additional challenge, as one needs a separate component to draw posterior predictive samples from.

3. Now we would like to update the deterministics for channel contributions, or create a set of new ones for the indirect contribution to sales, upper funnel contribution to lower funnel spend, and total channel contribution (direct + indirect). Also the total_media_contribution deterministic would need to get redefined.

[TeemuSailynoja]

Here is my implementation of a funnel using the MaskedPrior class from #1945:

class FunnelComponent:
    def __init__(
        self,
        lower_funnel_channels: list[str],
        saturation_upper_funnel,
        saturation_lower_funnel,
        likelihood_lower_funnel: Prior | None = None,
    ):
        self.lower_funnel_channels = lower_funnel_channels
        self.saturation_upper_funnel = saturation_upper_funnel
        self.saturation_lower_funnel = saturation_lower_funnel
        self.lower_funnel_likelihood = likelihood_lower_funnel

    def create_data(self, mmm):
        # Add lower funnel coordinate to the model
        mmm.model.add_coord("lower_funnel", self.lower_funnel_channels)

    def create_effect(self, mmm):
        # Create likelihood for lower funnel channels if not provided,
        # would be clearer to implement a default_likelihood for the class.
        if not self.lower_funnel_likelihood:
            self.lower_funnel_likelihood = Prior(
                "Normal",
                sigma=Prior("HalfNormal", sigma=1, dims="lower_funnel"),
                dims=(mmm.date_column, "lower_funnel"),
            )

        scaled_upper_spend = pt.expand_dims(
            mmm.model["channel_data"] / mmm.scalers._channel.values, -2
        )

        # Each upper funnel channel contributes to each lower funnel channel,
        # the MaskedPrior allows also creating separate masks per lower funnel channel.
        upper_to_lower = pm.Deterministic(
            "upper_to_lower_spend_contribution",
            self.saturation_upper_funnel.apply(scaled_upper_spend),
            dims=("date", "country", "lower_funnel", "channel"),
        )

        # Sum contributions from all upper funnel channels to get total spend
        # going into each lower funnel channel.
        # This is the mu for the lower funnel likelihood.
        lf_spend = self.lower_funnel_likelihood.create_likelihood_variable(
            "lower_funnel_spend",
            mu=upper_to_lower.sum(axis=-1),
            observed=(
                mmm.xarray_dataset._channel.sel(
                    channel=self.lower_funnel_channels
                )
                / mmm.scalers._channel
            ).values,
        )

        # Apply saturation to get contribution to sales from the lower funnel channels.
        lower_contribution = pm.Deterministic(
            "lower_funnel_contribution",
            self.saturation_lower_funnel.apply(lf_spend),
            dims=("date", "country", "lower_funnel"),
        )

        return lower_contribution.sum(axis=-1)

    def set_data(self, mmm, pymc_model, dataset_xarray):
        # No extra data to set
        pass

With a minimal MMM

mmm = MMM(
    target_column="sales",
    date_column="date",
    dims=("country",),
    channel_columns=upper_funnel + lower_funnel,
    adstock=NoAdstock(l_max=1),
    saturation=saturation_direct,
)
mmm.mu_effects.append(
    FunnelComponent(
        lower_funnel,
        saturation_upper_funnel,
        saturation_lower_funnel,
        Prior(
            "Normal",
            sigma=Prior("HalfNormal", sigma=1, dims="lower_funnel"),
            dims=("date", "country", "lower_funnel"),
        ),
    )
)
mmm.build_model(X, X["sales"])

where I mask away the lower funnel channels from saturation_direct and saturation_upper_funnel, it defines the following model graph

[timbo112711]

@TeemuSailynoja I really like this approach here. It's simpler than what we've built in the past!

For your second hurdle,

Often, the demand marketing spend on the demand generation channels includes a maximum budget per month or week, this would then require adding a censored likelihood into the model, which offers an additional challenge, as one needs a separate component to draw posterior predictive samples from.

If we were to roll this out without that functionality it should be enough, at least as a starting point. I wonder how often the users will have access to this data.