Causal models have no complete axiomatic characterization

Causal mo dels ha v e no complete axi omatic c haracterization Sanjiang Li Departmen t of Computer Scienc e and T ec hnology , Tsingh ua Univ ersit y , Beijing 1000 84, China lisanjiang@tsingh ua.edu.cn Abstract Mark ov netw orks and Bay esian netw orks are effective gra p hic representations of the dep en - dencies em b edded in probabilistic models. It is w ell known that ind ep endencies captured by Mark ov netw orks (called graph-isomorphs) h a ve a finite axiomatic characterization. This pa- p er, ho wev er, sho ws that indep endencies captured b y Ba yesian net works (ca lled causal mo dels) hav e no ax iomatization by using even countably many Horn or disjunctive clauses. This is b e- cause a sub-indep end ency mod el of a causal mo del ma y b e not causal, while graph-isomorphs are closed under sub-mo dels. Keywords: causal model; axiomatization; sub-mo del; conditional indep endence; graph-isomorph 1 In tro d uction The notion of conditio na l indep endence (CI) plays a fundamental role in pro babilistic reas oning. In traditional theories of proba bility , to decide if a CI statemen t holds, we need to ch eck whether tw o conditional pro babilities a r e equal, which req uire summations over exp o nent ia lly large n umber o f v ariable combinations. This numerical approach is clea rly impractica l. An alterna tive qua litative approach is very p opula r in artificial intelligence, wher e new CI statements ca n be der ived logica lly without reference to n umerical qua ntities. Given an initial set of indep endence r elations, a fixe d (finite) set o f axioms ca n b e used to infer new indep endencies by log ical manipulations. A natural question arises: can CI relations b e completely characterized by a finite set of axio ms (or ca lled infer ence rules )? Pearl and Paz [5] in tro duced the co ncept o f se mi- graphoid a s an independenc y mo del that sa tisfies four sp ecific a xioms, a nd show ed that each CI relation is a semi- graphoid. L a ter, Studen´ y [6] gave a negative answer to this ques tion. But, mor e p ositively , he also show ed that (i) CI rela tions hav e a characteriza tio n by a countable set of axio ms [6]; and (ii) ev er y probabilistically so und axiom with a t most tw o a nt ece dent s is a co nsequence of the semi-grapho id axioms [7]. Although CI relatio ns in genera l hav e no co mplete axiomatic c ha racteriza tion, Geiger and Pearl [2] develop ed complete ax io matizations for satura ted indep endence and marginal independenc e – t wo sp ecia l families of CI rela tions. Graphs are the most common metaphors for co mm unica ting a nd re a soning a b out dependen- cies. It is not surprising that gr aphical mo dels is a very p o pular wa y of sp ecifying indep endence constraints. There ar e in genera l tw o kinds o f graphica l mo dels: Mar ko v net works and Bay esian net works. A Markov net work is an undirected gra ph, while a Bayesian net work is a directed acyclic graph (DA G). Geig er and Pearl [2] develope d an axioma tic bas is for the relationships b etw een CI and gr aphic mo dels in s tatistic ana lysis. They s how ed in particular that (i) every axiom for con- ditional indep endence is also an a xiom for g raph separ a tion; and (ii) every gra ph r e pr esents a consistent set of indep endence and depe ndence constraints. Mo r eov er, an early work of Pearl and Paz [5] gav e an ax iomatic characterization for CI r elations captured b y undir ected gr aphs (called gr aph-isomorp hs ). It w as also c onjectured [3] that CI re la tions captured by DA Gs (called c ausal mo dels ) may hav e no finite axiomatic characterization. In