1 Introduction
Machine learning is increasingly being used to help inform consequential decisions in healthcare, law, and finance. In these applications, the goal is often to predict the effect of some intervention (called a treatment effect)—e.g., the efficacy of a drug on a given patient (Consortium, 2009; Kim et al., 2011; Bastani & Bayati, 2015; Henry et al., 2015)
, the probability that a defendent in a court case is a flight risk
(Kleinberg et al., 2017), or the probability that an applicant will repay a loan (Hardt et al., 2016). There are two important properties that these machine learning models must satisfy: (i) they must be must be causal (Rubin, 2005; Pearl, 2010), and (ii) they must be interpretable.First, to predict treatment effects, our model must predict outcomes when the world is modified in some way (called a counterfactual outcome
). For example, to predict the efficacy of a drug on a patient, we need to know the patient’s outcome both when given the drug and when not given the drug. One way to predict counterfactual outcomes is to use randomized controlled experiments (RCTs)—by randomly assigning individuals to treatment and control groups, we can ensure that the model generalizes to predicting counterfactual outcomes. Indeed, RCTs are frequently used to estimate
average treatment effects (e.g., whether the drug is effective for the population as a whole). However, they are unsuitable for predicting individual treatment effects (ITEs)—such models have many more parameters, so much more training data is required. ^{1}^{1}1Individual treatment effects are also known as heterogeneous treatment effects, or conditional average treatment effects. Yet, the promise of machine learning is exactly to predict ITEs, which can be used to tailor decisions to specific individuals.Instead, we consider the more common approach of predicting counterfactual outcomes based on observational data. In contrast to RCT data, individuals are selected into treatment and control groups by unknown mechanisms (Rubin, 2005; Shalit et al., 2017)
. For example, in observational data, sicker patients are more likely to receive drugs. Thus, our model may incorrectly conclude that drugs are ineffective, since individuals who do not take drugs are healthier than those who do. The problem is that supervised learning can only guarantee predictive performance on data that comes from the same distribution as the training data, but counterfactual outcomes do not satisfy this assumption.
To make progress, we have to make assumptions about the distribution of the observational data. Several algorithms along these lines have been proposed, including honest trees (Athey & Imbens, 2016), causal forests (Wager & Athey, 2017), propensity score weighting (Austin, 2011), instrumental variables (Wooldridge, 2015), and causal representations (Johansson et al., 2016; Shalit et al., 2017).
Second, the learned model must be interpretable—i.e., a human domain expert (e.g., a doctor) must be able to validate the model. Interpretability is important since there are often defects in the training data that cause the model to make preventable errors. Indeed, it has been shown that these issues often arise in practice (Caruana et al., 2015; Ribeiro et al., 2016; Bastani et al., 2017). Learning interpretable models is particularly important when there may be causal issues. In particular, there is often no way to validate the assumptions made by causal learning algorithms. For example, many approaches assume strong ignorability, which says that probability of selecting into treatment can be fully predicted from the covariates. However, this assumption often fails in practice (Louizos et al., 2017). Interpretability provides a way for experts to identify causal issues.
Many algorithms have been proposed for learning interpretable models, including decision trees
(Breiman, 2017; Bastani et al., 2017), sparse linear models (Tibshirani, 1996; Ustun & Rudin, 2016), generalized additive models (Lou et al., 2012; Caruana et al., 2015), rule lists (Wang & Rudin, 2015; Yang et al., 2017; Angelino et al., 2017), decision sets (Lakkaraju et al., 2016), and programs (Ellis et al., 2015; Verma et al., 2018; Valkov et al., 2018; Ellis et al., 2018).Thus, while there has been a variety of work on learning causal models and on learning interpretable models, there has been relatively little work on designing algorithms that are capable of achieving both desirable properties.
Our contributions.
We propose a general framework for learning interpretable models with causal guarantees. In particular, given any supervised learning algorithm for learning interpretable models, our framework converts into an algorithm for learning interpretable models that predict the ITE of an individual with covariates . Furthermore, we provide guarantees on the performance of the models learned using .
