Bayesian Inference of Individualized Treatment Effects using Multi-task Gaussian Processes
Predicated on the increasing abundance of electronic health records, we investigate the problem of inferring individualized treatment effects using observational data. Stemming from the potential outcomes model, we propose a novel multitask learning framework in which factual and counterfactual outcomes are modeled as the outputs of a function in a vector-valued reproducing kernel Hilbert space (vvRKHS). We develop a nonparametric Bayesian method for learning the treatment effects using a multi-task Gaussian process (GP) with a linear coregionalization kernel as a prior over the vvRKHS. The Bayesian approach allows us to compute individualized measures of confidence in our estimates via pointwise credible intervals, which are crucial for realizing the full potential of precision medicine. The impact of selection bias is alleviated via a risk-based empirical Bayes method for adapting the multi-task GP prior, which jointly minimizes the empirical error in factual outcomes and the uncertainty in (unobserved) counterfactual outcomes. We conduct experiments on observational datasets for an interventional social program applied to premature infants, and a left ventricular assist device applied to cardiac patients wait-listed for a heart transplant. In both experiments, we show that our method significantly outperforms the state-of-the-art.
The authors propose a method of estimating treatment effectiveness T(x) from a vector of patient features x.  Treatment effectiveness is defined as  (health outcome with treatment Yw) - (health outcome without treatment Y(1-w)).  Presumably a health outcome might be something like survival time.  If a patient survives 27 months with the treatment and only 9 without then the effectiveness T(x) would be 18 months?
The authors estimate models of "outcome with treatment" and "outcome without treatment" jointly using RKHS kernel approximations on the whole dataset (I think there is a shared kernel).  
For a specific patient the effectiveness is based on the actual outcome of the patient which will be based on their features and their treatment condition minus the population model for the features of the opposite or counterfactual treatment condition.  
The population model is represented by an RKHS approximation which generates Bayesian posterior confidence bounds allowing the algorithm to integrate over uncertainty in the treatment effectiveness.  I like that the authors call this out in line 139: integrating over the uncertainty in the counterfactual condition improves their results. They employ an empirical Bayes framework in which they optimize for hyperparameters that control distributions over health outcomes. They use a loss function that tries to maximize accuracy of the actual outcome observed for a patient with specific features and minimizing the variance of the counterfactual ... 
This seemed a bit counterintuitive to me - wouldn't you want to maximize the variance of the counterfactual in order to be consistent with the principle of least commitment / maximum entropy? Don't make conclusions that are stronger than the data supports??
They end up using a mixture of two kernels to provide additional flexibility in modeling. 
The algorithm does a matrix inversion which I think is common in RKHS style approximation which may limit its scalability somewhat (also true of Gaussian Process based approaches). The example they show has only 1000 patients which is very small for an EHR dataset generally which can have millions of patients.
The goal of the authors is to be able to handle situations with strong selection bias (that is, the treatment is correlated with the patient features so we don't get an unbiased selection of treatments for any given condition).  They end up having to filter the datasets they work with to create stronger selection bias.  It is not clear whether this was a necessary step for them to demonstrate advantage over the other algorithms they benchmark against.
For their filtered dataset they show metrics with around 1/3 the error rate of competing state of the art methods. They attribute the gains to the ability of their algorithm to compensate for selection bias by integrating over uncertainty in the counterfactual???
The paper has a lot of technical jargon which is not  obviously necessary to explaining the intuitions clearly. Do we really need the details of the representer theorem in the paper? I would prefer using the space to describe intuitions on why it works more clearly.
line 27: "glints" is used as a verb in an awkward way
I liked Figure 1, but it took me a while to understand it. It might be worth explicitly calling out the roles of the 3 subfigures in the caption.  If you used fewer data points it might be more obvious that the realized factual outcomes and the counterfatual distributions are supposed to align ... maybe it would make sense to stack these graphs vertically so they share the same x axis?
line 53: "helps creating" is an awkward expression
lines 251 ... it would be nice if the authors could describe a bit more about the competing bench mark methods and why they perform the way they do? There is a short related work section but it doesn't use any of the names of the algorithms.  I don't know if these are the same as the comparison benchmarks or not.
Would the ordering be the same with a dataset that has low selection bias?