Observational-Interventional Priors for Dose-Response Learning
Controlled interventions provide the most direct source of information for learning causal effects. In particular, a dose-response curve can be learned by varying the treatment level and observing the corresponding outcomes. However, interventions can be expensive and time-consuming. Observational data, where the treatment is not controlled by a known mechanism, is sometimes available. Under some strong assumptions, observational data allows for the estimation of dose-response curves. Estimating such curves nonparametrically is hard: sample sizes for controlled interventions may be small, while in the observational case a large number of measured confounders may need to be marginalized. In this paper, we introduce a hierarchical Gaussian process prior that constructs a distribution over the doseresponse curve by learning from observational data, and reshapes the distribution with a nonparametric affine transform learned from controlled interventions. This function composition from different sources is shown to speed-up learning, which we demonstrate with a thorough sensitivity analysis and an application to modeling the effect of therapy on cognitive skills of premature infants.
This paper proposes a two-layered prior model for dose-response learning. A Gaussian process is first learned from observational data and then is reshaped with a nonparametric affine transformation from controlled interventions. The key modelling idea is expressed in Eq. (5), where a nonstationary covariance function is constructed from the observational data. The advantages of the newly proposed model is empirically demonstrated by two examples involving simulation. 
Generally speaking the paper is written pretty clearly (except some questions and minor issues that I will list at the end). The underlying reason of the proposed model can be easily understood and the two case studies (which both involve simulations) have shown some advantages of the proposed model.   My main concern is that the idea of the paper does not seem significantly novel. The “elementwise affine” transform a . f_{obs} + b is simple and borrowing the information from K_{obs} to k_a also seems a natural way to address the problem. Overall this is more like developing a model suitable for one particular kind of problems. In addition, since both of the two examples involve simulation, the performance of the model has not been evaluated in any real data. (After all, the competitors to the proposed model in the paper seem naturally worse just from their semantics.)  Some questions and minor issues that I have found in the paper are listed as follows. (1). In Line 110: Why do you choose a Matern 3/2 covariance function for g(.,.)? Have you tried other (common) covariance options or Matern with different parameters for the comparison before choosing a Matern 3/2 covariance?  (2). In Line 120: What is the reason that “the cost of computing the covariance matrix K_{obs} of f_{obs}(X) is O(T^2 N^2)”? Do you assume that the TN * TN covariance matrix of the joint predictive distribution of {g(x, z^{(i)}} (mentioned at the end of Line 114) has already been computed? What is the running time of obtaining the posterior over functions g(.,.) (in your two studied cases)?  (3). For Eq. (6) (which is right after Line 135) 10^{-5} seems a magic number. Should it depend on the learned value of \lambda_a? How do you choose such a number?   (4). In Line 136: In “(\lambda_a, \sigma_h) are hyperparameters”, \sigma_h should be \sigma_a.  (5). Right after Line 146: Again, 0.5 in N(0, 0.5) and 0.1 in N(0, 0.1) are magic numbers. Should they be fixed or adjusted according to different data cases?    (6). In the third line of the caption for Figure 2: What is the meaning of “adjusted” in “adjusted \mu_{obs}”?  What is the meaning of “unadjusted” in “unadjusted Gaussian process regression”?