Discovering Cyclic Causal Models with Latent Variables: A General SAT-Based Procedure

Transcription

1 Discovering Cycic Causa Modes with Latent Variabes: A Genera SAT-Based Procedure Antti Hyttinen Patrik O. Hoyer HIIT & Dept. of Computer Science University of Hesinki Finand Frederick Eberhardt Phiosophy Caifornia Institute of Technoogy Pasadena, CA, USA Matti Järvisao HIIT & Dept. of Computer Science University of Hesinki Finand Abstract We present a very genera approach to earning the structure of causa modes based on d-separation constraints, obtained from any given set of overapping passive observationa or experimenta data sets. The procedure aows for both directed cyces feedback oops and the presence of atent variabes. Our approach is based on a ogica representation of causa pathways, which permits the integration of quite genera background knowedge, and inference is performed using a Booean satisfiabiity SAT sover. The procedure is compete in that it exhausts the avaiabe information on whether any given edge can be determined to be present or absent, and returns unknown otherwise. Many existing constraint-based causa discovery agorithms can be seen as specia cases, taiored to circumstances in which one or more restricting assumptions appy. Simuations iustrate the effect of these assumptions on discovery and how the present agorithm scaes. 1 INTRODUCTION One of the main goas in many fieds of science is to identify the causa reations existing among some set of variabes of interest. Such causa knowedge may be inferred both from experimenta data randomized controed trias and passive observationa measurements. In genera the information avaiabe from mutipe such studies may need to be combined to obtain an accurate picture of the underying system. In recent years, many approaches to this causa discovery probem have been suggested Spirtes et a., 1999; Richardson and Spirtes, 1999; Schmidt and Murphy, 2009; Caassen and Heskes, 2010; Peters et a., 2010; Triantafiou et a., 2010, buiding on the framework of causa Bayes networks Spirtes et a., 1993; Pear, In this framework, causa reations among a set of variabes V are represented by a directed graph G in which each variabe is represented by a node in the graph, and an arrow from node x to node y indicates that x is a direct cause of y with respect to V. Athough causa modes based on directed graphs are often restricted to be acycic, causa feedback can be represented by permitting directed cyces in G, i.e. directed paths from a node back to itsef. In addition, unmeasured common causes of two or more nodes in V are commony represented by aowing bi-directed arrows between any pair of confounded nodes. 1 If there are no such confounders, the set V is said to be causay sufficient. Thus, in the most genera case of cycic causa structures with atent variabes, any pair of nodes x, y V, with x y, can be connected by any combination of the edges x y, y x, and x y see Figure 1 for exampes. One of the key theoretica concepts in causa modes based on directed graphs is the notion of d-separation, due to Geiger et a This is a graphica separation criterion that provides the structura counterpart to conditiona independencies in the probabiity distribution generated by the mode. D-separation is based on paths in the graph. Since a singe pair of nodes can be connected by mutipe edges, in our mode space a path is defined as a sequence of consecutive edges in the graph, without any restrictions on the types or orientations of the edges invoved. Definition 1 D-separation A path p is said to be d-separated or bocked by a set of nodes C if and ony if i p contains a chain i m j or a fork i m j such that the midde node m is in C, or ii p contains an inverted fork or coider i m j 1 In this representation a atent variabe affecting more than two observed variabes is represented by two-way confounders bi-directed edges between a pairs of nodes corresponding to the affected observed variabes.

