on 22-Apr-2022 (Fri)

the fundamental problem of causal inference

It is fundamental because if we cannot observe both 𝑌 _𝑖 (1) and 𝑌 _𝑖 (0) , then we cannot observe the causal effect 𝑌 _𝑖 (1) − 𝑌 _𝑖 (0) . This problem is unique to causal inference because, in causal inference, we care about making causal claims, which are defined in terms of potential outcomes. For contrast, consider machine learning. In machine learning, we often only care about predicting the observed outcome 𝑌 , so there is no need for potential outcomes, which means machine learning does not have to deal with this fundamental problem that we must deal with in causal inference.

We get the average treatment effect (ATE) by taking an average over the ITEs: 𝜏 , 𝔼[𝑌 𝑖 (1) − 𝑌 𝑖 (0)] = 𝔼[𝑌(1) − 𝑌(0)] , where the average is over the individuals 𝑖 if 𝑌 𝑖 (𝑡) is deterministic. If 𝑌 𝑖 (𝑡) is random, the average is also over any other randomness

The main reason for moving from exchangeability (Assumption 2.1) to conditional exchangeability (Assumption 2.2) was that it seemed like a more realistic assumption. However, we often cannot know for certain if conditional exchangeability holds. There may be some unobserved confounders that are not part of 𝑋 , meaning conditional exchangeability is violated. Fortunately, that is not a problem in randomized experiments

An “estimator” is a function that takes a dataset as input and outputs an estimate. We discuss this statistics terminology more in Section 2.4.

That’s the math for why we need the positivity assumption, but what’s the intuition? Well, if we have a positivity violation, that means that within some subgroup of the data, everyone always receives treatment or everyone always receives the control. It wouldn’t make sense to be able to estimate a causal effect of treatment vs. control in that subgroup since we see only treatment or only control. We never see the alternative in that subgroup

The Positivity-Unconfoundedness Tradeoff

Although conditioning on more covariates could lead to a higher chance of satisfying unconfoundedness, it can lead to a higher chance of violating positivity. As we increase the dimension of the covariates, we make the subgroups for any level 𝑥 of the covariates smaller.

The Positivity-Unconfoundedness Tradeoff

This is related to the curse of dimensionality. As each subgroup gets smaller, there is a higher and higher chance that either the whole subgroup will have treatment or the whole subgroup will have control. For example, once the size of any subgroup has decreased to one, positivity is guaranteed to not hold.

Assumptions of causal inference:

1. Unconfoundedness (Assumption 2.2)

2. Positivity (Assumption 2.3)

3. No interference (Assumption 2.4)

4. Consistency (Assumption 2.5)

the main assumptions that we need for our causal graphical models to tell us how association and causation flow between variables are the following two:

1. Local Markov Assumption (Assumption 3.1)

2. Causal Edges Assumption (Assumption 3.3)

the flow of association and causation in DAGs. We can understand this flow in general DAGs by understanding the flow in the minimal building blocks of graphs. The minimal building blocks of DAGs consist of chains (Figure 3.9a), forks (Figure 3.9b), immoralities (Figure 3.9c), two unconnected nodes (Figure 3.10), and two connected nodes (Figure 3.11)

By “flow of association,” we mean whether any two nodes in a graph are associated or not associated. Another way of saying this is whether two nodes are (statistically) dependent or (statistically) independent.

Additionally, we will study whether two nodes are conditionally independent or not.

Answer: association

In defining the backdoor criterion (Definition 4.1) for the backdoor adjustment (Theorem 4.2), not only did we specify that the adjustment set 𝑊 blocks all backdoor paths, but we also specified that 𝑊 does not contain any descendants of 𝑇 . Why? There are two categories of things that could go wrong if we condition on descendants of 𝑇: 1. We block the flow of causation from 𝑇 to 𝑌. 2. We induce non-causal association between 𝑇 and 𝑌.

If we condition on a descendant of 𝑇 that isn’t a mediator, it could unblock a path from 𝑇 to 𝑌 that was blocked by a collider. For example, this is the case with conditioning on 𝑍 in Figure 4.13. This induces non-causal association between 𝑇 and 𝑌 , which biases the estimate of the causal effect. Consider the following general kind of path, where → · · · → denotes a directed path: 𝑇 → · · · → 𝑍 ← · · · ← 𝑌 . Conditioning on 𝑍 , or any descendant of 𝑍 in a path like this, will induce collider bias. That is, the causal effect estimate will be biased by the non-causal association that we induce when we condition on 𝑍 or any of its descendants (see Section 3.6).

Unfortunately, even if we only condition on pretreatment co- variates, we can still induce collider bias. Consider what would happen if we condition on the collider 𝑍 2 in Figure 4.16. Doing this opens up a backdoor path, along which non-causal association can flow. This is known as M-bias due to the M shape that this non-causal association flows along when the graph is drawn with children below their parents. For many examples of collider bias, see Elwert and Winship [19]

SUTVA is satisfied if unit (individual) 𝑖 ’s outcome is simply a function of unit 𝑖 ’s treatment.

The minimal building blocks of DAGs consist of chains, forks, immoralities, two unconnected nodes, and two connected nodes.

The potential outcomes that you do not (and cannot) observe are known as counterfactuals because they are counter to fact (reality)

An “estimator” is a function that takes a dataset as input and outputs an estimate.

The Positivity-Unconfoundedness Tradeoff

In machine learning, we often only care about predicting the observed outcome 𝑌 , so there is no need for potential outcomes, which means machine learning does not have to deal with this fundamental problem that we must deal with in causal inference

the fundamental problem of causal inference

It is fundamental because if we cannot observe both 𝑌 _𝑖 (1) and 𝑌 _𝑖 (0) , then we cannot observe the causal effect 𝑌 _𝑖 (1) − 𝑌 _𝑖 (0) .

To identify a causal effect is to reduce a causal expression to a purely statistical expression.

The main reason for moving from exchangeability (Assumption 2.1) to conditional exchangeability (Assumption 2.2) was that it seemed like a more realistic assumption

Exchangeability means that the treatment groups are exchangeable in the sense that if they were swapped, the new treatment group would observe the same outcomes as the old treatment group

We do not have exchangeability in the data because 𝑋 is a common cause of 𝑇 and 𝑌

No interference means that my outcome is unaffected by anyone else’s treatment.

By “flow of association,” we mean whether any two nodes in a graph are associated or not associated. Another way of saying this is whether two nodes are (statistically) dependent or (statistically) independent.

We denote by 𝑌(1) the potential outcome of happiness you would observe if you were to get a dog ( 𝑇 = 1 ). Similarly, we denote by 𝑌(0) the potential outcome of happiness you would observe if you were to not get a dog ( 𝑇 = 0 )

More generally, the potential outcome 𝑌(𝑡) denotes what your outcome would be, if you were to take treatment 𝑡