Under the exchangeability and consistency assumptions, the conditional
expectation of any KPI potential outcome \(\overset \sim Y_{g,t}^{
\left(\left\{ x_{g,t,m}^{(\ast)} \right\}\right) }\) can be written in terms of
a conditional expectation that can be estimated by a regression model.
From the definitions described in Input
data, this can be written as:
$$
\begin{align*}
\overset \sim Y_{g,t} &= u_{g,t}^{(y)} \overset {\cdot \cdot} Y_{g,t} \\
&= u_{g,t}^{(y)}L_g^{(y)-1}(Y_g,t)
\end{align*}
$$
Meridian also makes use of the fact that the pre-modeling sales
transformation function \(L_g^{(y)}(\cdot)\) is linear and therefore can be
passed outside the conditional expectation operator. This ends up with the
following equality, where the result is a quantity that can be estimated from a
regression model, such as the Meridian model:
$$
\begin{align*}
E\left(\overset \sim Y_{g,t}^{(\left\{ x_g,t,m^{(\ast)} \right\})} \Big|
\bigl\{ z_{g,t,c} \bigr\} \right)
&= E\left( \overset \sim Y_{g,t} \Big|
\bigl\{x_{g,t,m}^{(\ast)}\bigr\}, \bigl\{z_{g,t,c}\bigr\} \right) \\
&= E\left( u_{g,t}^{(y)}L_g^{(y)-1}(Y_{g,t}) \Big|
\bigl\{ x_{g,t,m}^{(\ast)} \bigr\}, \bigl\{z_{g,t,c}\bigr\} \right) \\
&= u_{g,t}^{(y)}L_g^{(y)-1} E\left( Y_{g,t} \Big|
\bigl\{ x_{g,t,m}^{(\ast)} \bigr\}, \bigl\{z_{g,t,c}\bigr\} \right)
\end{align*}
$$
Based on this, regression can be used to estimate the incremental KPI
between any two counterfactual scenarios
\(\left\{ x_{g,t,m}^{(1)} \right\}\) and
\(\left\{ x_{g,t,m}^{(0)} \right\}\):
$$
\begin{align*}
\text{IncrementalSales} \left( \bigl\{ x_{g,t,m}^{(1)} \bigr\},
\bigl\{ x_{g,t,m}^{(0)} \bigr\} \right)
&= E\left( \sum\limits_{g,t}\left( \overset \sim Y_{g,t}^{
\left( \left\{ x_{g,t,m}^{(1)} \right\} \right)
} - \overset \sim Y_{g,t}^{
\left( \left\{ x_{g,t,m}^{(0)} \right\} \right)
} \right) \Bigg| \bigl\{ z_{g,t,c} \bigr\} \right) \\
&= \sum\limits_{g,t}u_{g,t}^{(y)}L_g^{(y)-1}
\left( E\left( Y_{g,t} \Big| \bigl\{ x^{(1)} \bigr\},
\bigl\{ z_{g,t,c} \bigr\} \right)\right) -
\sum\limits_{g,t}u_{g,t}^{(y)}L_g^{(y)-1}
\left( E\left( Y_{g,t} \Big| \bigl\{ x^{(0)} \bigr\},
\bigl\{ z_{g,t,c} \bigr\}
\right) \right)
\end{align*}
$$
Under the Meridian model specification:
$$
\begin{align*}
E\left( Y_{g,t} \Big|
\bigl\{ x_{g,t,m}^{(\ast)} \bigr\}, \bigl\{ z_{g,t,c} \bigr\} \right) =
\mu_t &+ \tau_g + \sum\limits_{c=1}^C \gamma_{g,c}z_{g,t,c} \\
&+ \sum\limits_{m=1}^M \beta_{g,m} \text{HillAdstock} \left(
\bigl\{ x_{g,t-s,m}^{(\ast)} \bigr\}^L_{s=0};\ \alpha_m, ec_m, \text{slope}_m
\right)
\end{align*}
$$
This quantity is a function of the model parameters, and therefore has a
posterior distribution which Meridian can sample using Markov Chain
Monte Carlo (MCMC). ROI, mROI, and response curves can all be calculated based
on the incremental KPI
definition, and each of
these quantities also has a posterior distribution.