2 such that the midde node m is not in C and such that no descendant of m is in C. A set C is said to d- separate x from y if and ony if C bocks every path from x to y. Pear, 2000 a x z w y x z b y w When appying Definition 1 to graphs with bi-directed edges such as in Figure 1b, the bidirected edge z w can be viewed as a atent structure z zw w. In acycic modes, such as causa Bayes networks, if two nodes x and y are d-separated given a set C then the corresponding random variabes are statisticay independent when conditioning on C in the probabiity distribution generated by the mode. If there are no statistica independencies in the distribution other than those impied by d-separation appied to the underying graph, the distribution is said to be faithfu to the graph. Thus, under an assumption of faithfuness causa discovery procedures can use the outcomes of statistica independence tests, appied to the observed data, to infer d-separation and hence structura properties of the underying graph. For exampe, if in a set of four variabes V = {x, y, z, w} it is found that i x is unconditionay independent of y, ii x is independent of w given z, iii y is independent of w given z, and iv no other unconditiona independencies are found, then the we-known PC-agorithm Spirtes et a., 1993 wi infer that the underying causa structure is the one in Figure 1a. Whie the correspondence between probabiistic independence and d-separation is known to hod generay for acycic modes even when there are atent variabes, the case is not as cear for cycic modes. The correspondence is known to hod for inear causa reations with Gaussian error terms, i.e. non-recursive inear Gaussian structura equation modes Spirtes, 1995, and can be extended to modes with correated error terms, which is one way to account for causay insufficient sets of variabes. A genera characterization of the parameterizations of cycic modes with atent variabes, for which the correspondence between d-separation and probabiistic independence hods, is not known Pear and Dechter, 1996; Nea, Foowing the standard approach of non-parametric causa discovery agorithms, we use d-separation reations as the basic input to our procedure, but acknowedge that in the cycic case ony the inear Gaussian modes are known to provide the appropriate correspondence with statistica independence. We aow for a set-up simiar to the overapping data sets approach of the ION-procedure Timan et a., 2009 in that we do not restrict ourseves to a singe data set measured over some set of observed nodes, but can hande d-separation reations that were obtained from different overapping sets V i of nodes. Anaogousy to Figure 1: Exampe graphs see text for detais. Hyttinen et a we generaize the overapping data sets case to aow that the V i can contain nodes that are known to have been subject to a randomized experiment. Nevertheess, the target of our discovery procedure is the underying causa graph G over the set of nodes V = i V i, impying that G may contain edges between nodes that are never measured together in the same data set. 2 PROBLEM SETTING We consider the space of cycic causa modes G over the jointy causay insufficient set of nodes V = i V i, where each V i specifies the nodes present in experiment E i = J i, U i. J i and U i form a partition of V i such that the nodes in J i are randomized simutaneousy and independenty and the nodes in U i are passivey observed J i can be empty to aow for passive observationa settings. We use the foowing simpification of the d-separation criterion: Definition 2 D-separation A path is d-connecting with respect to a conditioning set C if every coider c on the path is in C and no other nodes on the path are in C, otherwise the path is d-separated or bocked. Definition 2 is equivaent to Definition 1 when an edge can be used mutipe times in a path Studený, 1998; Koster, For exampe, the sequence of edges x z w z y in the graph in Figure 1a is d-connecting with respect to conditioning set C = {w}. The extension of d-separation to experimenta settings is straightforward: a d-connecting path may ony contain a node x J if x / C and x is a fork common cause on the path or the source of the path. We write x y C J resp. x y C J to denote that x is d-separated from resp. d-connected to y given C in the experiment with intervention set J. We assume we have a d-separation orace that returns the truth vaues of statements of the form x y C J i in the true graph G, for any pair of distinct nodes x, y and set of nodes C that occur together in some V i. It is we known that even in the presence of randomized experiments the set of a d-separation reations over the set of nodes in genera underdetermines the true causa structure even for much more restricted mode spaces than we consider here. So the discov-

