Causal Model¶

A causal/scientific model \(M\) for a set of RVs \(\{ x_1, \dots, x_n \}\) is a model of the joint distribution \(p(x_1, \dots, x_n)\) as well as the causal structure governing \(\{ x_1, \dots, x_n \}\) which describes the causal relationships among the vars $$ M: y \leftarrow x $$

Causal Graph Model¶

Causal diagram/graph is a graph that can be used to represent a causal structure and hence, describe our quantitative knowledge about a causal mechanism

Causal model that uses a causal graph to represent the causal structure is called as Causal Graph Model/Causal Bayesian/Belief network.

RCM was developed in statistics, causal graphical model is derived from CS/AI.

Parts of Causal Model¶

1. Causal Structure \(G\): Direction of causality (What causes what)
2. Statistical Model \(H\): Joint distribution function

Example¶

Consider a model \(y = 2x\). This is a statistical model.

If \(y \leftarrow 2x\), then it means that \(x\) causes \(y\). This is a causal model.

Let’s analyze the \(x,y\) pairs for the following sequential changes.

This is because, \(x\) causes \(y\); not the other way around.

Causal Diagram/Graph¶

Directed graph that represents the causal structure of a model. It is represented using a bayesian network: DAG (Directed Acyclic Graph) in which each node has associated conditional probability of the node given in its parents.

The joint distribution is obtained by taking the product of the conditional probabilities. $$ \begin{aligned} p(x_1, \dots, x_n) &= \Pi_{i=1}^n p( x_i \vert \text{pa}(x_i) ) \ x_i & {\tiny \coprod} \text{nd}(x_i) \vert \text{pa}(x_i) \end{aligned} $$ where

• pa = parent
• nd = not descendant

Useful when you are only interested in the causal structure, and not so much in the statistical model

flowchart TD
B([B]) & C([z]) --> A([A])

Why DAG?¶

• Directed, because we want causal effects directions
• Acyclic, because a variable cannot cause itself.

my question is: What about recursive loops, such as our body’s feedback loops

Parts¶

flowchart TB
a([a])
w([w])
x([x])
y([y])
z([z])

x --> y & z
z --> a

w --> y

Properties¶

Causal diagrams compatible with a joint distribution \(p(x_1, \dots, x_n)\) must satisfy

• \(\text{nd}:\) non-descendent
• \(\text{pa}:\) parents

Causal Relations¶

General Framework¶

flowchart TB
oc([Observed<br/>Confounders])
uc([Unobserved<br/>Confounders])
oce([Observed<br/>Common<br/>Effects])
uce([Unobserved<br/>Common<br/>Effects])
x([x])
y([y])

x --> y
oc ---> x & y
uc -..-> x & y

x & y --> oce
x & y -.-> uce

classDef dotted stroke-dasharray:5
class uc,uce dotted

linkStyle 0 stroke:green

Types¶

flowchart TB

subgraph Mediation
  subgraph ucm[Unconditioned]
    direction TB
    x1([x]) --> c1([z]) --> y1([y])
    %% x1 --> y1
  end

  subgraph cm[Conditioned]
    direction TB
    x2([x]) --> c2[z] --- y2([y])
    %% x2 --> y2
  end
end

subgraph mc2[Mutual Causation 2]
    subgraph ucmc2[Unconditioned]
        direction TB
        x8([x]) & y8([y]) --> w8([w]) --> c8([z])
        %% x8 --> y8
    end
    subgraph cmc2[Conditioned]
        direction TB
        x7([x]) & y7([y]) --> w7([w]) --> c7[z]
        %% x7 --> y7
    end
end

subgraph mc[Mutual Causation 1]
    subgraph ucmc[Unconditioned]
        direction TB
        x4([x]) & y4([y]) --> c4([z])
        %% x4 --> y4
    end
    subgraph cmc[Conditioned]
        direction TB
        x3([x]) --> c3
        y3([y]) --> c3[z]
        %% x3 --> y3
    end
end

subgraph md[Mutual Dependence]
    subgraph ucmd[Unconditioned]
        direction TB
        c6([z]) --> x6([x]) & y6([y])
        %% x6 --> y6
    end
    subgraph cmd[Conditioned]
        direction TB
        c5[z] --- x5([x]) & y5([y])
        %% x5 --> y5
    end
end

%%linkStyle 4,24,23 stroke:none;

Confounding¶

Self Selection Bias¶

Special case of confounding, when \(z\) affects the selection of \(x\) and also has a causal effect on \(y\), then \(z\) is confounder.

flowchart TB
c([Ability/Talent])
a([Education])
b([Earnings])

c -->|Affects<br/>selection| a
c -->|Directly<br/>Affects| b

a -->|Associated but<br/>not necessarily causes| b

Here, education may not necessarily causally affect earnings.