We build on recent work on causal representations (Johansson et al., 2016; Shalit et al., 2017), a general framework for converting any supervised learning algorithm into an algorithm for learning models that predict ITEs. Their key idea is to first learn a causal representation , where is an embedding space. Intuitively, is designed to eliminate the bias from using observational data. In particular, they then use to train a model on the embedding of the original dataset , where is the treatment and is the outcome. Finally, assuming strong ignorability, they prove bounds on the error of the following model for predicting ITEs:
The reason we cannot directly use their approach is that the causal representation is uninterpretable. In particular, their approach would use the interpretable learning algorithm to train an interpretable model on . However, remains uninterpretable since is uninterpretable—the problem is that the inputs to are the uninterpretable features .
We propose a solution to this problem inspired by model compression (Bucilua et al., 2006; Hinton et al., 2015). First, we use (Shalit et al., 2017) to learn an uninterpretable function . We refer to the function defined by as the oracle model. Then, we use to learn an interpretable model to approximate —i.e., for some distribution of our choosing,
(1) 
where are i.i.d. samples. Then, we propose to use to predict ITEs.
It remains to choose in (1). We make a simple and intuitive choice—namely, the distribution over treatments that would have been induced by running an RCT (which we call the RCT distribution), where treatments are randomly assigned and are independent of the covariates . This choice amounts to using to label the unobserved counterfactual for each covariate in the original observational dataset, and then running on the combined dataset to train .
Intuitively, since RCTs can be used to predict ITEs, should have good performance as long as has good performance and is a good approximation of on the RCT distribution. Indeed, under these conditions, we prove a performance guarantee for analogous to the one available for the causal representations approach (Johansson et al., 2016; Shalit et al., 2017). Finally, in an experimental study, we show how our approach can be used to improve the performance of a wide range of interpretable models.
Related work.
There has been prior work proposing the “honest tree” algorithm for learning decision trees for prediting ITEs (Athey & Imbens, 2016). This work builds on the CART algorithm (Breiman, 2017)—in particular, they reduce the bias of CART by using different subsets of the training data to estimate the internal nodes and the leaf nodes. In contrast, our framework can be used to convert any interpretable learning algorithm into one for learning models for predicting ITEs. Furthermore, unlike their work, our approach comes with provable performance guarantees. Finally, we show in our experiments that our approach can substantially outperform theirs.
There has also been work using interpretability to identify causal issues in learned predictive models (Caruana et al., 2015; Ribeiro et al., 2016; Bastani et al., 2017). However, there is currently no way to fix these causal issues except by having an expert manually correct the model.
Finally, there has been a wide range of work using an uninterpretable oracle model to guide the learning of an interpretable model (Lakkaraju et al., 2017; Bastani et al., 2017; Verma et al., 2018; Frosst & Hinton, 2017; Bastani et al., 2018). Our work is the first to leverage this approach in the context of learning causal models.
2 Preliminaries
In this section, we give background on causal inference for estimating individual treatment effects (ITEs). Then, we summarize the approach of causal representations proposed in (Johansson et al., 2016; Shalit et al., 2017), as well as a bound they prove on the estimation error for their approach.
Potential outcomes framework.
We begin by describing the RubinNeyman potential outcomes framework (Rubin, 2005)
. Suppose we have a set of units, and we want to estimate the efficacy of a treatment for a given unit. We assume that each unit is associated with a covariate vector
(e.g., encoding patientspecific characteristics such as their healthcare history). Each unit is either assigned to the control group (denoted ) or to the treatment group (denoted ). Furthermore, each unit is associated with two potential outcomes—the outcome if the unit is assigned to control (i.e., ), and the outcome if the unit is assigned to treatment (i.e., ). The object of interest is the treatment effect , which informs the decision maker whether the unit would experience a better outcome under the treatment or under the control.For example, units may be patients, and covariates may be patientspecific features such as biomarkers and healthcare history. The treatment may be prescribing a drug to the patient (so the control is not prescribing the drug). Then, may be how quickly the patient recovers when prescribed the drug, and is how quickly the patient recovers without the drug. Then, the treatment effect measures whether the drug helps the patient recover more quickly. Ideally, the patient would only be given the drug if .
