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Model interpretability has become an important problem in machine learning (ML) due to the increased effect algorithmic decisions have on humans. Counterfactual explanations can help users understand not only why ML models make certain decisions, but also how these decisions can be changed. We frame the problem of finding counterfactual explanations as an optimization task and extend previous work that could only be applied to differentiable models. In order to accommodate non-differentiable models such as tree ensembles, we use probabilistic model approximations in the optimization framework. We introduce an approximation technique that is effective for finding counterfactual explanations for predictions of the original model and show that our counterfactual examples are significantly closer to the original instances than those produced by other methods specifically designed for tree ensembles.

The nephrotoxic mycotoxin ochratoxin A (OTA), a common contaminant of cereals, has been implicated in the etiology of endemic nephropathy. It was also frequently found in low concentrations in blood of healthy populations in countries where endemic nephropathy is not known. However, data on regional and seasonal differences in the frequency and concentration of OTA in human blood are scarce. In June, September and December 1997, and March 1998, about 50 human blood samples were collected randomly from blood donors for blood banks in the Coatian cities of Osijek, Rijeka, Split, VaraZdin and Zagreb. OTA was measured in the total of 983 samples using an HPLC technique with fluorescent detection. The daily intake of OTA was estimated from the mean concentration found in different cities and at different times of year. Samples containing OTA above the detection limit (0.2 ng/ml of plasma) were found in populations from all Croatian cities at all collecting periods. The highest frequency (59%) of samples containing OTA above the detection limit and the highest mean concentration (0.39 ng/ml) were found in June. Both the frequency and the mean concentration were lowest in all samples in December (36% and 0.19 ng OTA/ml, respectively). Osijek was the city with the highest frequency of OTA-positive samples (81%) and the highest mean OTA concentration (0.56 ng/ml). The total mean concentration of OTA in blood of healthy population in Croatia is lower (0.30 ng/ ml) than the mean concentration in European countries as a whole (0.90 ng/ml). The estimated daily intake, calculated from the mean concentration in all blood samples, is 0.40 ng OTA/kg body weight, which is much lower than that proposed by World Health Organization as the tolerable daily intake (16.0 ng/kg body weight). Healthy populations of Croatia are exposed to low, but seasonally and regionally variable amounts of OTA.