3 ery task is to determine for each pair of nodes in V and for each edge type,, whether the edge is present, absent or if its existence is unknown. In addition we determine possibe indirect ancestra causa reations: for each ordered pair of nodes x, y, whether a directed path x y exists, does not exist or if its existence is unknown. 3 SAT AND BACKBONES Our agorithm for causa structure discovery is based on computing the so-caed backbone of a given formua in propositiona ogic. We empoy a Booean satisfiabiity SAT sover Biere et a., 2009 to determine the backbone, which can be directy interpreted as the soution to the structure discovery task. This section provides an overview on SAT and backbones. Propositiona formuas are buit from Booean variabes by repeated appication of the connectives negation, disjunction, ogica OR, conjunction, ogica AND, impication and equivaence. Any propositiona formua can be represented in conjunctive norma form CNF using a standard inear-size encoding Tseitin, For a Booean variabe X, there are two iteras, X and X. A cause is a disjunction of iteras; a CNF formua is a conjunction of causes. A truth assignment is a function τ from Booean variabes to {0, 1}. A cause C is satisfied by τ if τx = 1 for some itera X in C, or τx = 0 for some itera X in C. A CNF formua F is satisfiabe if there is an assignment that satisfies a causes in F, and unsatisfiabe otherwise. The NP-compete Booean satisfiabiity SAT probem asks whether a given CNF formua F is satisfiabe. Impementations of decision procedures for SAT, socaed SAT sovers, can in practice not ony determine satisfiabiity of CNF formuas, but aso produce a satisfying truth assignment for satisfiabe formuas. The most efficient SAT sovers are based on the compete confict-driven cause earning CDCL search agorithm Marques-Siva and Sakaah, 1999; Moskewicz et a., 2001; Eén and Sörensson, Centra to CDCL is the abiity to derive emmas in terms of new CNF causes based on non-soutions detected during search, which makes the search performed by CDCL SAT sovers differ from standard depth-first backtracking search. In many cases, the state-of-the-art CDCL SAT sovers can sove SAT instances consisting of miions of causes and variabes Järvisao et a., If a Booean variabe X takes the same vaue in a satisfying truth assignments of a given CNF formua F, X is caed a backbone variabe of F ; the vaue X is assigned to in a satisfying assignments is caed the poarity of X. The set of backbone variabes or simpy, the backbone of a formua F can be computed by a inear number of cas in the number of variabes in F to a SAT sover: if exacty one of F X and F X is satisfiabe, then X is in the backbone of F. 4 ENCODING D-SEPARATION Figure 2 shows our propositiona encoding for the d- connection property. The encoding aows to represent both d-separation and d-connection reations as constraints directy on the edges present or absent in the underying causa graph. In essence, the encoding spes out Definition 2 extended to experiments by expressing the conditions for paths being bocked or unbocked. In the encoding, Booean variabes [x y] and [x y] represent the underying causa graph. For each pair of distinct nodes x, y V, the Booean variabe [x y] variabe [x y], respectivey takes the vaue 1 if and ony if the edge x y edge x y, respectivey is present in the graph. 2 The Booean variabe [x y C J] is 1 if and ony if x and y are d-connected in the underying graph when conditioning on C and intervening on J. To encode the different types of d-connecting paths of ength between pairs of nodes x, y when conditioning on C and intervening on J Eqs. 1 7, Booean variabes [x [x y], and [x y], y] are introduced, with the respective arrowheads and edge-tais as indicated. In genera, d-connecting paths in a cycic graph can have infinite ength, ength of a path being the number of its edges. However, as shown in Appendix B, ony paths of a maximum ength max = 2 V 4 need to be considered. These Booean variabes are hence defined for a paths of the four types of ength = 1,..., max and for a pairs of nodes in V. The constraint requiring that a specific d-connection x y C J is present is constructed by taking the conjunction of the variabe [x y C J] and Equations 1 7. Simiary, the constraint requiring that a specific d-separation x y C J is present is the conjunction of [x y C J] and Equations 1 7. From a causa perspective, for a d-connection x y C J the encoding spits the d-connecting paths into four groups Eq. 1: i paths that start with an edge-tai at x and end with an arrowhead at y, ii paths that start with an arrowhead at x and end with an edge-tai at y, iii paths that start with an arrowhead at x and end with an arrowhead at y, and iv 2 We omit sef-oops, i.e. edges from a node to itsef, as they do not affect the d-connectedness of a graph.