Formally, each unit is associated with a tuple of random variables
. We assume that the covariate vector takes values in , and the potential outcomes take values in (of course, the treatment takes values in ). Furthermore, we assume that for each unit, this tuple is drawn i.i.d. from a distribution .The fundamental challenge in causal inference is that for each unit, we only observe either or , but never both—in particular, for each unit, we only observe .
Definition 2.1.
The observed outcome is the factual outcome, and the unobserved outcome is the counterfactual outcome.
For example, if we give a patient the drug, we cannnot observe what would have happened without the drug.
Thus, we can only estimate the average over multiple units. If we average over the entire population, then we obtain average treatment effect (ATE)
However, the ATE does not yield any information about the efficacy of treatment on an individual unit. Instead, our goal is to estimate the efficacy of a treatment for an individual units based on their covariates.
Definition 2.2.
The individual treatment effect (ITE) is
To estimate the ITE, we make the following standard assumption about the treatment assignment mechanism (Johansson et al., 2016; Shalit et al., 2017).
Assumption 2.3.
We assume that the treatment assignment is strongly ignorable, i.e.,
For example, this assumption eliminates the possibility that we only observe for which . We also make the standard assumption that each unit has a nonzero probability of being assigned to each the control and the treatment.
Assumption 2.4.
We assume that for all ,
For example, this assumption eliminates the possibility that we never get observations of for a particular .
Our goal is to obtain an estimate of the ITE . A natural metric is our accuracy for predicting for a unit chosen at random from distribution .
Definition 2.5.
The expected precision in estimation of heterogenous effect (PEHE) (Hill, 2011) is
(2) 
Causal representations.
Now, we describe the causal representations approach to estimating (Johansson et al., 2016; Shalit et al., 2017). Suppose that we have observational data that we want to use to estimate . One way to do so is by estimating
and then using . Naïvely, we can use supervised learning to fit one model to predict on samples for which , yielding an estimate , and a second model to predict on samples for which , yielding an estimate .
This approach corresponds to fitting on samples from , and fitting on samples from . However, when evaluating the PEHE, we are also concerned with the errors of and on the counterfactual distributions and , respectively—i.e., when fitting , we also need samples , and when fitting , we also need samples . Otherwise, our estimate may be poor.
Thus, the error contains a term that comes from the discrepancy between the factual and counterfactual distributions. More precisely, by strong ignorability,
Comparing this with
we observe that the difference between these factual and counterfactual distributions are captured by the difference in the distributions and .
Definition 2.6.
The distribution of control units is , and the distribution of treated units is .
For this source of error to be small, we need to be similar to . However, for observational data, unlike RCT data, these distributions are given to us, and are not ones that we can choose.
As proposed by (Johansson et al., 2016; Shalit et al., 2017), one solution is to split the prediction problem into two steps: (i) learn a representation for some embedding space , and (ii) fit a predictive model on rather than on . Then, we can bound the error coming from the discrepancy between and by the discrepancy between and .
Assumption 2.7.
The representation is a twicedifferentiable, onetoone function. Without loss of generality, we assume that is the image of under , so that we can define an inverse .
Next, we define the distributions on induced by the distributions of treated units and of control units.
Definition 2.8.
For , define to be the density at of , and define to be the density of .
In other words, is the distribution of treated units on induced by , and is the distribution of control units on induced by .
We can now combine the estimates of into a single function. In particular, consider hypotheses of the form , where we estimate by and by . We are interested in the case where is derived from an estimator .
Definition 2.9.
Given a representation , we say a hypothesis factors through if there exists such that .
Then, we consider the following estimate of :
Definition 2.10.
The treatment effect estimate of the hypothesis for a unit with covariate is
We let . When factors through a representation —i.e., —we let .
Bound on causal error.
Our goal is to bound . We describe a bound on proven in (Shalit et al., 2017) for approaches to estimating the ITE
based on causal representations. We have two derived loss functions, one corresponding to the factual loss
and another corresponding to the counterfactual loss .^{2}^{2}2We assume that we are using the squared loss.Definition 2.11.
Given , the expected loss for the unit and treatment pair is
and the expected factual and counterfactual losses of are
We break up the factual loss into two parts based on the following definition.
Definition 2.12.
The expected factual treated and control losses are
It follows immediately that
One term in the bound on from (Shalit et al., 2017) quantifies the quality of , through the discrepancy between two distributions and . We use the following metric to measure this discrepancy:
Definition 2.13.