Unmeasured Confounding¶

We need to find new ways to identify causal effect

However, in some cases, there can exist a set of observed vars conditioned such that it satisfies the back-door criterion, by blocking all non-causal paths b/w \(x\) and \(y\)

flowchart TB

subgraph Conditioned
    direction TB
  wc([w]) -.- zc & yc
  zc[z] --- xc([x]) --> yc([y])
end

subgraph Unconditioned
    direction TB
  wu([w]) -.-> zu & yu
  zu([z]) --> xu([x]) --> yu([y])
end

linkStyle 3,7 stroke:green
linkStyle 0,1,2 stroke:none

classDef dotted stroke-dasharray:5
class wu,wc dotted

\(w\) is a confounder to \(x\) and \(y\), but we do not need to observe it, as causal effect of \(x\) on \(y\) is identifiable by conditioning on \(z\)

Conditioning¶

• Condition common causes
• Do not condition common effects

If we condition on a set of vars \(z\) that block all open non-causal paths between treatment \(x\) and outcome \(y\), then

• the causal effect of \(x\) on \(y\) is identified (estimated from observed data)
• \(z\) is said to satisfy the ‘back-door criterion’
• Conditioning on z makes \(x\) exogenous to \(y\)

Covariate Balance¶

In a sample where is \(z\) is a conditioned confounder $$ P(z \vert x = x_i) = P(z \vert x = x_j) \quad \forall i, j $$

If \(x\) is binary $$ \begin{aligned} & x \in { 0, 1 } \ \implies & P(z \vert x=1) = P(z \vert x=0) \end{aligned} $$

Matching¶

Converting non-RCT observed sample into a sample that satisfies Covariate Balance, ie behaves similar to a RCT sample through control of observed confounding.

Effect Modifiers¶

Confounders \(s\) that change the causal effect of a treatment \(x\), since their causal effect on the outcome \(y\) interacts with treatment’s causal effect on \(y\)

Controlling them help control conditionally randomized experiments.

Consider the following causal diagram.

flowchart TD

subgraph After Simplification
    others([s<sup>y</sup>]) & x2([X]) --> y2([Y])
end

subgraph Before Simplification
    a([A]) & b([B]) & c([C]) & w([W]) & x1([X]) --> y([Y])
end

where \(s^y\) is anything other than \(x\) that can causally affect \(y\); ie, the set of potential effect modifiers

An example could be gender, temperature, etc.

Mathematically, \(s\) is an effect modifier if $$ \begin{aligned} P(y) &\ne P(y \vert s) \ E[y \vert \text{do}(x)] &\ne E(y \vert \text{do}(x), s) \end{aligned} $$

Instrumental Variable¶

Variable \(iv\) that satisfies

1. \(iv\) correlated with \(x\)
2. Every open path connecting \(iv\) w/ \(y\) has an arrow to \(x\)
3. \(iv\) exogenous to \(y\) (desired but not strictly necessary)
4. \(iv\) affects \(y\) only though correlation with \(x\)

If we use a model $$ \begin{aligned} \hat y &= \beta_0 + \beta_1 x + u\ \implies \beta_1 &= \dfrac{\text{Cov}(y, iv)}{\text{Cov}(x, iv)} \end{aligned} $$

Fixed Effects¶

Variable that captures effect of unobserved confounders $$ \hat y = \tau + \beta_0 + \beta_1 x + u \ \implies \widehat{\text{ATE}} = \beta_1 $$

Example¶

flowchart TB
pe(["Parents<br/>Education<br/>(IV)"]) --> ce(["Child<br/>Education<br/>(x)"]) --> ci(["Child<br/>Income<br/>(y)"])

pe -.- ag(["Child<br/>Age, Gender<br/>(s)"]) --> ce & ci
city(["City<br/>(FE)"]) --> pe & ce & a & ci

a(["Ability<br/>(s)"]) -..-> ce & ci

classDef dotted stroke-dasharray:5
class a dotted

classDef highlight fill:darkred,color:white
class pe highlight

linkStyle 1 stroke:green

Parent’s education is an instrument variable for child’s wage through child education.¶

College-educated parents have +ve impact on children’s college attainment, either through better home education or because they are more capable of affording college education.

If we assume that more educated parents

• don’t produce children with higher unobserved abilities/preferences that affect education and wages
• don’t directly help children obtain higher wage jobs

The statement holds true as the only way parent’s education affects individual’s earnings is through effect on child’s education.

City is a Fixed Effect¶

City captures ability, school quality, productivity and other unobserved factors

Types of Intermediary vars¶

flowchart TB

subgraph Mediator
direction TB
z3((z)) --> y3
x3((x)) --> y3((y))
x3((x)) --> z3
end

subgraph Collider
direction TB
x2((x)) --> y2((y))
x2((x)) & y2((y)) --> z2((z)) 
end

subgraph Confounder
direction TB
x1((x)) --> y1((y))
z1((z)) --> x1((x)) & y1((y))
end

Intervention¶

\(\text{do}(x_i = a)\) implies removing all arrows entering \(x_i\) while setting \(x_i = a\), hence removing the effect of all parents of \(x_i\)

Hence \(x_i\) becomes endogenous after intervention

flowchart TB

subgraph Post-Intervention
direction TB
a2((a)) & b2((b)) --- c2((c)) --> d2((d))
end

subgraph Pre-Intervention
direction TB
a1((a)) & b1((b)) --> c1((c)) --> d1((d))
end

linkStyle 0,1 stroke:none

IDK¶

Limitation of Causal Graph¶

There is no way for causal graph to represent the causal effect of input \(x\) on output \(y\) due to some effect modifier. For example

flowchart TD
f(("Fertilizer<br/> (x)")) & t(("Temperature<br/> (s<sup>y</sup>)")) --> y(("Yield<br/> (y)"))

Temperature does not affect how much usage of fertilizer, but we know that the temperature affects the effectiveness of the fertilizer, but we have no way of representing that here. However, the Causal Effect Formula will take care of that.

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