4 Encoding of d-connection between nodes x, y given conditioning set C and intervention set J. [x y C J] max =1 [x y] [y x] [x y] [x y] 1 Paths of ength 1: [x Paths of ength = 2,..., max : [x [x 1 y] 1 y] [x 1 [x y] z / C { [x y] if y / J 0 otherwise { [x y] if x / J and y / J 0 otherwise y] 0 4 [x 1 y] [z z / C [x z / C [x y] z / C 1 1 [x 1 z] [z y] x] [z y] z] [z > y] z] [z y] z C [x 1 [z z / C 1 [x z C z C [x 1 z] [z x] [z 1 y] y] z] [z y] z] [y z] Figure 2: Encoding d-connection via paths between pairs of nodes paths that start with an edge-tai at x and end with an edge-tai at y. The paths are buit up recursivey in terms of ength Eqs. 5, 6, and 7. By keeping track of the path engths we ensure that each path bases out through Eqs. 2 and 3 on the actua edges in the graph, whose presence is represented by Booean variabes [x y] and [x y]. There are no paths of type iv with ength 1, as such a path must invove at east one coider in C to have tais at both ends hence Eq. 4. The shortest vaid case is of ength = 2 and resuts from the second haf of Eq. 7. By expicity keeping track of the termina edge-marks in each path variabe, the encoding ensures that a coiders on a d-connecting path are in the conditioning set C, and a non-coiders are not in C. The base cases Eqs. 2 and 3 ensure that there is no path with an edge into a variabe that is intervened on into y J. For each given d-separation reation x y C J or simiary each d-connection reation x y C J, the whoe encoding, incuding Eqs. 1 7, is cubic in the number V of nodes. Furthermore, it is important to notice that our agorithm, as described next, does not generate the constraints in Eqs. 1 7 for a possibe d-separation and d-connection reations at once. The constraints for individua reations are generated ony on demand during the execution of the agorithm, in many cases avoiding generating an exponentia number of constraints needed to represent a possibe d- separation and d-connection reations. The SAT-based approach to causa structure discovery by Triantafiou et a uses an encoding based on partia ancestra graphs PAGs, a particuar form of equivaence cass. Their encoding does not suffice for our purposes, since it is restricted to acycic causa structures in non-experimenta settings, and given experiments it is often possibe to distinguish between different graphs that for passive observationa data beong to the same PAG. 5 ALGORITHM The encoding of d-separation reations presented in the previous section can be used for a variety of discovery appications. For the purpose of iustration we wi present here one agorithm for a common discovery setting. The extension to other settings is then easiy expained. Agorithm 1 iterates over three steps unti a d-separation reations are known: 1 finding a set of d-separation/connection tests T c in order of increasing conditioning set size with currenty unknown resut, and determining those reations D c, 2 generating the additiona constraints encoding the reations in D c reca the encoding in Figure 2, and 3 computing the backbone over the propositiona formua

5 Agorithm 1 SAT-based causa structure discovery Initiaize soution S for the edge variabes [x y], [x y] of each pair of distinct nodes x, y V to status unknown. Initiaize ϕ to be the empty propositiona formua. For conditioning set size c from 0 to V 2: 1: Determine d-separation/connection reations. Find a set T c of d-separation/connection tests with conditioning set size c that are undetermined given S. Determine the d-separation/connection reation for each test in T c, and et set D c consist of these reations. 2: Refine the working formua ϕ. For each x y C J in D c: Encode x y C J using equations 1-7: ϕ := ϕ Encodex y C J. For each x y C J in D c: Encode x y C J using equations 1-7: ϕ := ϕ Encodex y C J. 3: Incrementa backbone computation with SAT sover Compute B: the set of edge-variabes [x y], [x y] in the backbone of ϕ. For each edge variabe e in B: If e B with poarity 1, set status of e to present in S. If e B with poarity 0, set status of e to absent in S. Output S: the status of each edge. consisting of the constraints generated so far. In Step 1 we appy a pruning heuristic described in Appendix A that guarantees that a unknown d- separation reations are found, but remains computationay tractabe. We use a d-separation orace see Section 2 to determine the resut of each test. In Step 2, given a d-connection reation x y C J, the subroutine Encode returns the conjunction of [x y C J] and the formuas in Eqs Simiary, given a d-separation reation x y C J, Encode returns the conjunction of [x y C J] and the formuas in Eqs Note that for each combination of C and J, Eqs. 2 7 need to the added ony once into ϕ aso guaranteed by our current impementation. This is important in practice, so that the SAT sover is not suffocated with many copies of the same constraints. In Step 3, a SAT sover is used incrementay for determining which of the edge-variabes in the current working formua ϕ are in the backbone of ϕ. The poarity of these backbone variabes determines whether the corresponding edges are present or absent. Like other constraint based causa discovery agorithms, Agorithm 1 considers d-separation reations in order of the size of the conditioning set C. For sparse graphs, this enabes a rapid pruning of the constraint generation on the basis of the simpest tests. But