To obtain guarantees, we require the following assumption on the function family :
Assumption 2.14.
The family satisfies
for some .
Then, one desirable property of the representation is for to be small. The other term in the bound on the error
comes from the variances of
.Definition 2.15.
Given a distribution on , we denote the counterfactual density of by , defined by .
Definition 2.16.
Given a distribution on , the expected variances of and with respect to are
Furthermore, we let
We have the following bound on (Shalit et al., 2017):
Theorem 2.17.
For any factored as for some ,
This theorem shows that the error of our estimate of can be bounded by two terms. The first term
captures the error due to the test error of on the observational dataset. The second term
captures the error due to the mismatch between the distributions of treated units and of control units in the embedding space.
3 Interpretable Models for Individual Treatment Effect Estimation
Our learning framework can convert any algorithm for learning interpretable models in the supervised setting into an algorithm for learning interpretable models to predict individual treatment effects. Recall that the key issue with applying the causal representations approach is that we cannot simply train an interpretable model on the causal representation —in particular, the representation function is uninterpretable, so the composed model is uninterpretable.
Learning algorithm.
We propose an approach where we first train an uninterpretable oracle model using the causal representation approach, and then train an interpretable model to approximate . In particular, we prove that using our approach, as long as closely approximates , we can obtain a bound on the error of analogous to Theorem 2.17.
Let be the space of interpretable models considered by . Given observations from the distribution of , our goal is to learn an interpretable model for which we can provide causal guarantees. Let
be the set of datasets of any finite size (i.e., of size for ). Suppose we have a learning algorithm for interpretable models—i.e., given a dataset , then (usually approximately) solves the supervised learning problem
(3) 
We use to denote the model returned by .
In addition, suppose we also have an oracle model that is not interpretable (so ), but whose associated estimate of is good. We assume that is learned using the causal representation approach described in Section 2—in particular, that it factors as .
Our approach is to train to approximate —i.e., , where for some set of covariatetreatment pairs. The key question is how to choose so that produces a good estimate of —i.e., so that is small.
Intuitively, when we have control over the treatment assignment—e.g., in a randomized controlled trial (RCT)—a good distribution to use is to uniformly randomly assign treatments. In particular, consider the following distribution:
Definition 3.1.
Given a distribution on , the RCT distribution derived from is the distribution on defined by
and
In other words, the random variables
have joint distribution
if is distributed as and is independent from .Letting be the empirical distribution over covariates , we show below that is a good candidate for . In particular, with this choice, we can prove a bound on analogous to Theorem 2.17.
Given an observational dataset , our algorithm (shown in Algorithm 1) first uses the causal representations approach to learn an oracle model based on that has provable guarantees on (the subroutine LearnCR). Then, our algorithm constructs the distribution , where is the empirical distribution of covariates in . Next, our algorithm uses to label the points in , producing a dataset ; this step amounts to using to label the unobserved counterfactual for each covariate in . Finally, our algorithm runs the interpretable learning algorithm on the training set , and returns the result .
