Under the hood, Recast fits a statistical model to your data. The model is a Bayesian hierarchical time series model and the fitting happens via a Hamiltonian Monte Carlo (HMC) algorithm in Stan.

The documentation below covers the core parts of the statistical model as they are specified mathematically. Since there’s a lot of very dense information here, we’ll start by laying out the model's assumptions in prose before getting into the formal specification.

To start, let’s lay out a diagram of how we believe marketing performance actually works. This causal diagram is designed to help us think from first principles about 1) what are the true causal relationships we are trying to understand and 2) how we can build a statistical model that estimates the relationships of interest.

The causal diagram looks like this:

It’s complicated! Marketing is a complicated domain and there are lots of factors that impact each other in non-obvious and non-measurable ways. Our job as modelers is to build a statistical that can estimate the most important relationships in this causal diagram in a way that’s robust and unbiased.

The Recast statistical model is a fully Bayesian generative model that attempts to capture all of these relationships in code. The core model includes thousands of lines of Stan code which we won’t walk through, and instead, we’ll outline the core components and how the pieces fit together.

Core assumptions:

• The dependent variable is impacted by
  • Paid media performance.
  • Baseline non-media sales or the “intercept”.
  • “Spikes”: exogenous shocks, usually from holidays, promotions, or new product launches.
  • “Contextual variables” which have their impacts through the intercept and paid media variables.
• Paid media performance has the following characteristics
  • Marketing performance can change over time, but the changes must be smooth according to a Gaussian process
  • The effect of marketing is shifted in time according to a negative binomial distribution
  • There are diminishing marginal returns to increased investment in Paid media. I.e., marketing channels “get saturated”
  • Marketing performance may be impacted by seasonal trends (i.e., channels might get more performance in the summer vs. the winter).
• The intercept has the following characteristics
  • The intercept changes over time, but the changes must be smooth according to a Gaussian process
  • The intercept may be impacted by seasonal trends (i.e., it may be higher in the summer vs. the winter).
• Spikes have the following characteristics
  • Spikes may have pull-forward and pull-backward effects (i.e., they distort sales that were going to happen before or after the spike date)
  • Spikes may interact with marketing performance (you can spend more money in certain marketing channels when a promotion is happening)

Now, let’s get into the math!

Preliminaries

Time-varying coefficients

Several things change over time in Recast’s model:

• The intercept