Argument definitions in MMM.allocate_budget_to_maximize_response

[scottlyden]

From the documentation:

budget[float]
The total budget to be allocated.

time_granularity[str]
The granularity of the time periods (e.g., ‘daily’, ‘weekly’, ‘monthly’).

num_days[int]
The number of days over which the budget is to be allocated.

These definitions plus the discussion in the example notebook (https://www.pymc-marketing.io/en/stable/notebooks/mmm/mmm_budget_allocation_example.html) leave me completely confused.

The labelling of the graphs in the notebook suggests that the total budget is the amount allocated on each "day". (See the plots labelled Comparison of Optimized and Initial Budget Responses.)

Is budget the budget for the whole analysis period or for each period within the campaign? Does num_days really mean a number of days or a number of periods, where period is expressed in units of time_granularity? In the graphs, the labels say that the budget is an amount that's allocated each day even though granularity is specified as weekly!

To put it concretely, let's say I want to optimize a campaign over 8 week-long periods with a total budget of 8 million dollars for the whole campaign using a time_granularity of weekly. Is my budget 8 or 1 or 1/7 million? Granularity is presumably weekly, but is num_days 8 or 56?

A related question concerns the discussion around the second plot titled "Comparison of Optimized and Initial Budget Responses" in the example notebook. The discussion claims:

By using less than half of the budget with an optimal distribution, we could achieve the same outcomes as our initial setup, which distributed the budget evenly across all channels.

Please explain how that graph shows this. The outcomes look very different to me. Indeed, the 'optimized' outcome looks inferior in terms of total response. Please tell me what I'm missing. Is it that the original orange line exceeds the blue line by less than three (the savings from spending less on the campaign)?

PS

Since my initial posting, I have since read the notebook https://github.com/pymc-labs/pymc-marketing/blob/main/docs/source/notebooks/mmm/mmm_case_study.ipynb . That notebook adds yet another possible interpretation of the budget variables, ie, that they are vector-valued per channel per period quantities.