this pap er , we confirm this conjecture and show that ca usal mo dels have no complete char- acterization by a ny (finite or countable) set of (Horn o r disjunctive) axioms. W e achiev e this by 1 showing that a sub-mo del of a causa l mo del ca n b e not ca usal. This is contrasted by CI r elations and g raph-isomo rphs. Both ar e closed under sub-mo dels. It came to us very late that the same observ ation has b een made in [8 , Remark 3.5], where Studen´ y gav e just ba s ic a r gument. This pap er will provide a complete pro o f for this obser v ation. The remainder par t of this pap er pro ceeds as follows. Section 2 pr ovides pr e liminary defini- tions for indepe ndenc y mo de ls , CI r elations, graph-is omorphs, and causal mo dels. Sec tion 3 gives syntactic a nd semantic descr iptions of indep endency logic, and then formalizes the notion of ax- iomatization. Then in Section 4 w e discuss heredity pro p erty of indep endency mo dels. F urther discussions are given in the last s ection. 2 Preliminaries In this section we intro duce the basic notions used in this pap er. Our refer e nce is [3, 4]. In what follows, if not o therwise stated we assume U is a finite set, and write ℘ ( U ) for the p owerset of U . The notion of conditional ind ep endency (CI) pla ys a fundamental role in probabilistic r easoning. Definition 2.1 (co nditional independency , CI) . Let U b e a finite set of v ariables with discrete v alues. Let P ( · ) be a joint pr o bability function ov er the v ariables in U . F or three dis jo int subsets X , Y , Z o f U , X a nd Y are said to b e c ondi t ional indep endent given Z if for all v alues x, y and z such that P ( y , z ) > 0 we hav e P ( x | y , z ) = P ( x | z ). W e use the notation I ( X, Z , Y ) P to denote the conditional indep endency of X and Y giv en Z . The set of all these CI statements for m a terna ry relation on ℘ ( U ), called a CI relation. In general, we hav e Definition 2.2 (independency mo del [3]) . An indep endency mo del M defined on U is a ternar y relation on ℘ ( U ) which satisfies the following condition: ( A, C, B ) ∈ M ⇒ A, B , C a r e pa irwise disjoint. (1) A tuple ( A, C , B ) in M (out o f M , re sp.) is called an independence statement (a dep endence statement, resp.). W e write I ( A, C, B ) M to indicate the fact that ( A, C, B ) is in M . Two o ther cla sses o f indep endency mo dels arise from gr aphs, where the notion of separ ation plays a key role. Definition 2.3 (gra ph separa tio n [3]) . If A, B and C are three disjoint s ubsets of no des in a n undirected graph G , then C is said to separa te A from B , denoted h A | C | B i G , if along every path betw ee n a no de in A and a no de in B there is a no de in C . The indep endency mo del consisting of a ll g raph sepa ration insta nces in G is a gr a ph-isomor ph. Definition 2 .4 (gr a ph-isomor ph [3]) . An indep endency mo del M is sa id to b e a gr aph-isomor ph if there exists an undirected gr aph G = ( U, E ) s uch that for every three disjoint subsets A, B , C of U , we hav e I ( A, C , B ) M ⇔ h A | C | B i G . (2) F or directed acyclic graphs, a s imilar separatio n prop erty was defined. Definition 2.5 ( d -separa tion [3]) . If A, B and C are three disjoint subsets o f no des in a DA G D , then C is said to d -separa te A from B , deno ted h A | C | B i D , if along every pa th b etw een a no de in A and a node in B there is a no de w satisfying one of the following tw o conditions: • w ha s co nv erging ar rows a nd none of w or its descendants ar e in C ; or • w do es not have con verging arr ows and w is in C . The indep endency model c o nsisting of all d -separa tio n instances in a DA G D is a causal mo del. 2 Definition 2.6 (causal model [3]) . An indep endency model M is said to b e c ausal if there is a D AG D such that for every three disjoint subsets A, B , C of U , we hav e I ( A, C , B ) M ⇔ h A | C | B i D . ( 3 ) It was prov ed by Geiger and Pearl that, for every graph-iso morph (causal mo del) M o n U , there is a probability distribution P on U such that M is pre cisely the CI relation induced by P [1, 2 ]. 