Using background knowledge for the estimation of total causal e ects
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1 Using background knowledge for the estimation of total causal e ects Interpreting and using CPDAGs with background knowledge Emilija Perkovi, ETH Zurich Joint work with Markus Kalisch and Marloes Maathuis Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 1 / 26
2 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: DAG D. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 2 / 26
3 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: DAG D. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 3 / 26
4 Goal Estimate the total causal e ect of X on Y do(x): anoutsideinterventionthatsetsvariablesx to x. Observational data Randomized control studies Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 4 / 26
5 Goal Estimate the total causal e ect of X on Y -theaveragechangeiny due to do(x) - do(x): anoutsideinterventionthatsetsvariablesx to x. Observational data Randomized control studies Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 4 / 26
6 Goal Estimate the total causal e ect of X on Y -theaveragechangeiny due to do(x) - from observational data. do(x): anoutsideinterventionthatsetsvariablesx to x. Observational data Randomized control studies Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 4 / 26
7 Framework Observational data Learn the causal structure Causal graph Graphically find Estimate total causal e ect Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 5 / 26
8 Framework Observational data Learn the causal structure Causal graph Graphically find There is no Estimate total causal e ect Estimate the set of possible total causal e ects Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 5 / 26
9 Framework Observational data: Learn the causal structure Causal graph: Graphically find GAC, GBC, AC, BC Estimate total causal e ect e.g., Y X + S no cycles, no latent variables. PC, GES, ARGES DAG, CPDAG. There is no IDA, joint-ida Estimate the set of possible total causal e ects PC (Spirtes et al, 1993), GES (Chickering, 2002), ARGES (Nandy et al, 2016). Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 6 / 26
10 Framework Observational data: Learn the causal structure Causal graph: Graphically find GAC, GBC, AC, BC Estimate total causal e ect e.g., Y X + S no cycles, no latent variables. PC, GES, ARGES DAG, CPDAG. There is no IDA, joint-ida Estimate the set of possible total causal e ects BC (Pearl, 1993), AC (Shpitser et al. 2012; van der Zander et al. 2014), GAC (Perkovic et al, 2015,2017a), GBC (Maathuis and Colombo, 2016). IDA (Maathuis et al., 2009), joint-ida (Nandy et al, 2017). Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 7 / 26
11 Framework Observational data: Learn the causal structure Causal graph: Graphically find GAC, GBC, AC, BC Estimate total causal e ect e.g., Y X + S no cycles, no latent variables. PC, GES, ARGES DAG, CPDAG. There is no IDA, joint-ida Estimate the set of possible total causal e ects Causal e ects are often estimated by adjusted regression. Adjustment sets depend on the causal structure, which can be represented by a graph. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 8 / 26
12 Framework Observational data: Learn the causal structure Causal graph: Graphically find GAC, GBC, AC, BC Estimate total causal e ect e.g., Y X + S no cycles, no latent variables. PC, GES, ARGES DAG, CPDAG. There is no IDA, joint-ida Estimate the set of possible total causal e ects Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 9 / 26
13 Framework Observational data: Learn the causal structure Causal graph: Graphically find GAC, GBC, AC, BC Estimate total causal e ect e.g., Y X + S no cycles, no latent variables. PC, GES, ARGES DAG, CPDAG. There is no IDA, joint-ida Estimate the set of possible total causal e ects If the total causal e ect is di erent between DAGs in the equivalence class, then no adjustment set. Then often the set of possible total causal e ects will contain zero. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 9 / 26
14 Framework Observational data Background knowledge Learn the causal structure with background knowledge Causal graph Graphically find?? Estimate total causal e ect Estimate the set of possible total causal e ects How much does background knowledge help to identify total causal e ects? Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 10 / 26
15 Sources of background knowledge Applications - Expert knowledge of some causal relations, previous studies etc. Using a mix of observational and interventional data Model restrictions Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 11 / 26
16 Sources of background knowledge Applications - Expert knowledge of some causal relations, previous studies etc. (Meek, 1995; Scheines et al., 1998) Using a mix of observational and interventional data (Hauser and Bühlmann, 2012; Wang et al., 2017) Model restrictions (Hoyer et al., 2008; Ernest et al. 2016; Eigenmann et al., 2017) Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 11 / 26
17 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: DAG D. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 12 / 26
18 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: CPDAG C of DAG D. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 13 / 26
19 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: CPDAG C with background knowledge. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 14 / 26
