Bayesian nonparametric modeling for causal inference.

 

 

1. Context of the Problem (Causal Inference in Nonexperimental Settings)

• Researchers face challenges when trying to identify causal effects without experimental data, where treatment assignment is not random.
• Many causal inference methods assume "ignorability" (that the treatment assignment is independent of the outcomes, conditional on observed covariates). These methods typically require the researcher to fit two models: one for the treatment assignment mechanism (how the treatment is assigned) and one for the response surface (how the outcome depends on the treatment and covariates).

2. The Proposed Strategy (Focus on Response Surface)

• Instead of focusing on both the treatment assignment mechanism and the response surface, this article suggests an approach that focuses entirely on flexibly modeling the response surface using BART.
• BART is a Bayesian nonparametric modeling technique that excels at capturing complex, nonlinear relationships in data without requiring the researcher to specify those relationships in advance.

Recall : Ignorability(Unconfoundedness) Assumption

 

Recall : Response surface

 

Recall : overlap

 

 

Selling point : BART를 사용해서 Y = f(X,Z)에 대해 model f()를 flexible 하게 만들겠다는 것. 여기서 Y는 potential outcome, X: confounder, Z:Treatment.

기존의 response surface에 대해 nonparametric and semiparametric method는 robust 하지만 분석가의 개입이 많이 들어감. 예를들어 각 구간(bandwidth)을 어느정도로 설정할거냐 그 구간 안에서 어떤 함수로 fitting할거냐. BART를 사용할경우 이러한 분석가의 많은 개입이 들어가지 않아도 됨.

interaction term을 넣지 않아도 변수 사이의 interaction을 알수 있음

예를들어

Imagine we have a tree that predicts the price of a house based on two features:

• x1x_1x1​ = square footage of the house
• x2x_2x2​ = number of bedrooms

Here’s how a tree might work:

1. The first split might be based on square footage (x1x_1x1​):
   • If x1x_1x1​ > 2000, then split further; if not, go to another branch.
2. Then, within the group of houses where square footage is greater than 2000, the second split might be based on the number of bedrooms (x2x_2x2​):
   • If x2x_2x2​ > 3, split further or make a prediction; if not, go to another branch.

• The tree has now created a prediction based on both x1x_1x1​ and x2x_2x2​. This is an interaction, because the effect of the number of bedrooms (x2x_2x2​) on the house price is conditioned on the size of the house (x1x_1x1​). This interaction is automatically handled by the tree structure.

 

BART의 장점 : 왼쪽 - nonlinearity를 잘 잡아냄, 오른쪽 - 검정색선은 true treatment effect (E[Y(1) - Y(0)|X]) 파란색 수직선은 marginal 95% posterior intervals for the treatment effect at each X value]

 

 

 

Simulations based on real data

CATT(conditional average treatment effect on the treated)와 CATC(conditional average treatment effect on the controls)을 타겟으로 각 방법을 비교. 전자는 overlap이 만족된 경우 후자는 overlap이 만족되지 않은 경우에대해

 

1. Response Surface:

• Figure 3: This figure uses a linear, parallel response surface (denoted as Surface A in the paper). In this case, the relationship between the covariates and the outcome is linear, making it easier for simpler models like linear regression to perform well.
• Figure 4: This figure involves a nonlinear, non-parallel response surface (denoted as Surface B). Here, the relationship between the covariates and the outcome is more complex and nonlinear, which is more challenging for traditional methods like linear regression and propensity score matching.

여기서 x축은 각 method의 신뢰구간 y축은 true값과 추정값의 deviation. BART가 좁은 신뢰구간에서 covarage를 잘함.

 

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BART의 장점 : continuous treatment variable에 대해서도 확장이 가능함