3 Indep endency logic T o forma lize the no tion of axio matization, we in tro duce the indep endency logic I L . Although I L is a fragment o f first-orde r logic, we are ma inly concerned with its prop ositional counterpart. The lang uage of I L has as its alpha bet of symbols : • v ariables X 1 , X 2 , · · · ; • the constant ∅ ; • the ternary predicate I ; • three function le tters: − , ∪ , ∩ ; • the punctuation s y m b o ls (,) and ,; • the connectives ¬ , ∨ , ∧ T erms in the indep endency logic are defined as follows. Definition 3 .1 (term) . A term in I L is defined as follows. (i) Consta nt and V a riables a re terms. (ii) If T 1 , T 2 are terms in I L , then − T 1 , T 1 ∪ T 2 , T 1 ∩ T 2 are terms in I L . (iii) The set of all terms is g enerated as in (i) and (ii). Using the unique pr edicate I , we ca n form atomic formulas. Definition 3. 2 (atom, literal, clause ) . An atom in I L is defined by: if T i ( i = 1 , 2 , 3 ) ar e terms in I L , then I ( T 1 , T 2 , T 3 ) is a n atom. A literal is defined to b e an ato m (called p ositive literal) or its negation (ca lled negative literal). A clause is the disjunction o f a finite set of literals . F ormulas in I L ar e defined in the standar d wa y . Definition 3.3 (formula) . A for mula in I L is an expression inv o lving ato ms and connectives ¬ , ∧ , ∨ , which can b e for med us ing the r ules: (i) Any atom is a formula. (ii) If A and B are fo rmulas, then so are ( ¬A ), ( A ∧ B ), and ( A ∨ B ). In the rest of this pap er, we sha ll so metimes omit pare ntheses, as lo ng as no a mbiguit y is int r o duced. Since implication statemen t are conv enient for expressing inference rules, we define ( A → B ) as an a bbr eviation o f (( ¬A ) ∨ B ). As a c o nsequence, each clause k _ i =1 ¬ I ( T 1 i , T 2 i , T 3 i ) ∨ l _ j =1 I ( T 1 ,k + j , T 2 ,k + j , T 3 ,k + j ) (4) can b e eq uiv alen tly repr e s ented as an implication (or r ule) k ^ i =1 I ( T 1 i , T 2 i , T 3 i ) → l _ j =1 I ( T 1 ,k + j , T 2 ,k + j , T 3 ,k + j ) . ( 5 ) Clauses are o f particular imp o r tance in axiomatizatio n of indep endency mo dels. 3 Definition 3. 4 (Horn and disjunctive cla uses) . F or a cla use C o f form Eq. 5, C is called a Horn clause if l ≤ 1, and ca lled disjunctive o therwise. Above we in tro duce d the syntactic par t of I L . Next we turn to s emantic notions. Definition 3. 5 (v aluation) . Let M b e a n independency mo del defined on U . A v aluation in M is a function v : { X 1 , X 2 , · · · } → 2 U . V aluations ca n b e extended in a natural way to terms in I L . Definition 3.6 (v alid v aluation) . Let A b e a for mu la , and le t M b e an independency mo del defined on U . A v aluation v in M is valid for A if for each atom I ( T 1 , T 2 , T 3 ) a pp e ared in A , v ( T 1 ) , v ( T 2 ), and v ( T 3 ) are pair wise dis joint , where v ( T ) is the v aluatio n of T in M . The notion of sa tis fa ction is defined in the standa rd wa y . No te that if v is v alid for A in M , then it is a lso v alid for any sub-formula B o f A in M . The following definition is therefore well-defined. Definition 3 .7 (satisfaction) . Let A b e a formula, a nd let M b e an indep endency mo del defined on U . A v aluation v in M is s aid to satisfy A if v is v alid for A and it can be shown inductiv ely to do so under the following co nditions. • v satisfies atom I ( T 1 , T 2 , T 3 ) if ( v ( T 1 ) , v ( T 2 ) , v ( T 3 )) ∈ M . • v satisfies ¬B if v do es not s atisfies B . • v satisfies B ∨ C if either v sa tisfies B or v sa tisfies C . • v satisfies B ∧ C if v satisfies b oth B and C . W e say M satisfies A , in notation M | = A , if all v alid v aluations o f A in M satisfy A . The following pr op osition is a co ns equence o f the definition of → . Prop ositi o n 3. 