4 Model Based Inference for CRE

Fisher, Neyman, Reg 三种方法都是非参方法：不需要模型假设。共性：填补，以应对核心挑战。

有了模型假设，$\tau$ 就是模型参数的函数，我们应该估计参数并对参数估计量做推断。频率方法推导困难，贝叶斯方法更普适。贝叶斯框架中，参数天然就有分布。

搭建基础的分析框架很重要

Slides.

Motivation

Background:

• A random sample from a super population
• CRE

Goal

• Finite sample ACE (Average Causal Effect).
• Super population ACE.

Let's switch our point of view:

If we know the distribution of $Y(1)$ and $Y(0)$, then the ATE we concern is just a function of the parameters of the distribution. All we need to do is to estimate the parameters and conduct inference about them. So we introduce model-based methods here.

Bayesian Framework

Key opinion: treat unknowns (model parameters $\theta$) as random variables.

Maybe the parameter is fixed in fact. But we don't know it. We are uncertain about it. We describe our uncertainty by treat the parameter as a random variable.

Main elements:

• Prior distribution $p(\theta)$.
• Likelihood function $p(data\mid \theta)$.
• Joint distribution $p(data,\theta)$.
• Posterior distribution $p(\theta\mid data)$.

Bayesian Model-Based Imputation

Finite Sample

Theory

Suppose we know the distribution of

1. $\boldsymbol{W}\mid \boldsymbol{Y}(1),\boldsymbol{Y}(0)$.
2. $(\boldsymbol{Y}(1),\boldsymbol{Y}(0))\mid \theta$.
3. $\theta$.

Note that they are all joint distribution. If i.i.d., marginal distribution is enough.

With $\boldsymbol{Y}^{obs},\boldsymbol{W}$ at hand, we can infer $\boldsymbol{Y}^{miss}$ by $f(\boldsymbol{Y}^{miss}\mid \boldsymbol{Y}^{obs},\boldsymbol{W})$. Then use them to estimate the finite sample ATE

$$\tau=\tau(\boldsymbol{Y}(1),\boldsymbol{Y}(0))=h(\boldsymbol{Y}^{mis},\boldsymbol{Y}^{obs},\boldsymbol{W}).$$

That is, we want to know $\tau\mid \boldsymbol{Y}^{obs},\boldsymbol{W}$. The randomness comes from $\boldsymbol{Y}^{mis}$.

So we need to deduce

$$f(\boldsymbol{Y}^{miss}\mid \boldsymbol{Y}^{obs},\boldsymbol{W})$$

$\theta$ is the bridge. See $\boldsymbol{Y}^{obs},\boldsymbol{W}$ as pendants and we get

$$f(\boldsymbol{Y}^{miss}\mid \boldsymbol{Y}^{obs},\boldsymbol{W})=\int _{\theta}f(\boldsymbol{Y}^{miss}\mid \boldsymbol{Y}^{obs},\boldsymbol{W},\theta)p(\theta\mid \boldsymbol{Y}^{obs},\boldsymbol{W})d\theta$$

So we need $f(\boldsymbol{Y}^{miss}\mid \boldsymbol{Y}^{obs},\boldsymbol{W},\theta)$ and $p(\theta\mid \boldsymbol{Y}^{obs},\boldsymbol{W})$.

Easy to deduce $f(\boldsymbol{Y}^{mis},\boldsymbol{Y}^{obs}\mid \boldsymbol{W},\theta)$ because given $\boldsymbol{W}$, $(\boldsymbol{Y}^{mis},\boldsymbol{Y}^{obs})$ is a linear combination of $(\boldsymbol{Y}(1),\boldsymbol{Y}(0))$.

And $f(\boldsymbol{Y}^{mis},\boldsymbol{Y}^{obs},\boldsymbol{W}\mid \theta)$ can be known because it is the multiplication of $P(\boldsymbol{W}\mid \theta)$ and $f(\boldsymbol{Y}^{mis},\boldsymbol{Y}^{obs}\mid \boldsymbol{W},\theta)$. The former one is known because CRE.

An Example

Read the slides from page 38 to 47. The results are the same as imputation with the mean of the observed value 😂.

But the gain is that we know the distribution of $\tau\mid \boldsymbol{Y}^{obs},\boldsymbol{W}$, which allows us to easily conduct inference.

See Homework 3 where there are details.

Simulation

Challenges in Bayesian Inference:

• computation of the posterior distribution $p(\theta\mid data)$.
• computation of the posterior predictive distribution $p(\tilde{y}\mid y)$.

They are usually impossible to compute analytically in close form.

We can repeat:

1. Impute $\theta$.
2. Impute $Y^{mis}$.
3. Calculate $\tau$.

Then we can get a distribution of $\tau$.

In View of Super Population

$$\begin{align} \tau_{sp} & =\mathbb{E}[Y_{i}(1)-Y_{i}(0)] \\ & = \tau(\theta) \end{align}$$

Note that $\tau_{sp}$ is also a random variable because of $\theta$.

We want to know $f(\theta\mid \boldsymbol{Y}^{obs},\boldsymbol{W})$, which equals to

$$\dfrac{p(\theta)f(\boldsymbol{Y}^{obs},\boldsymbol{W}\mid \theta)}{f(\boldsymbol{Y}^{obs},\boldsymbol{W})}$$

We can know the numerator, and the denominator is an integration of the numerator.

An analytic example:

See Homework 3 for more detail.