unike other constraint based agorithms, Agorithm 1 can expicity incude known d-connections, rather than assuming that there is a d-connection whenever no d-separation is found see aso Section 7. Agorithm 1 is easiy adjusted to consider an arbitrary set of d-separation/connection reations as input, as ong as the set of nodes V is specified from the outset. If the set is sma, one can just run step 2 and 3 to compute the backbone using a avaiabe reations, otherwise one can run the fu procedure, simpy omitting reations from D c that are not avaiabe in the set. It wi terminate when a reations are encoded or when no more are needed, as determined by step 1. In Agorithm 1 we use the status on each edge as the output. If other aspects of the graphs are of interest, one can easiy define other variabes and compute the backbone over them. In Section 6 we use this feature to determine which ancestra reationships are known. 5.1 BACKGROUND KNOWLEDGE AND MODEL SPACE ASSUMPTIONS Athough we have considered a very genera mode space, restricting the procedure to smaer spaces is simpe. Focusing on just one data set rather than a set of overapping data sets, or ony considering passive observationa data and no experiments, requires no adjustments of Agorithm 1. If one has reason to beieve that there are no unmeasured nodes, i.e. that V is jointy causay sufficient, then setting [x y] 0 8 for a pairs of nodes in the encoding wi enforce this restriction. If one is ony interested in acycic causa structures, then adding the constraint [x y {x}] [x y {y}] 9 for each pair of nodes, together with the respective path definitions Eqs. 2 7, is sufficient. Eq. 9 disaows cyces by enforcing that there cannot be a directed path from x to y and a directed path from y to x. Since the conditioning set in each of the d- connection caims in Eq. 9 is empty, there cannot exist any coiders in the d-connecting paths. The intervention on x and y, respectivey, in each of the caims in Eq. 9 ensures that d-connections due to common causes are excuded. Ony directed paths are invoved in x y {x} and x y {y}. In Section 5.2 we use this fexibiity to generate the same causa inferences as other d-separation based agorithms. More generay, the encoding aows for incuding various types of background knowedge. One can enforce that a particuar edge is present or absent, that particuar ancestra reations are maintained or disaowed,

6 that specific paths with, if needed, particuar waypoints and of a specific ength are present or absent. The type of knowedge that can be encoded is more genera than any other constraint based procedure we are aware of, incuding the additions to the csat+ agorithm by Borboudakis et a One is in principe ony imited by what can be encoded in terms of a Booean constraint over the edge and path variabes. We think this coud be of enormous utiity to appications with significant domain knowedge or when quaitative causa reations are discovered by other means e.g. using the additive noise or non-gaussian techniques of Peters et a and Hoyer et a COMPLETENESS For more restricted mode spaces, graphica representations of the casses of d-separation-equivaent graphs have been deveoped e.g. partia ancestra graphs. We do not have a simiar representation for our more genera mode space and it is uncear whether an easiy interpretabe representation is possibe, since there can be graphs that share the same d-separation reations, but differ in adjacencies, orientations and ancestra reationships. By ony providing the status of each edge as output of Agorithm 1, we foow Triantafiou et a who used this soution format in ight of the often even computationay unmanageabe output of the ION-agorithm which does not consider cycic graphs. The downside is that this output is not fuy informative about the soution space. For exampe, if d-separation reations were obtained from a passive observation of the graph x y z, then the current output does not represent that x y z is not a soution. Instead, it woud among other things mark a edges of adjacent nodes as unknown, since x y z is aso a soution. Nevertheess, it is trivia to query our procedure about graphica aspects that are not represented in the output. Since the compete soution space is impicity represented by the working formua ϕ, the SAT-sover can easiy determine that x y z is not a vaid soution in this exampe. Simiary, one can query the status of any other structura proposition by constructing a Booean variabe X for it using the edge or path variabes in the encoding, and determining whether X is in the backbone of ϕ or not. If it is, then poarity 1 indicates that X is true for a graphs that satisfy ϕ, whie poarity 0 indicates that X is fase for a graphs that satisfy ϕ. If X is not in the backbone, then there is a graph G 1 that satisfies ϕ, for which X is true, and a graph G 2 that satisfies ϕ, but for which X is fase. This is one, given the encoding perhaps trivia, sense in which our procedure is compete for any propositiona query given the d- separation/connection reations and any mode space restrictions as input. We ca this query-competeness. A different type of competeness is used in the context of other constraint based agorithms. Given the d- separation tests that an agorithm performs, we say that an agorithm is d-separation compete if a d- separation reations over the set of nodes are known. The