1. L et A , B b e two formulas, and let M b e an indep endency mo del define d on U . Then M | = A → B iff for any valid valuation v of A → B in M , v s atisfies A implies v satisfies B . F or a clause we hav e the following characterizatio n. Corollary 3.1. Le t C b e a clause of form Eq. 5, and let M b e an indep endency mo del define d on U . Then M | = C iff the fol lowing c ondition holds: • for any valid valuation v of C in M , if ( v ( T 1 i ) , v ( T 2 i ) , v ( T 3 i )) ∈ M for al l 1 ≤ i ≤ k , then ( v ( T 1 j ) , v ( T 2 j ) , v ( T 3 j )) ∈ M for some k + 1 ≤ j ≤ k + l . Given a family o f independency mode ls M and a (finite o r countable) s et of for mulas F in I L , we now formalize the no tion that B ca n b e ax io matically characterize d by F . Definition 3.8 (axiomatiza tion) . A family of indep endency mo de ls M can be completely charac- terized by a se t of formulas F in I L if the following c ondition holds for a ny indep endency mo del M : M ∈ M ⇔ ( ∀B ∈ F ) M | = B . (6) W e say M has a finite (countable, resp.) axio matization if it ca n b e completely characterized b y a finite (countable, resp.) set of formulas in I L . Since each formula in I L is semantically equiv alent to the conjunction of a set of finite clauses, we need o nly co nsider clauses. Prop ositi o n 3.2 . A family of indep endency mo dels M has a fin ite (c ountable, r esp.) axiomatiza- tion iff it c an b e c ompletely char acterize d by a finite (c ountable, re sp.) set of clause s in I L . Analogous to pro p ositional ca lculus, we hav e the completenes s theor em for I L . 4 Theorem 3.1. S upp ose M is axiomatic al ly char acterize d by F . L et Σ b e a set of formulas, A b e a formula. Then the fol lowing two c onditions ar e e quivalent. (1) Σ | = M A : for any mo del M in M , if M satisfies al l formulas in Σ , it also satisfies A ; (2) Σ ⊢ F A : A is de ducible fr om 1 Σ by using axioms in F . In par ticular, we hav e Corollary 3 .2. L et M and F b e as in the ab ove the or em. F or a s et Γ of indep endenc e st atements { I ( T i 1 , T i 2 , T i 3 ) : 1 ≤ i ≤ k } and an indep endenc e statement γ = I ( T k +1 , 1 , T k +1 , 2 , T k +1 , 3 ) , we have Γ | = M γ iff γ is de ducible fr om Γ by u sing axioms in F . 4 Sub-mo dels In this section, we consider sub-indep endency mo dels. Definition 4.1 (sub- mo del) . Let M be an indepe ndency mo del defined o n U , a nd let V be a subset of U . W e call M | V = { ( A, C, B ) ∈ M : A, B , C ⊆ V } the sub-indep endency mo del (or simply sub-mo del) of M on V . The following result asser ts that if an indep endency mo del satisfies a formula, so do es its sub-mo del. Prop ositi o n 4. 