인과와 인과추론 | Causality and Causal inference

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인과, 인과추론의 개념과 관련 이론 (Back-door, Do-calculus) 들을 알아봅시다.

구조적 인과 모델(SCM)은 관측과 개입을 통해 인과 관계를 설명하는데 도움을 줍니다.

Back-door 기준은 인과 관계를 확인하고 혼동을 줄이는 데 도움이 되며, Do-계산법은 확률을 처리하는 데 유용한 규칙을 제공합니다.

     

Causality

• Influence by shich one event, process, state, or object a contributes to the production of another event, process, state, or object where the cause is partly responsible for the effect, and the effect is partly dependent on the cause.
• Causality in various academic disciplines
  • Physics, chemistry,biology, climate science
  • Psychology, social science, economics
  • Epidemiology, public health
• Relation to AI, ML, DS
  • AI : a rational agent performing actions to achieve a goal (reinforcement learning)
  • ML : currently focused on learning correlations
  • DS : capture, process, analyze, communicate with data

     

Structural Causal Model (SCM)

• SCM \(M = <U,V,F,P(U)>\) provides a formal framework.
• SCM induces observational, interventional, and counterfactual distributions.
• SCM induces a causal graph \(g\), which implies conditional independencies testable via d-separation (blockage).
• The underlying model \(M\) is unknown but the causal graph \(g\) can be given from common sense or domain knowledge.
• Intervention do(X=x) as a submodel M_x, which induces a manipulated causal graph \(g_\bar{x}\).
• Causal effect of \(X=x\) on \(Y=y\) is defined as \(P(y\mid{do(x)})\).

Remark

• Identifiability : causal effect may be computable from existing observational data for some causal graphs.
• In a Markovian case an singleton X, a causal effect can be easily derivable by canceling output \(P(x\mid{pa_x})\)

     

Back-door Criterion

• DefinitionㅣBack-door

  • Find a set \(Z\) s.t. it can sufficiently explain ‘confounding’ between \(X\) and \(Y\). Then,
  \[P(y|do(x))=\sum_Z{P(y|x,z)P(z)}\]

• DefinitionㅣBack-door criterion

  • A set \(Z\) satisfies the back-door criterion with respect to a pair of variables \(X, Y\) in causal diagram \(g\) if;
    • (i) no node in \(Z\) is a descendant of \(X\); and
    • (ii) $Z$ blocks every path between X ∈ \(X\) and Y ∈ \(Y\) that contains an arrow into X.

• A back-door adjustment formula is simple and widely used but limited.

Back-door sets as substitutes of the direct parents of X

• Rain satisfies the back-door criterion relative to Sprinkler ans Wet:
  • (i) Rain is not descendant of Sprinkler, and
  • (ii) Rain blocks the only back-door path from Sprinkler to Wet.
• Adjusting for the direct parents of Sprinkler, we have:

\[P(\text{wt}|do(\text{sp}))=\sum_\text{sn}P(\text{wt}|\text{sp,sn})P(\text{sn})=\cdots=\sum_\text{rn}P(\text{wt}|\text{sp,rn})P(\text{rn})\]

     

Rules of Do-calculus

• Backdoor criterion results in a very specific form of indentification formula.

• Do-calculus (Pearl, 1995) provides general machinery to manipulate observational and interventional distributions.

• TheoremㅣRules of Do-calculus (simplified)

  • Rule 1 : Adding/removing observations
  \[P(y|do(x),z)=P(y|do(x))\,\,\,\text{if}\,\,(Z\perp{Y|X})\,\,\text{in}\,\,g_{\bar{X}}\]
  • Rule 2 : Action/observation exchange
  \[P(y|do(x),do(z))=P(y|do(x),z)\,\,\,\text{if}\,\,(Z\perp{Y|X})\,\,\text{in}\,\,g_{\bar{X}\underline{Z}}\]
  • Rule 3 : Adding/removing actions
  \[P(y|do(x),do(z))=P(y|do(x))\,\,\,\text{if}\,\,(Z\perp{Y|X})\,\,\text{in}\,\,g_{\bar{X}\bar{Z}}\]
• Do-calculus is sound and complete but it has no algorithmic insight

• A graphical condition and an efficient algorithmic procedure have developed for identifiability.

• Do-calculus is a set of rules to manipulate observational or interventional probabilites. (Do-calculus is complete)

     

Modern Identification Tasks

• Experimental conditions ➔ Generalized identification

  • Combining datasets of different experimental conditions

  • The identifiability of any expression of the form \(P(y\mid{do(x), z})\) can be determined given any causal graph \(g\) and an arbitrary combination of observational and experimental studies.
  • If the query is identifiable, then its estimand can be derived in polynomial time.

• Environmental conditions ➔ Transportability

  • Combining datasets from different sources

  • Non-parametric transportability can be determined provided that the problem instance is encoded in selection diagrams.
  • When transportability is feasible, the transport formula can be derived in polynomial time.
  • The causal calculus and the corresponding transportation algorithm are complete.

• Sampling conditons ➔ Recovering from selection bias

  • Nonparametric recoverability of selection bias from causal and statistical settings can be determined provided that an augmented causal graph is available.
  • When recoverability is feasible, the estimated can be derived in polynomial time.
  • The result is complete for pure recoverability, and sufficient for recoverability with external information.

• Responding conditons ➔ Recovering from missingness

     

Reference

본 포스팅은 LG Aimers 프로그램에서 학습한 내용을 기반으로 작성된것입니다. (전체 내용 X)

1. LG Aimers AI Essential Course Module 5. 인과추론, 서울대학교 이상학 교수