目录

• 10.1 Identification versus estimation
• 10.2 Estimation of causal effects
• 10.3 The myth of the super-population
• 10.4 The conditionality "principle"
• The curse of dimensionality
• Fine Point
  • Honest confidence intervals
  • Uncertainty from systematic bias
• Technical Point
  • Bias and consistency in statistical inference
  • A formal statement of the conditionality principle
  • Approximate ancillarity
  • Comparison between adjusted and unadjusted estimators
  • Most researchers intuitively follow the extended conditionality principle
  • Can the curse of dimensionality be reversed

Hern\(\'{a}\)n M. and Robins J. Causal Inference: What If.

在之前, 一直假设样本数量足够大, 从而没有随机因素的影响(即把以个体看成一亿或者更多个体的集合).

但是这种假设在实际中显然是不合理的, 往往我们只有少量的数据.

即使样本很多的一致性estimator也有可能离其正确的值相差很远.

另外, 这一节还提了提Wald confidence.

似乎用的就是一般的大样本的区间估计, 就是:

\[\frac{\bar{X} - \mu}{\sigma} \sim \mathcal{N} (0, 1).
\]

对于伯努利的情况,

\[\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}.
\]

在我们估计类似上面讲的置信区间的时候,

randomness 有两个来源:

1. 本章将的采样的随机性;
2. 来自于不确定的conterfactuals.

实际上, 我们能这么估计置信区间的原因是, 这些样本的确来源于一个binomial分布.

但是实际上, 有可能是每一个样本有一个独立的概率分布\(p_i\), 然后我们最后所观测到的\(p\)是一个均值而已(好浮夸).

\[\mathrm{Var} (\bar{X}_1 - \bar{X}_2)=
\mathrm{Var} (\bar{X}_1) +
\mathrm{Var} (\bar{X}_2).
\]

在confounders并不多的时候, 选择adjust for这些confounders是一个不错的主意.

Honest confidence intervals

uniform, honest: 存在一个样本数量n, 能够确保95%置信区间在95%的实验中发生.

Uncertainty from systematic bias

除了采样的误差, 置信区间的随机性也有可能是confounding, selection, measurement这些系统偏置带来的.

Bias and consistency in statistical inference

consistent estimator:

\[\mathrm{Pr}_P [|\hat{\theta}_n- \theta(P)| > \epsilon] \rightarrow 0 \quad \mathrm{as} \: n \rightarrow \infty \: \mathrm{for} \: \mathrm{every} \: \epsilon > 0, P \in \mathcal{M}.
\]

A formal statement of the conditionality principle

Approximate ancillarity

不想看.

Comparison between adjusted and unadjusted estimators

Most researchers intuitively follow the extended conditionality principle

Can the curse of dimensionality be reversed