18d-separation in factor graphs Tests whether X independent of Y given Z.Criterion 1: Observed node on pathCriterion 2: No observed descendant
19d-separation with gates Gate selector acts like another parent𝑿𝑿𝑊𝑿FTY𝑍FF𝑍𝑊𝑊𝑍TTYYCriterion 1: Observed node on pathCriterion 2: No observed descendant
20d-separation with gates Paths are blocked by gates that are off, but pass through gates that are on.𝒁=T𝒁=FFF𝑌𝑋𝑌𝑋TTCriterion 3 (context-sensitive): Path passes through off gate
21d-separation summary New! Criterion 1: Observed node on path Criterion 2: No observed descendantCriterion 3: Path passes through off gateNew!Allows new independencies to be detected, (even if they apply only in particular contexts)
23Inference in Gated Graphs Extended forms of standard algorithms:belief propagationexpectation propagationvariational message passingGibbs samplingAlgorithms become more accurate + more efficient by exploiting conditional independencies.Free software at[Minka & Winn, Gates. NIPS 2009]
32do calculusRules for rewriting P(y| 𝑥 ) in terms of P(𝑦|𝑥) etc. where 𝑥 stands for “an intervention on 𝑥”.P y 𝑥 ,𝑧 =𝑃(𝑦| 𝑥 ) if y independent of z in graph with parent edges of x removed.P y 𝑧 =𝑃(𝑦|𝑧) if y independent of z in graph with child edges of z removed.P y 𝑧 =𝑃(𝑦) if y independent of z in graph with parent edges of z removed if no descendent of z is observed.[Pearl, Causal diagrams for empirical research, Biometrika 1995]
33Rule 1: deletion of observations do calculusgatesP y 𝑥 ,𝑧 =𝑃(𝑦| 𝑥 )P(y│𝑑𝑜𝑋=𝑇,𝑧)=𝑃(𝑦|𝑑𝑜𝑋=𝑇)𝑑𝑜𝑋 =Tparents(𝑥)𝑥Criterion 3: Gate is offFRemove parent edges of xparents(𝑥)𝑥Tparents(𝑥)𝑥
34Rule 2: action/observation exchange do calculusgatesP y 𝑧 =𝑃(𝑦|𝑧)P(y│𝑑𝑜𝑍=𝑇,𝑧)=𝑃(𝑦|𝑑𝑜𝑍=𝐹,𝑧)Criterion 1: Observed node on path𝑑𝑜𝑍𝑧children(𝑧)FRemove child edges of zparents(𝑧)𝑧T𝑧children(𝑧)children(𝑧)
35Rule 3: deletion of actions do calculusgatesP y 𝑧 =𝑃(𝑦)P(y│𝑑𝑜𝑍)=𝑃(𝑦)Criterion 2: No observed descendentparents(𝑧)𝑧𝑑𝑜𝑍Fparents(𝑧)𝑧parents(𝑧)𝑧Tdesc(𝑧)desc(𝑧)
36Rule 3: deletion of actions do calculusgatesP y 𝑧 =𝑃(𝑦)P(y│𝑑𝑜𝑍)=𝑃(𝑦)parents(𝑧)𝑧𝑑𝑜𝑍Fparents(𝑧)𝑧parents(𝑧)𝑧Tdesc(𝑧)desc(𝑧)
37do calculus equivalence The three rules of do calculus are a special case of the three d-separation criteria applied to the gated graph of an intervention.
48ConclusionsCausal reasoning is a special case of probabilistic inference:The rules of do-calculus arise from testing d-separation in the gated graph.Causal inference can be performed using probabilistic inference in the gated graph.Causal structure can be discovered by using gates in two ways:to model interventions and/orto compare alternative structures.
49Future directions Imperfect interventions Counterfactuals Partial complianceMechanism changeCounterfactualsVariables that differ in the real and counterfactual worlds lie in different gatesVariables common to both worlds lie outside the gates