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Explaining black box decisions by Shapley cohort refinement

23 March 2021

Masayoshi Mase

Masayoshi Mase
Research & Development Group, Hitachi, Ltd.

As AI prediction models continue to make increasingly accurate predictions, there is growing demand to apply them to mission critical tasks such as credit screening, crime prediction, fire risk assessment, and emergency medicine demand prediction. For such applications, recent complex prediction models are often criticized as black boxes as it is difficult to understand why the prediction model produces a given output. Explainable AI (XAI) has been intensively studied to resolve this dilemma, and ensure transparency. Post-hoc explanation methods such as LIME [1] and SHAP [2] has been proposed for explaining which input variables are important. The Shapley value [3] from cooperative game theory is gaining popularity for the explanation. It has a mathematically reasonable definition for globally consistent local explanation, and widely used, especially for tabular data. However, conventional Shapley based methods are potentially evaluating unlikely or even logically impossible synthetic data points where the machine learning prediction might be unreliable. Our cohort Shapley uses observed data points only. It realizes reliable explanations that matches human intuition.

Shapley values

Shapley value [3] is used in cooperative game theory to define a fair allocation of rewards to a team of players that has cooperated to produce a value. Lloyd Shapley introduced it in his 1953 paper. The Shapley value is the unique method that satisfies the following four properties:

  1. Efficiency: contributions of all players should sum up to the total value of the team.
  2. Symmetry: if participation of a player gains the same value as another player in for all combinations of other players coalitions, the contribution of the two players should be same.
  3. Dummy: if participation of a player does not gain for all combinations of another player coalitions, the contribution of the player should be zero.
  4. Additivity: if values of two games are additive, the contributions of a player in these games should be additive.

To understand black box predictions, we can see that each input variable is each player, and the predicted value is the produced value of the game. The problem then is how do we define the partial participation of input variables for the prediction.

Baseline Shapley

Conventional baseline Shapley [4] (Figure 1) defines the partial participation of input variables by changing the input value and predicting on these synthetic data points. For example, when explaining the target subject (x1,x2,x3 ), it uses a baseline input (z1,z2,z3) as all absent, then the presence of the 1st input variable is defined as predicted values when the 1st variable is changed to the value of the resulting target subject (x1,z2,z3), and so on. Then we apply the Shapley value to the prediction on the synthetic data points. The process is potentially creating unlikely or even logically impossible combinations where the prediction model is likely to generate unreliable output.

Figure1

Figure 1: Illustration of baseline Shapley usage. Baseline Shapley uses input data and model to generate synthetic prediction by changing input value to apply Shapley value.

Cohort Shapley

Our cohort Shapley (Figure 2) uses only observed data points and their predicted values. It operates by including and excluding data subjects. Here, all absence, or baseline cohort consists of all data points, then the presence of the 1st variable is defined as a prediction average of data points where the value of the 1st variable is similar to the 1st variable of the target subject x_1, and so on. We then apply the Shapley value to the cohort prediction averages. The process does not create synthetic data points.

Figure1

Figure 2: Illustration of cohort Shapley usage. Cohort Shapley uses input data and their predictions to calculate cohort prediction averages by refining cohort with similarity.

Example

Here is an example on a popular Boston housing data set. Figure 1 shows cohort Shapley (CS) and baseline Shapley (BS) values for this target subject. As in the figure, baseline and cohort Shapley values are very different.

Figure3

Figure 3: Baseline and cohort Shapley value for 205th subject of Boston housing data set. The baseline point is sample average in baseline Shapley.

In baseline Shapley ‘CRIM’, for example, was the most important variable, while in cohort Shapley it is one of the least important variables. We think that the explanation can be found in the way the baseline Shapley uses the synthetic data point at the upper orange cross in the left plot of Figure 4. The predicted value at the synthetic point is much smaller than that of the target subject. This leads to the impact of ‘CRIM’ being very high. Data like those synthetic points were not present in the training set and so the value represents an extrapolation where we do not expect a good prediction. We believe that an unreliable prediction there gives the extreme baseline Shapley value that we see for ‘CRIM’.

Figure4

Figure 4: Two scatterplots of the Boston housing data. It marks the target (blue), baseline points (red), and synthetic points (orange) in baseline Shapley, depicts the cohort boundaries and shows predicted value in gray scale.

Discussion

Baseline and cohort Shapley values have different characteristics. We should carefully consider about when and how to use these explanation methods. Especially, when a data set has dependence among input variables, we think that the cohort Shapley provides a more reliable and intuitive explanation. Also, cohort Shapley works with observed data points and their predicted values even if the target black box model itself is not available. This property looks favorable for auditing the model.

For more details, we encourage you to read our paper "Explaining black box decisions by Shapley cohort refinement."

Acknowledgements

This work was conducted in collaboration with Prof. Art B. Owen and Benjamin Seiler, Stanford University.


References

[1]
] M.T. Ribeiro, S. Singh and C. Guestrin, C. (2016). Why should I trust you?: Explaining the predictions of any classifier. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 1135-1144.
[2]
S.M. Lundberg and S.-I. Lee (2017). A unified approach to interpreting model predictions. Proceedings of the 31st International Conference on Neural Information Processing Systems, in Advances in Neural Information Processing Systems 30 (NIPS 2017), 4768-4777.
[3]
Shapley, L. S. (1953). A value for n-person games. In Harold William Kuhn and Albert William Tucker, editors, Contributions to the Theory of Games (AM-28), vol. II, 307-317, Princeton University Press.
[4]
M. Sundararajan and A. Najmi (2020). The many Shapley values for model explanation. Proceeding of the 37th International Conference on Machine Learning, in Proceedings for Machine Learning Research 119: 9269-9278.