• 본 포스팅은 인과, 인과추론의 개념과 관련 이론 (Back-door, Do-calculus) 들을 소개하고 있습니다.
• Keyword : Causality, SCM, Back-door, Do-calculus
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인과 | Causality 와 인과추론 | Causal inference

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 Mx, 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:

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

References

• 본 포스팅은 LG Aimers 프로그램에 참가하여 학습한 내용을 기반으로 작성된것입니다. (전체내용 X)

➔ LG Aimers 바로가기

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