Under the exchangeability and consistency assumptions, the conditional
expectation of any potential outcome \\(\\overset \\sim Y_{g,t}\^{
\\left(\\left\\{ x_{g,t,i}\^{(\\ast)} \\right\\}\\right) }\\) can be written in terms of
a conditional expectation that can be estimated by a regression model, where
\\(x_{g,t,i}\^{(\\ast)}\\) represents the set of intervenable treatment variables:
media, organic media, and non-media treatments. For demonstration purposes, we
assume the paid and organic media channels here are impression based, although
the following also holds for reach and frequency based channels.

> [!NOTE]
> **Note:** The following equations rely heavily on causal inference notation, including potential outcomes ($\\overset \\sim Y$) and counterfactual scenarios ($x\^{(1)}, x\^{(0)}$). If you need help interpreting these symbols, see the [Causal inference \& budget optimization notation](https://developers.google.com/meridian/docs/basics/math-notation#causal-optimization-notation) guide.

From the definitions described in [Input
data](https://developers.google.com/meridian/docs/advanced-modeling/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,t}\^{\[Y\]-1}(Y_{g,t}) \\end{align\*} $$

Meridian also makes use of the fact that the pre-modeling KPI
transformation function \\(L_{g,t}\^{\[Y\]}(\\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,i}\^{(\\ast)} \\right\\})} \\Big\| \\bigl\\{ z_{g,t,i} \\bigr\\} \\right) \&= E\\left( \\overset \\sim Y_{g,t} \\Big\| \\bigl\\{x_{g,t,i}\^{(\\ast)}\\bigr\\}, \\bigl\\{z_{g,t,i}\\bigr\\} \\right) \\\\ \&= E\\left( u_{g,t}\^{\[Y\]}L_{g,t}\^{\[Y\]-1}(Y_{g,t}) \\Big\| \\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\}, \\bigl\\{z_{g,t,i}\\bigr\\} \\right) \\\\ \&= u_{g,t}\^{\[Y\]}L_{g,t}\^{\[Y\]-1} E\\left( Y_{g,t} \\Big\| \\bigl\\{ x_{g,t,i}\^{(\\ast)} \\bigr\\}, \\bigl\\{z_{g,t,i}\\bigr\\} \\right) \\end{align\*} $$

Based on this, regression can be used to estimate the incremental outcome
between any two counterfactual scenarios \\(\\left\\{ x_{g,t,i}\^{(1)} \\right\\}\\)
and \\(\\left\\{ x_{g,t,i}\^{(0)} \\right\\}\\):
$$ \\begin{align\*} \\text{IncrementalOutcome} \\left( \\bigl\\{ x_{g,t,i}\^{(1)} \\bigr\\}, \\bigl\\{ x_{g,t,i}\^{(0)} \\bigr\\} \\right) \&= E\\left( \\sum\\limits_{g,t}\\left( \\overset \\sim Y_{g,t}\^{ \\left( \\left\\{ x_{g,t,i}\^{(1)} \\right\\} \\right) } - \\overset \\sim Y_{g,t}\^{ \\left( \\left\\{ x_{g,t,i}\^{(0)} \\right\\} \\right) } \\right) \\Bigg\| \\bigl\\{ z_{g,t,i} \\bigr\\} \\right) \\\\ \&= \\sum\\limits_{g,t}u_{g,t}\^{\[Y\]}L_g\^{\[Y\]-1} \\left( E\\left( Y_{g,t} \\Big\| \\bigl\\{ x_{g,t,i}\^{(1)} \\bigr\\}, \\bigl\\{ z_{g,t,i} \\bigr\\} \\right)\\right) - \\sum\\limits_{g,t}u_{g,t}\^{\[Y\]}L_{g,t}\^{\[Y\]-1} \\left( E\\left( Y_{g,t} \\Big\| \\bigl\\{ x_{g,t,i}\^{(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,i}\^{(\\ast)} \\bigr\\}, \\bigl\\{ z_{g,t,i} \\bigr\\} \\right) = \\mu_t \&+ \\tau_g + \\sum\\limits_{i=1}\^{N_C} \\gamma\^{\[C\]}_{g,i}z_{g,t,i} \\\\ \&+ \\sum\\limits_{i=1}\^{N_N} \\gamma\^{\[N\]}_{g,i}x\^{\[N\] (\\ast)}_{g,t,i} \\\\ \&+ \\sum\\limits_{i=1}\^{N_M} \\beta\^{\[M\]}_{g,i} \\text{HillAdstock} \\left( \\bigl\\{ x\^{\[M\] (\\ast)}_{g,t-s,i} \\bigr\\}\^L_{s=0};\\ \\alpha\^{\[M\]}_i, ec\^{\[M\]}_i, \\text{slope}\^{\[M\]}_i \\right) \\\\ \&+ \\sum\\limits_{i=1}\^{N_{OM}} \\beta\^{\[OM\]}_{g,i} \\text{HillAdstock} \\left( \\bigl\\{ x\^{\[OM\] (\\ast)}_{g,t-s,i} \\bigr\\}\^L_{s=0};\\ \\alpha\^{\[OM\]}_i, ec\^{\[OM\]}_i, \\text{slope}\^{\[OM\]}_i \\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 outcome
definition](https://developers.google.com/meridian/docs/post-modeling/roi-mroi-response-curves#incremental_outcome),
and each of these quantities also has a posterior distribution.