Model  IHDP  Jobs  

Ours  Baseline  Ours  Baseline  Ours  Baseline  Ours  Baseline  
CFRNet  –  0.926 0.02  –  0.271 0.01  –  0.235 0.02  –  0.086 0.03 
CART (depth 6)  3.668 0.17  4.305 0.20  0.485 0.03  0.679 0.04  0.241 0.01  0.271 0.02  0.086 0.03  0.067 0.02 
CART (depth 5)  3.824 0.18  4.436 0.21  0.492 0.02  0.725 0.05  0.241 0.01  0.280 0.02  0.086 0.03  0.069 0.02 
CART (depth 4)  4.086 0.19  4.605 0.22  0.530 0.03  0.717 0.05  0.241 0.01  0.281 0.02  0.086 0.03  0.064 0.01 
CART (depth 3)  4.462 0.21  4.930 0.23  0.585 0.03  0.795 0.05  0.241 0.01  0.285 0.02  0.086 0.03  0.067 0.02 
Honest Tree (depth 6)  3.694 0.17  4.086 0.19  0.481 0.02  0.483 0.03  0.235 0.02  0.223 0.01  0.086 0.03  0.073 0.02 
Honest Tree (depth 5)  3.760 0.17  4.098 0.19  0.488 0.02  0.486 0.03  0.235 0.02  0.216 0.01  0.086 0.03  0.074 0.02 
Honest Tree (depth 4)  3.875 0.18  4.128 0.19  0.498 0.02  0.488 0.03  0.235 0.02  0.223 0.02  0.086 0.03  0.084 0.02 
Honest Tree (depth 3)  4.090 0.19  4.237 0.20  0.535 0.03  0.498 0.03  0.235 0.02  0.236 0.01  0.086 0.03  0.080 0.02 
LASSO  5.725 0.26  5.777 0.26  0.671 0.04  0.942 0.05  0.235 0.02  0.226 0.02  0.086 0.03  0.080 0.02 
Kernel Ridge  2.077 0.09  3.190 0.14  0.361 0.02  0.562 0.02  0.235 0.02  0.234 0.02  0.086 0.03  0.077 0.02 
GBM  1.845 0.09  2.799 0.14  0.352 0.02  0.453 0.03  0.241 0.01  0.223 0.02  0.086 0.03  0.080 0.02 
Random Forest  2.905 0.14  3.653 0.19  0.439 0.02  0.621 0.04  0.241 0.01  0.239 0.01  0.086 0.03  0.073 0.02 
the standard error. We bold the better of the two values between ours and the baseline.
Bound on causal error.
We prove that as long as is close to on the distribution , where is the true covariate distribution, then is small.
Definition 3.2.
The relative error of to is
In other words, captures the test error of relative to the oracle model . Now, we can bound the generalization error by a combination of and the bound on .
Theorem 3.3.
For any function , and any function factored as for some , we have
We give a proof in Appendix A. Our bound has three terms—the first term captures the test error of relative to . The second two terms are from Theorem 2.17—the second term is the test error of on the observational dataset, and the third term captures the error due to the mismatch between the distributions of treated units and of control units in the latent representation.
While the bound in Theorem 3.3 is stated according to the exact error of with respect to , it can be straightforwardly converted to a finite sample bound using standard assumptions—e.g., that the model family has finite Rademacher complexity (Bartlett & Mendelson, 2002) and that solves (3) exactly. The other terms can similarly be converted into finitesample bounds (Shalit et al., 2017).
Finally, note that we can estimate on a heldout test set of observational data—it is simply the loss of on the dataset constructed from constructed from the same way Algorithm 1 constructs from . As discussed in (Shalit et al., 2017), the remaining terms in the bound can similarly be estimated on . Thus, we can obtain an test set estimate of the bound in Theorem 3.3.
4 Experiments
Evaluating the performance of causal models is a challenging task, since ground truth data on individual treatment effects (ITEs) is difficult to obtain. Following previous work (Shalit et al., 2017), we evaluate our framework on the IHDP (Hill, 2011) and Jobs (LaLonde, 1986) datasets.
IHDP dataset.
We use a dataset for causal inference evaluation based on the Infant Health and Development Program, from (Hill, 2011) and preprocessed by (Shalit et al., 2017) using the NPCI package (Hill, 2016). The dataset has 747 units (139 treated, 708 control) and 25 covariates of children and their mothers. This dataset contains 1000 realizations of the outcomes with 63/27/10 train/validation/test splits. The outcomes in this dataset are simulated—i.e., we have ground truth values of the ITE for each unit. Using this ground truth, we can obtain a test set estimate of the error in the predicted ITE. Then, we report the mean and standard errors of , as well as the absolute error in the average treatment effect (ATE)
over the 1000 realizations. Our primary metric of interest is , which measures predictive accuracy of ITEs, whereas measures predictive accuracy of the ATE.
Jobs dataset.
