概述
文章目录
- 11.1 Data cannot speak for themselves
- 11.2 Parametric estimators of the conditional mean
- 11.3 Nonparametric estimators of the conditional mean
- 11.4 Smoothing
- The bias-variance trade-off
- Fine Point
- Fisher consistency
- Model dimensionality and the relation between frequentist and Bayesian intervals
- Technical Point
- A taxonomy of commonly used models
HernKaTeX parse error: Can't use function ''' in math mode at position 1: ̲'̲{a}n M. and Robins J. Causal Inference: What If.
前10章介绍了一些基本概念, 从这一章开始, 将通过模型进一步分析.
11.1 Data cannot speak for themselves
我们要估计
E
[
Y
∣
A
=
a
]
mathbb{E}[Y|A=a]
E[Y∣A=a], 但是可能由于数据有限, 或者
A
A
A甚至是一个连续的变量, 则我们没有办法对每一个
a
a
a进行估计.
这厮我们可以引入模型, 比如假设
E
[
Y
∣
A
=
a
]
=
θ
0
+
θ
1
A
mathbb{E}[Y|A=a]=theta_0 + theta_1 A
E[Y∣A=a]=θ0+θ1A, 然后去估计
θ
^
0
,
θ
^
1
hat{theta}_0, hat{theta}_1
θ^0,θ^1.
11.2 Parametric estimators of the conditional mean
11.3 Nonparametric estimators of the conditional mean
当
A
∈
{
0
,
1
}
A in {0, 1}
A∈{0,1}的时候, 我们可以发现:
E
[
Y
∣
A
=
0
]
=
θ
0
,
E
[
Y
∣
A
=
1
]
=
θ
0
+
θ
1
.
mathbb{E}[Y|A=0] = theta_0, \ mathbb{E}[Y|A=1] = theta_0 + theta_1.
E[Y∣A=0]=θ0,E[Y∣A=1]=θ0+θ1.
我们的有参模型这个时候就相当于是无参模型.
11.4 Smoothing
实际上, 我们可以把我们的模型假设得更加复杂一点:
E
[
Y
∣
A
]
=
θ
0
+
θ
1
A
+
θ
2
A
2
.
mathbb{E}[Y|A] = theta_0 + theta_1A + theta_2A^2.
E[Y∣A]=θ0+θ1A+θ2A2.
一个很自然的结论是, 这种线性模型, 参数越少模型越光滑.
The bias-variance trade-off
一般来说, 选择复杂的模型会有更小的bias, 但是又更大的variance.
Fine Point
Fisher consistency
That is, an estimator of a population quantity that,
when calculated using the entire population rather than a sample,
yields the true value of the population parameter.
就是说一个模型, 用了全部的population就能获得正确的参数, 那么这个模型就是非参数模型.
就像均值一样?
Model dimensionality and the relation between frequentist and Bayesian intervals
Technical Point
A taxonomy of commonly used models
g { E [ Y ∣ X ] } = ∑ i = 0 p θ i X i . g {mathbb{E}[Y|X]} = sum_{i=0}^p theta_i X_i. g{E[Y∣X]}=i=0∑pθiXi.
最后
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