PC-agorithm for acycic graphs over a causay sufficient set of nodes, the FCI-agorithm for acycic graphs over a causay insufficient set of nodes and the CCD-agorithm for cycic graphs over a causay sufficient set of nodes are a d-separation compete for their mode spaces, respectivey Spirtes et a., 1993; Richardson, 1996; Spirtes et a., Reying on the mode space assumptions, the agorithms conduct just enough d-separation tests to determine a d-separation reations of the graphs in the soution space, even reations that the agorithms did not expicity test. None of these agorithms are d-separation compete when their mode space assumptions are vioated: Figure 1b gives a cycic graph for which FCI is not d-separation compete, since it does not test whether x y {w, z}. The graph with atent confounders in Figure 3 is an exampe for which CCD is not d-separation compete, because it does not determine the d-separation x 1 x 5 {x 2, x 3, x 4 }. PC does not hande either graph. These imitations iustrate that achieving d-separation competeness without performing a tests is a non-trivia probem in the genera mode space we consider containing both graphs. Once we consider overapping data sets, there are d- separation reations invoving nodes that do not occur together in any V i. Sometimes these can be determined from the other d-separation reations, but often they remain undetermined even when a the d- separation reations within each V i are estabished. For this setting we adjust the definition of d-separation competeness to require that exacty those reations that cannot be determined in the sense just described are eft unknown and a others are determined. For cycic modes with atent variabes in overapping experimenta or observationa data sets, Agorithm 1 is d-separation compete, and in genera it wi not test a avaiabe d-separation reations. But in the present impementation of step 1 we resort to simpe safe heuristics to avoid some redundant tests, and otherwise appy brute force see Appendix A. It is an open chaenge to further reduce the number of tests performed whie preserving d-separation competeness. We cannot empoy a simpe variant of the efficient test schedues of FCI and CCD, as they seect subsequent tests on the basis of a graphica representation of the knowedge acquired so far that is specific to their restricted mode spaces. But given those restrictions, we can adopt the test schedues: Using FCI as basis, the ION-agorithm which aso assumes acycicity is d-separation compete for pas-

7 x 1 x 2 x 3 x 4 x 5 Figure 3: An acycic graph with atents for which the CCD-agorithm is not d-separation compete. sivey observed overapping data sets Timan et a., Simiary, if we assume acycicity, we can use instead of our heuristic the test schedue of FCI in Agorithm 1 when anayzing overapping experimenta data sets: run FCI on each individua data set experimenta or not and input to Agorithm 1 the resuts of the tests that FCI considered on each individua data set together with the acycicity constraints in 9 for the FCI mode space. Agorithm 1 wi then combine the information across data sets and output a information avaiabe on the status of each edge in the true graph. The set of d-separation reations tested by FCI is sufficient for d-separation competeness for the V i in that data set. Interventions do not affect the d-separation competeness, since the manipuated graph in any experiment sti satisfies a mode space assumptions of FCI. One coud avoid some tests by further book-keeping of the information about the interventions, but for d-separation competeness it is unnecessary. Given FCI s d-separation competeness on each data set, the constraints generated by feeding the test resuts to Agorithm 1 impy that a d- separations reations that coud be tested, are aready determined. Any d-separation reation sti unknown cannot be determined. By assuming acycicity we thus obtain d-separation competeness using the efficient FCI schedue of tests for overapping data sets with experiments. As Agorithm 1 is aso query-compete, we have a genera procedure for the approaches of Lagani et a An anaogous argument for cycic graphs without atent nodes, using the test schedue of CCD, can ony be made if we assume that the nodes in V are a observed in a possiby experimenta data sets. In the overapping setting, causa sufficiency can be vioated in the individua data sets and, as shown above, CCD is not d-separation compete for such a mode space. 6 SIMULATIONS To determine the effectiveness of the proposed approach, we impemented Agorithm 1 and investigated the properties of the method empiricay. Our impementation is based on the MiniSAT sover Eén and Sörensson, 2004, The code is avaiabe at First, we investigated the extent to which our approach, and in particuar the SAT sover used, is abe to sove the arge probem instances generated by nontrivia graphs. We generated random directed graphs of size n = nodes, in which each of the edges both directed and bidirected was independenty incuded with probabiity 0.2. We then generated a random set of 10 overapping experiments, in each of which each node was independenty and with equa probabiity chosen to be either intervened, passivey observed, or unobserved. Finay, we computed a observabe d-separation/connection reations; these constituted the input to our procedure. Figure 4a gives, for each vaue of n, the median