Counterfactual

Local Explanation of changing a datapoint in a minimal way such that it achieves a different predefined output. Often a really good Explanation which explains what is most important locally. Often Model-Agnostic.

If X had not occurred, Y would not have occurred

Unlike prototypes , counterfactuals do not have to be actual instances from the training data, but can be a new combination of feature values

• human-friendly

  • Contrastive → to the current instance via a “what-if” scenario
  • Selectivity → small number of feature changes

• Rashomon Effect

  • there are multiple different counterfactuals that all change the prediction equally well

• solve by

  • showing all counterfactuals
  • selecting only the best ones (by some measure)
    • produce the predefined prediction as closely as possible
    • as similar as possible to the original instance regarding feature values
      • Manhattan Distance
      • Gower Distance
    • change as few features as possible
    • generate multiple diverse explanations
    • likely feature values (see correlations etc.)

How to generate counterfactuals

Minimize the Loss Function $L\left(x,x^{\prime },y^{\prime },\lambda \right)=\lambda \cdot {\left(\hat{f}\left(x^{\prime }\right)-y^{\prime }\right)}^2+d\left(x,x^{\prime }\right).$

• original instance $x$
• current counterfactual $x^{\prime }$
• predefined output $y^{\prime }$
• regularisation $\lambda$

Distance between original instance and counterfactual should be minimized $d\left(x,x^{\prime }\right)={\sum }_{j=1}^p\frac{|x_j-x_j^{\prime }|}{MA{D}_j}.$ Stop when constraint is reached $\left|\hat{f}\left(x^{\prime }\right)-y^{\prime }\right|\le ϵ.$ So in total $\arg {\min }_{x^{\prime }}{\max }_{\lambda }L\left(x,x^{\prime },y^{\prime },\lambda \right).$

1. Select instance $x$, desired outcome $y^{\prime }$, tolerance $ϵ$ and a low initial value for $\lambda$
2. Sample random instance as initial counterfactual $x^{\prime }$
3. Optimize loss
4. While constraint is not fullfilled
   1. Increase $\lambda$ (more attention on the constraint part)
   2. Optimize loss
   3. Return counterfactual with minimal loss
5. Repeat 2-4 to generate multiple counterfactuals

Pros

• clear interpretation
• two results
  • counterfactual
  • what was changed
• Model-Agnostic and no access to the data is needed (privacy)
• works for systems without ML
• easy to implement

Cons

• Instability via the Rashomon Effect

Software

• https://github.com/SeldonIO/alibi