We use the Jobs dataset from (Shalit et al., 2017) based on (LaLonde, 1986), where the binary outcome is employment (versus unemployment). This dataset (3212 individuals) is a combination of data from a randomized trial (297 treated and 425 control) and data from an observational study (2490 control). A difficulty with the Jobs dataset is that we do not have ground truth on the ITEs. Instead, we use a metric based proposed in (Shalit et al., 2017), which evaluates a policy that makes treatment decisions based on the predictions of . In particular, recall that is the predicted outcome for a unit with covariates and treatment . We consider the policy that assigns this unit to treatment if the predicted treatment effect is positive—i.e., if . Then, the policy risk
measures the quality of outcomes on average over the test population. For any predictor , we can estimate on the randomized subset of the Jobs data as follows:
We also use the randomized subset to estimate the “ground truth” effect. In particular, let be the set of units in the treated subgroup, the randomized study, and in the control subgroup, respectively (note that ). We report the treatment effect on the treated by
and use as one metric
We report the mean and standard error of and over 10 outcomes with 56/24/20 train/validation/test splits. For this study, our primary outcome of interest is the , since it to some degree measures the predictive accuracy of ITEs; in contrast, similar to , measures the predictive accuracy of a population average effect.
Oracle model.
For , we train a CFRnet from (Shalit et al., 2017), which has 3 fully connected exponentiallinear layers for each the representation function and for the prediction function , with layer sizes 100 for all layers used for Jobs and 200 and 100 for the representation and hypothesis layers for IHDP. For IHDP, we used mean squared loss; for Jobs, we use logistic loss.
Interpretable models.
We evaluate the performance of our approach on a variety of models with a range of interpretability: CART trees (Breiman, 2017), honest trees (Athey & Imbens, 2016)
(Tibshirani, 1996), kernel ridge regression
(Murphy, 2012), gradient boosted models (GBMs)
(Friedman, 2001), and random forests (Breiman, 2001). For each model family, we train one model using our approach, and a baseline model using only the observational data for training.Of these models, only honest trees are designed to handle causality; however, their focus is on obtaining unbiased estimates rather than lowvariance estimates. In particular, they split the dataset into two, using the first part to estimate splits and the second to estimate values at the leaf nodes. This approach ensures that the estimates at the leaf nodes are unbiased, but also greatly increases variance since they are only using half the data at each point.
Results.
Discussion.
On the IHPD dataset, our approach uniformly outperforms the baseline approach in terms of , which measures performance on predicting ITEs. Even on predicting ATEs, our approach mostly outperforms the baseline; the only exception are honest trees, which are interpretable models tailored towards estimating treatment effects. As we discussed before, honest trees are focused on reducing bias at the expense of increased variance. Otherwise, we observe the usual trends—more complex models (e.g., GBMs and random forests) outperform more interpretable models (LASSO, CART, honest trees).
On the Jobs dataset, our performance was more mixed. Our approach significantly benefited CART in terms of , as well as honest trees of depth 3. However, for the remaining models (including honest trees of depth ), the baseline approach outperformed ours.
The problem is that the oracle model CFRNet did not perform as well as even some of the simpler models—indeed, the baseline honest tree of depth 5 was the best performing model on the dataset. In particular, we were unable to replicate the results of (Shalit et al., 2017), despite using their available code and obtaining the original train/validation/test splits from the authors. The gap in our performance ( 0.235) relative to ones reported in (Shalit et al., 2017) ( 0.21) is not very large; however, even in their results, a number of baseline models perform very similarly (or even better) than CFRNet.
As a consequence, many of the models trained using our approach achieved performance equal to that of CFRNet—in particular, since we are training our models using labels provided by CFRNet as ground truth, we cannot expect to do better than than their performance (i.e., and ). Furthermore, CFRNet appears to have learned a relatively simple function, since LASSO and kernel ridge regression both performed exactly as well CFRNet when trained to imitate it; similarly, none of the CART and honest trees trained to imitate CFRNet grew beyond depth 3.
In summary, while our approach proved less useful for the Jobs dataset, where simple models already perform as well as (or better than) more expressive models, our results on the IHDP dataset clearly demonstrate the potential for our approach to substantially improve the performance of interpretable learning algorithms used to predict ITEs.
5 Conclusion
We have proposed a general framework for learning interpretable models with causal guarantees. A number of directions remain for future work. Most importantly, as with previous work, our approach makes the strong ignorability assumption. The predominant approach to avoiding this assumption is to use instrumental variables. Incorporating these ideas with the instrumental variables framework could enable causal guarantees without strong ignorability.
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