MCCE: Monte Carlo sampling of valid and realistic counterfactual explanations for tabular data

Abstract

We introduce MCCE: ${{{\underline{\varvec{M}}}}}$onte ${{{\underline{\varvec{C}}}}}$arlo sampling of valid and realistic ${{{\underline{\varvec{C}}}}}$ounterfactual ${{{\underline{\varvec{E}}}}}$xplanations for tabular data, a novel counterfactual explanation method that generates on-manifold, actionable and valid counterfactuals by modeling the joint distribution of the mutable features given the immutable features and the decision. Unlike other on-manifold methods that tend to rely on variational autoencoders and have strict prediction model and data requirements, MCCE handles any type of prediction model and categorical features with more than two levels. MCCE first models the joint distribution of the features and the decision with an autoregressive generative model where the conditionals are estimated using decision trees. Then, it samples a large set of observations from this model, and finally, it removes the samples that do not obey certain criteria. We compare MCCE with a range of state-of-the-art on-manifold counterfactual methods using four well-known data sets and show that MCCE outperforms these methods on all common performance metrics and speed. In particular, including the decision in the modeling process improves the efficiency of the method substantially.

Appendix A Further experiments

1.1 A.1 How is MCCE’s performance when the categorical features are not binarized?

Since none of the on-manifold methods investigated in the main paper handle categorical features with more than two levels, we had to restrict the models in the method comparisons to models with only two levels. The MCCE method can, however, handle models with categorical features with an arbitrary number of levels. To exemplify this, we here provide performance results for the MCCE applied to the Adult data set without binarizing the seven categorical features. The results are shown in Table 7.

Since the explanation is carried out on different data/model, the performance scores are not directly comparable. Note, however, that the costs are slightly higher for these counterfactuals, and that the computation time also is higher. Both of these effects are expected since the data are much richer—one of the categorical features (country) has, for instance, 41 different levels. Changes in the categorical features are then more likely. Training the decision trees is also more time-consuming as many more splits need to be considered.

Table 7 Experiment 5: Average and standard deviation (in parentheses) of performance metrics for counterfactuals generated by MCCE when the categorical features are not binarized

1.2 A.2 How is MCCE’s performance when the prediction model is non-gradient based?

In addition to the restriction to categorical features with two levels, our method comparison required a gradient-based predictive model to be applicable to all the alternative methods. Again, MCCE is not restricted to gradient-based predictive models. Thus, below, we use MCCE to explain a random forest model on the Adult data set, to showcase that it is directly applicable to non gradient-based predictive models. We use a random forest model with 200 trees.

The counterfactual method C-CHVAE is the only other on-manifold method in our list of benchmark methods that supposedly handles non-gradient-based models. Thus, it would have been interesting to compare the performance with this method. Unfortunately, a non-gradient-based version of this method is not currently implemented in CARLA.

The average performance metrics for MCCE are reported in Table 8. As in Sect. A.1, the results are not directly comparable. Note, however, that the cost is quite similar to those for the ANN model. The computation time, on the other hand, is quite a bit higher for the random forest model. This is a result of increased prediction time (used when computing validity) for the random forest implementation compared to the ANN model.

Table 8 Average and standard deviation (in parentheses) of performance metrics for counterfactuals generated by MCCE when the predictive model is a random forest

1.3 A.3 Parameter values for competing methods

For all four data sets we used the default parameter values in CARLA for all the competing methods, except for CLUE where we had to use other values for the FICO and German Credit Data sets. The parameter values used for the different methods are shown in Fig. 10. The values in parenthesis are the ones used for the FICO and German Credit Data sets.

Fig. 10

Parameters used for the competing methods

1.4 A.4 How do the metrics and computation times compare when we do not condition on the desired decision?

Table 9 Average and standard deviation (in parentheses) of performance metrics for counterfactuals generated with MCCE when we do not condition on the decision

One of the novel contributions of this paper is the modeling of the decision alongside the mutable features, to then condition on the desired decision (and the immutable features) when generating the data set of potential counterfactuals. The idea behind this notion is to bias the method toward generating samples that are more likely to yield the desired decision. In this section, we investigate whether including the decision on the modeling and then conditioning on the desired decision really ensures a higher proportion of valid samples. For the test observations in the Adult data set we generate counterfactuals without conditioning on the decision for various K and show the usual metrics and run times in Table 9, to be compared to Table 4. As suspected, this variation of MCCE is much less effective in generating valid counterfactuals ($N_\text {CE}$ in Table 9 is much smaller than in Table 4).

Furthermore, the fitting and generation times exhibit a slight decrease compared to the regular MCCE approach. Although one might have expected the training time to decrease due to fewer candidate features per split, the omission of the decision variable can result in larger trees as the available features convey less information. This was precisely the case for the given dataset, also leading to a slight increase in generation time.

On the other hand, the post-processing time is significantly reduced with this alternative approach, as it only generates around 25% unique and valid samples (compared to 86% with the normal MCCE), requiring fewer $L_0$ and $L_1$ calculations. In total, the computation time is smaller for the MCCE approach which does not condition on the desired decision.

However, it is essential to highlight that this alternative approach requires a significantly higher value of K in order to generate valid counterfactuals for all 1000 test observations. When not conditioning on the desired decision, a K value of 5, 000 is needed compared to only 50 when conditioning on the desired response. Consequently, this negates the initial speed-up of the former method, making the original MCCE a clearly superior alternative.

1.5 A.5 Quality of generated data

In Sect. 3.3 we showed the histograms for the generated data for four of the variables in the FICO data set. Here, we show the histograms for the rest of the variables. The histograms for the generated data are in white while the histograms for the real data are in dark grey. As we can see, the marginal distributions of the generated data match the original ones very well also for these features (Figs. 11, 12, 13, 14, 15).

Fig. 11

Histograms for four of the variables in the generated data set (white) with the histograms for the real data superimposed (dark grey). Where the histograms overlap, the blend of white and dark grey gives a light grey color

Fig. 12

Histograms for four of the variables in the generated data set (white) with the histograms for the real data superimposed (dark grey). Where the histograms overlap, the blend of white and dark grey gives a light grey color

Fig. 13

Histograms for four of the variables in the generated data set (white) with the histograms for the real data superimposed (dark grey). Where the histograms overlap, the blend of white and dark grey gives a light grey color

Fig. 14

Histograms for four of the variables in the generated data set (white) with the histograms for the real data superimposed (dark grey). Where the histograms overlap, the blend of white and dark grey gives a light grey color

Fig. 15

Histograms for three of the variables in the generated data set (white) with the histograms for the real data superimposed (dark grey). Where the histograms overlap, the blend of of white and dark grey gives a light grey color

1.6 A.6 Simulations mimicking real data experiments

In Sect. 3.4 we performed simulation experiments with a linear model when varying the dimension p (with no fixed features, i.e., $u=0$), $n_{\text {train}}$, $n_{\text {test}}$ and K. To showcase that these scalability results are relevant and valid in a broader context, we also ran simulations with quantities mimicking the four real data sets in Sect. 3.1. As seen from the table, the total computation times in the simulation (27, 50, 7, and 49 s) are similar in magnitude to those recorded in the real data experiments (25, 32, 6, and 34 s), despite the model, the feature dependence and most likely the tree depth, differing significantly. This indicates that the scalability in our basic simulation study in Sect. 3.4 generalizes roughly to real data settings as well.

Table 10 Computation times (in seconds) of the three steps of MCCE when mimicking $n_{\text {test}}$, $n_{\text {train}}$, and p/u/q, with $K=1000$ for the four real data experiments in Sect. 3.1