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Correcting the cost function formula for Counterfactual examples (Dandl et al.'s method) (S#9.3.1.2) #364

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Correcting the cost function formula for Counterfactual examples (Dandl et al.'s method) (S#9.3.1.2) #364

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Changed y' to Y' according to the paper
alishakiba Jan 13, 2023
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Merge pull request #1 from alishakiba/alishakiba-correct-dandl-cost-f…
alishakiba Jan 13, 2023
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8 changes: 4 additions & 4 deletions manuscript/06.1-example-based-counterfactual.Rmd
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Expand Up @@ -177,13 +177,13 @@ Let us now have a look on another approach overcoming these issues.

Dandl et al. suggest to simultaneously minimize a four-objective loss:

$$L(x,x',y',X^{obs})=\big(o_1(\hat{f}(x'),y'),o_2(x, x'),o_3(x,x'),o_4(x',X^{obs})\big) $$
$$L(x,x',y',X^{obs})=\big(o_1(\hat{f}(x'),Y'),o_2(x, x'),o_3(x,x'),o_4(x',X^{obs})\big) $$

Each of the four objectives $o_1$ to $o_4$ corresponds to one of the four criteria mentioned above.
where $Y'$ is a set of desired outcomes. Each of the four objectives $o_1$ to $o_4$ corresponds to one of the four criteria mentioned above.
The first objective $o_1$ reflects that the prediction of our counterfactual x' should be as close as possible to our desired prediction y'.
We therefore want to minimize the distance between $\hat{f}(x')$ and y', here calculated by the Manhattan metric ($L_1$ norm):
We therefore want to minimize the distance between $\hat{f}(x')$ and a set of desired outcomes Y', here calculated by the Manhattan metric ($L_1$ norm):

$$o_1(\hat{f}(x'),y')=\begin{cases}0&\text{if $\hat{f}(x')\in{}y'$}\\\inf\limits_{y'\in y'}|\hat{f}(x')-y'|&\text{else}\end{cases}$$
$$o_1(\hat{f}(x'),y')=\begin{cases}0&\text{if $\hat{f}(x')\in{}Y'$}\\\inf\limits_{y'\in Y'}|\hat{f}(x')-y'|&\text{else}\end{cases}$$

The second objective $o_2$ reflects that our counterfactual should be as similar as possible to our instance $x$.
It quantifies the distance between x' and x
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