1 . L et A b e a formula, and let M b e an indep en dency mo del define d on U . F or any su bset V of U , if M satisfies A , t hen so do es M | V . Pr o of. This is b ecaus e any v aluation v in M | V is a ls o a v aluation in M . An int er esting ques tion arise s naturally . Giv en a n independency mode l M o n U , suppos e M is a CI relation (or graph-isomor ph, or causal model), and V ⊆ U . Is sub-mo del M | V also a CI relation (or g r aph-isomo rph, or causa l mo del)? This is imp ortant for a family o f indep endency mo dels M to be axiomatizable. Actually , if M is not closed under sub-mo dels , then it cannot b e axiomatically characterized by any set of formulas. Given a join t probability P ( · ), write M for the CI relatio n on U induced by P ( · ), i.e. for any pairwise disjoint subsets A, B , C of U the tuple ( A, C, B ) is an instance of M if and only if A and B are conditionally indep endent given C (see Def. 2.1 and Def. 2.2). W e cla im that, for a none mpty subset V of U , M | V , the restriction of M on V , is a CI relation o n V . This is because M | V is induced by the join t pro bability P | V ( · ), which is obtained fro m P ( · ) by computing the mar ginal probability of P on V . A similar conclus ion ho lds for gr aph-isomor phs. Lemma 4.1. L et G = ( U, E ) b e an undir e cte d gr aph on U , and let V b e a nonempty pr op er su bset of U . Define an undir e cte d gr aph G ′ = ( V , E ′ ) as fol lows: for any t wo no des α, β ∈ V , ( α, β ) ∈ E ′ iff ther e is a p ath p fr om α t o β in G such that al l other no des in p ar e c ontaine d in U − V . Then h α | C | β i G ⇔ h α | C | β i G ′ (7) for any α, β ∈ V , and any C ⊂ V . Pr o of. Supp os e h α | C | β i G ′ . W e show C separates α from β in G . F or ea ch path p = αγ 1 γ 2 · · · γ m β ( m ≥ 0) in G , we show m ≥ 1 and some γ i is co nt a ined in C . Since h α | C | β i G ′ , ( α, β ) is no t an edge in G ′ . By definition, we know (i) ( α, β ) is not a n edge in G , hence m ≥ 1 ; and (ii) some no de γ i m ust b e contained in V . Suppose γ i 1 , γ i 2 , · · · , γ i k (1 ≤ i 1 < i 2 < · · · < i k ≤ m ) are all those no des in V . Since nodes b etw een γ i u and γ i u +1 (and those betw een α and γ i 1 , and betw een γ i k and β ) must b e 1 In the sense of logic deduct ion. 5 ❘ ✠ ❘ ✠ 1 0 4 2 3 Figure 1: A DA G D o n U = { 0 , 1 , 2 , 3 , 4 } . contained in U − V . By definition of G ′ , we know ( α, γ i 1 ) , ( γ i u , γ i u +1 ) , ( γ i k , β ) are all edges in G ′ . Therefore p ′ = αγ i 1 γ i 2 · · · γ i k β is a path in G ′ . By h α | C | β i G ′ , we k now some γ i u m ust be in C . T his shows tha t C s eparates α from β for every path p in G . On the o ther ha nd, suppo se h α | C | β i G . W e show C separ ates α fro m β in G ′ . F or each path p = αγ 1 γ 2 · · · γ m β ( m ≥ 0) in G ′ , we show some γ i is contained in C . W rite γ 0 and γ m +1 for α a nd β . Note tha t if ( γ i , γ i +1 ) is not an edge in G , then b y definition there is a path p i in G fro m γ i to γ i +1 such that all other nodes in p i are cont a ined in C . Concate- nating paths p 0 , p 1 , · · · , p m we obtain a new ‘path’ in G from α to β which sa tis fie s the fo llowing condition: Each node is either in p or in U − V . In this ‘path’ identical nodes may o ccur several times. With prop er mo difications, we obtain a shortened pa th p ′ in G whic h also satisfies condition (4). By our assumption that h α | C | β i G , w e know some no de in p ′ m ust b e contained in C . But by C ⊆ V , this s hows some γ i m ust b e in C . Hence C separates α from β for every path p in G ′ . Prop ositi o n 4.2. L et M b e a gr aph-isomorph on U . F or a nonempty subset V of U , M | V is also a gr aph-isomorph. Pr o of. Supp os e M is repres e nted by an undirected g raph G = ( U, E ). W e s how M | V can be represented by the undirected graph G ′ = ( V , E ′ ) constructed in Lemma 4.1, i.e. for for any pairwise disjoint subsets A, B , C of V , we hav e I ( A, C, B ) M | V iff h A | C | B i G ′ . By definition of gr aph separatio n, for a graph G ∗ we know h A | C | B i G ∗ iff ( ∀ α ∈ A )( ∀ β ∈ B ) h α | C | β i G ∗ . By Lemma 4.1, for any α, β ∈ V and any C ⊂ V we hav e h α | C | β i G ⇔ h α | C | β i G ′ . Ther efore h A | C | B i G ′ iff h A | C | B i G for any pair wise dis jo int subsets A, B , C o f V . Since M is representable by G , it is clear that M | V is a ls o r epresentable by G ′ . But the following ex a mple shows tha t this is not true for caus a l mo dels . Example 4.1. Let M b e the ca usal model r e pr esentable by the D AG D given in Fig. 1, and let V = { 1 , 2 , 3 , 4 } . The sub-indep endency mo del M | V is not r epresentable by any D AG. T o prov e this co nclusion, we us e the notatio n D ( α, β ) to expres s the fact that in M | V there is no C ⊂ V such that I ( α, C, β ) is true. It is clea r that the fo llowing indep endency statements holds in M | V : • D (1 , 2), D (3 , 4), D (2 , 3); • I (1 , ∅ , 3 ), I (1 , 4 , 3), I (2 , ∅ , 4), I (2 , 1 , 4). 6 In a D AG D = ( U, − → E ), for a ny tw o no des α, β ∈ U , it is w ell known that ( α, β ) ∈ − → E or ( β , α ) ∈ − → E iff no C ⊆ U ca n d -se pa rates α fr o m β . Suppo se M | V is representable by some DA G D ′ defined on V . By D (1 , 2 ), D (3 , 4 ), and D (2 , 3) we know in D ′ no de 1 is c o nnected to no de 2, no de 2 is co nnected to no de 3, and no de 3 is connected to no de 4. This shows that p = 1234 is a path fr om no de 1 to no de 4. But by I (1 , ∅ , 3) M | V and p ′ = 123 is a path from no de 1 to no de 3, we k now in D ′ we should hav e 1 → 2 ← 3 . Similar ly , for no des 2 and 4, we should als o have 2 → 3 ← 4 in D ′ . This is imp ossible since 2 → 3 and 2 ← 3 cannot app ear together in the s ame DA G. This pr oves that M | V has no DA G repr e sentation, hence is not ca us al. As a cor o llary of this exa mple and P rop. 4.1 we hav e Theorem 4.1 . Causal mo dels have no c omplete axiomatic char acterizatio n . Pr o of. Le t Γ = {A 1 , A 2 , · · · } b e the set of claus e s tha t ar e satisfied by all causal mo dels. In particular, the ca us al mo del M g iven in Example 4 .1 satisfies each A i . By Pro p. 4.1 we know M | V also satisfies each A i . Since M | V is not causa l, the infinite set Γ (let alone finite subs ets of Γ) cannot provide a complete characteriz ation for causal mo dels. 5 Discussion W e hav e shown tha t it is impo ssible to give a co mplete axioma tic ch a racteriza tion for causal mo dels. This is different from the res ults obtaine d in [5] and [6]. In [5], Pearl a nd P a z pr ov ed that graph-iso morphs hav e a co mplete c har acterizatio n b y five axioms (4 Horn, 1 dis junctive). Since a sub-mo del of a graph-is omorph a lso satisfies these axio ms, it is clear that sub-mo dels o f gr a ph- isomorphs are graph-is o morphs. W e g av e a metho d for constr ucting such a gra ph repr e sentation. Studen´ y [6] show ed that there is no finite a xiomatization for CI rela tions by using Horn clauses. More p ositively , he a ls o show ed that there exist an infinite set of Hor n clauses that completely characterize CI relations. But it is still unknown whether CI relations hav e finite a xiomatization by using arbitrar y clauses (Horn or disjunctive) . The cla ss of sub-mo dels of causal models seems us e ful when (unknown) hidden v ariables are inv olved. As for axiomatization, a result by Geiger (see [3, E xercises 3.7]) suggests that it may hav e no finite characteriza tion b y Horn axioms . References [1] D. Geiger and J. Pearl. On the logic of c ausal mo dels . In R. Shach ter, T. Levitt, L. Kanal, and J. Lemmer, editors, UA I ’88: Pr o c e e dings of t he F ourth Annual Confer enc e on Unc ertainty in Artificia l Intel ligenc e , page s 3–1 4 . North-Holla nd, 19 88. [2] D. Geiger and J. Pearl. 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