runtime based on 100 random probem instances, for the compete procedure soid curve, and when ony considering conditioning sets C with two or fewer eements dotted curve. Note that most instances invoving a reativey sma number of nodes on the order of 10 or ess can be soved by the compete procedure in minutes, if not seconds. We emphasize that these are not trivia probems: No other existing causa discovery procedure can hande our mode space aowing both atents and cyces, nor our very genera experimenta setup overapping data sets incuding interventions. At the same time, it is quite cear that, at east in its current impementation, the method does not scae to much arger numbers of variabes. Scaabiity coud be achieved with a more efficient search for unknown d-separations in Step 1 of the agorithm. An effective way to reduce the run-time of the agorithm is to imit the size of the conditioning sets considered dotted ine in Figure 4a. Whie this means that competeness is not guaranteed, Figure 4b shows that in most cases very itte is ost in terms of identifiabiity. We randomy samped 100 probem instances as above, except that we now fixed the number of nodes to n = 8. The red soid curve shows the proportion of true directed edges i.e. x y in the true graph which were identified as a direct edge as opposed to unknown, since no errors are made. Simiary, the red dashed curve shows the identification of absences of direct edges, and the remaining curves indicate the amount of bidirected edges and existence of directed paths ancestra reationships identified. A key observation is that tests of higher order roughy C 3 provide very itte additiona information over those invoving smaer conditioning sets. Finay, we investigated the extent to which our very genera mode space aowing both cyces and atents is detrimenta to identification when the true mode is more restricted. We generated a tota of 300 random probem instances using the same procedure as above, each with n = 8 nodes, where the first 100 modes were restricted to being acycic, the second 100 were restricted to contain no atents i.e. no edges of the form x y in the true graph over V, and the remaining

8 a b true mode true mode true mode run-time [s] amount identified [%] amount identified [%] presences absences n max C Figure 4: a Median run-time of the procedure as a function of the tota number of nodes in the mode. The dotted ine gives the median run-time when restricting to max C = 2. b Proportion of edges soid ines and absences of edges dashed ines identified, as a function of max C. Directed edges are shown in red, bidirected edges confounders in green, and directed paths ancestra reationships in bue. 100 were both acycic and contained no atents. Figure 5 shows the proportion of direct edges identified, and the proportion of absences of direct edges identified, as a function of the assumptions used assuming an acycic mode, assuming no atents, assuming both, or assuming neither. The genera message is that very itte identifiabiity seems to be ost when assuming the more genera mode spaces in this experimenta setup. 7 DISCUSSION By focusing excusivey on d-separation and d-connection reations obtained without errors we have so far taken the approach used by other constraint-based agorithms in the iterature PC, FCI, CCD, ION, IOD, csat+ etc. to separate the causa from the statistica inference. As an important direction for future work, we now briefy discuss integrating statistica inference. In most reaistic situations d-separation/connection reations are determined by independence tests from statistica data. Such tests, especiay when performed in arge numbers, produce errors due to the finite number of sampes avaiabe and probems of mutipe testing. A other constraint-based causa discovery agorithms face simiar probems. In our case, the errors can resut in d-separation/connection reations that are contradictory. Since the ogica encoding is simpy unsatisfiabe in such cases, no output is given. But there are more interesting features of the encoding and the agorithm that hod promise to be usefu with actua statistica data. First, since no definite answer is required of a d-separation test, we can enforce different p-vaue threshods to detect independencies and dependencies see Tsamardinos et a If assumptions assumptions assumptions Figure 5: Proportion of directed edge presences and absences identified, under various mode space assumptions, for acycic true modes without atents eft, acycic modes with atents center, and cycic modes without atents right. a p-vaue of a test fas between the threshods, the d- separation reation can be treated as unknown, by just not adding any constraints into the working formua ϕ. This approach does not competey avoid conficts, but reduces their number and aows for at east some more contro than many extant agorithms are abe to offer. A second approach to deaing with statistica issues woud be to expoit extensions of SAT, especiay Booean optimization in terms of maximum satisfiabiity MaxSAT of propositiona formuas Biere et a., 2009, where the task is to find a truth assignment that satisfies the maximum number of CNF causes. Hence a MaxSAT sover coud be used for discovering causa modes that entai a minima number of contradictory d-separation/connection reations in the input. 8 CONCLUSION We presented a causa discovery procedure for a very genera mode space: to our knowedge, it is currenty the ony nonparametric causa discovery agorithm that aows for a mode space that incudes graphs with cyces and atent confounders reca the discussion on cyces and d-separation in Section 1. The agorithm can be appied to overapping data sets, whether they are experimenta or passive observationa, and can incorporate a arge variety of different background information if avaiabe. It does not depend on parametric restrictions such as inearity Hyttinen et a., 2012, and requires ony the abiity to test for d- separation/connection reations. SAT-based procedures have been previousy proposed for the more restricted space of acycic causa modes Triantafiou et a., 2010; Borboudakis and Tsamardinos, However, ours is the first procedure that is compete with respect to overapping surgica experiments, and additionay handes a mode

9 space that aows for cyces. In order to capture the more genera mode space, we empoy a nove ogica encoding of d-separation and d-connection reations. The Booean constraints for individua reations are generated iterativey and ony on demand during the execution of our procedure, and an incrementa SAT sover is used for iterativey computing the backbone of the Booean constraints. Our procedure can aso be easiy used for the more restricted mode spaces by introducing additiona Booean constraints. By constraining the mode space to causay sufficient or acycic causa structures we can perform the inferences of the standard agorithms in the iterature, such as PC, FCI, ION, IOD, csat+ and CCD for moderatey sized graphs. The inferences made are compete in the most genera and in the more restricted settings. A PRUNING HEURISTICS In the intermediary soution S describing our current knowedge some edges are present, some are absent and the presence of some edges is unknown. We consider two graphs G 1 and G 2, such that they agree on a the edges that are determined, but G 1 omits a undetermined edges, whie G 2 incudes a undetermined edges as present. As removing edges can ony resut in more d-separation reations, a d-connection reation present in G 1 must be present in a soutions. Simiary, a d-separation reation present in G 2 must be present in a possibe soutions. Ony the remaining tests are possiby informative. This is a safe heuristic that turns out to be computationay feasibe, as forward cacuation of d-separation/connection reations for a fuy defined graph is fast for the mode sizes we are considering. In addition, we aso omit tests with conditioning sets that contain nodes that cannot be on a d-connecting path between the nodes in question. B LIMIT ON THE PATH LENGTH Written soey in terms of edge variabes, the righthand side of Eq. 1 is a arge disjunction of d-connecting paths up to ength max for the reation on the efthand side. As a path of arbitrary ength can be d- connecting, max shoud be infinite to guarantee soundness of the formuation. Here we show that ony paths of engths up to a certain upper bound need to be considered. The foowing emma, proven at the end of this appendix, is essentia in showing this. Lemma 1 If there exists a path that is d-connecting with respect to x \ y C J and onger than 2 V C J {x, y} 1 edges, then there exists a shorter path that is d-connecting with respect to the same reation. Consider a path p ong that is d-connecting for x y C J and onger than 2 V C J {x, y} 1. By Lemma 1 there is a path p short with at most ength 2 V C J {x, y} 1 edges that is d-connecting with respect to the same reation. Now the expanded version of the right hand side of Equation 1 has the form:... [p short ] [p ong ].... The ony situation where such a constraint may have a different vaue than... [p short ]... is when p ong exists and p short does not. This is impossibe by the construction of p short using Lemma 1. We can thus ignore [p ong ] and a paths onger than 2 V C J {x, y} 1. We can set max = 2 V 4, since if C =, then paths have at most ength V 1. Proof of Lemma 1 The foowing six rues aways give a shorter d-connecting path with respect to the same reation. The deeted part is underined on the eft. Circes indicate arrowhead or tai. x x y x y 10 x y y x y 11 >z< >z< >z< 12 >z z >z 13 z z< z< 14 z z z 15 The rues impy that if a midde node z appears three times on a d-connecting path, the path wi necessariy have at east one of the forms on the eft in A path can never be d-connecting if the same node appears both as a coider and a non-coider somewhere on the path. Thus a node can appear at most two times in paths that cannot be shortened. First, consider the case with no coiders on the path. The ony situation where a d-connecting path cannot be shortened and a node appears twice, occurs when the path has the form >z z<. This path cannot be d-connecting without a coider between the instances of z. Thus, without coiders a path that cannot be shortened has at most V nodes and thus ength V 1. Second, if the path cannot be shortened, each node in C J {x, y} can appear at most once due to The remaining nodes can appear at most twice. This makes a tota of 2 V C J {x, y} nodes. Hence the ength of the path is at most 2 V C J {x, y} 1. Acknowedgements This research was supported by the Academy of Finand under grants and POH, and MJ, by HIIT AH and by the James S. McDonne Foundation FE.

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