20 Causal e ects Experience of supervisor Other PhD Students Conference attendance Field of research #ofmeetings per month #ofpublished papers Figure: CPDAG C with background knowledge. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 15 / 26
21 Framework Observational data Background knowledge Learn the causal structure with background knowledge PC, GES, PC LINGAM, GIES, IGSP, AGES Causal graph: DAG, maximal PDAG, CPDAG Graphically find?? Estimate total causal e ect Estimate the set of possible total causal e ects PC (Spirtes et al, 1993) or GES (Chickering, 2002) + background knowledge (Meek, 1995; TETRAD, Scheines et al., 1998) PC LINGAM (Hoyer et al., 2008), GIES (Hauser and Bühlmann, 2012), AGES (Eigenmann et al., 2017), IGSP (Wang et al., 2017). Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 16 / 26
22 Framework Observational data Background knowledge Learn the causal structure with background knowledge PC, GES, PC LINGAM, GIES, IGSP, AGES Causal graph: DAG, maximal PDAG, CPDAG Graphically find?? Estimate total causal e ect Estimate the set of possible total causal e ects Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 17 / 26
23 Adjustment criterion for maximal PDAGs Theorem (Perkovi et al, 2017b): S is an adjustment set relative to (X, Y) and G if and only if: Amenability G is b-amenable relative to (X, Y). Forbidden Set S does not contain nodes in b-forbidden(x, Y, G). Blocking S blocks all proper b-non-causal definite status paths from X to Y in G. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 18 / 26
24 Adjustment criterion for maximal PDAGs Theorem (Perkovi et al, 2017b): S is an adjustment set relative to (X, Y) and G if and only if: Amenability G is b-amenable relative to (X, Y). Forbidden Set S does not contain nodes in b-forbidden(x, Y, G). Blocking S blocks all proper b-non-causal definite status paths from X to Y in G. Maximal PDAGs can contain partially directed cycles and do not have a chordal undirected component. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 18 / 26
25 Adjustment criterion for maximal PDAGs Theorem (Perkovi et al, 2017b): S is an adjustment set relative to (X, Y) and G if and only if: Amenability G is b-amenable relative to (X, Y). Forbidden Set S does not contain nodes in b-forbidden(x, Y, G). Blocking S blocks all proper b-non-causal definite status paths from X to Y in G. In a linear setting the total causal e ect of X on Y is then the linear regression coe cient of X in the regression Y X + S. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 19 / 26
26 Does an adjustment set always exist? 1.0 Fraction of identifiable effects via adjustment Randomly sampled DAGs: p 2{20, 30,...,100}, E[N] 2{3, 4,...,10}. X -randomlychosen, Y -connectedtox and Y 6! X. 0.7 TrueEffect All Non zero Proportion of background knowledge Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 20 / 26
27 Framework Observational data Background knowledge Learn the causal structure with background knowledge PC, GES, PC LINGAM, GIES, IGSP, AGES Causal graph: DAG, maximal PDAG, CPDAG Graphically find X There is no? Estimate total causal e ect E ciently estimate the set of possible total causal e ects Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 21 / 26
28 Framework Observational data Background knowledge Learn the causal structure with background knowledge PC, GES, PC LINGAM, GIES, IGSP, AGES Causal graph: DAG, maximal PDAG, CPDAG Graphically find X There is no? Estimate total causal e ect E ciently estimate the set of possible total causal e ects Modify IDA and joint-ida framework. Assume a linear Gaussian generating mechanism. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 21 / 26
29 IDA and joint-ida in maximal PDAGs Use sets of direct causes (parents) of X to estimate all possible total causal e ect of X on Y. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 22 / 26
30 IDA and joint-ida in maximal PDAGs Use sets of direct causes (parents) of X to estimate all possible total causal e ect of X on Y. Find all sets of parents of X in G Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 22 / 26
31 IDA and joint-ida in maximal PDAGs Use sets of direct causes (parents) of X to estimate all possible total causal e ect of X on Y. Find all sets of parents of X in G in an e cient way. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 22 / 26
32 IDA and joint-ida in maximal PDAGs Use sets of direct causes (parents) of X to estimate all possible total causal e ect of X on Y. Find all sets of parents of X in G in an e cient way. The same local algorithm cannot be used with added background knowledge, due to partially directed cycles. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 22 / 26
33 IDA and joint-ida in maximal PDAGs Use sets of direct causes (parents) of X to estimate all possible total causal e ect of X on Y. Find all sets of parents of X in G in an e cient way. The same local algorithm cannot be used with added background knowledge, due to partially directed cycles. Runtime Median Mean Max IDA w/o bg IDA w bg Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 22 / 26
![Identifiability gain with background knowledge. 4315 out of 20 000 sampled DAGs: p 2{20, 30,...,100}, E[N] 2{3, 4,.](/docs-images/94/118733359/images/34-0.jpg "..,10}, n = 200. X -randomlychosen, Y -connectedtox and Y 6! X. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 23 / 26")
34 Identifiability gain with background knowledge out of sampled DAGs: p 2{20, 30,...,100}, E[N] 2{3, 4,...,10}, n = 200. X -randomlychosen, Y -connectedtox and Y 6! X. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 23 / 26
35 Implementation Algorithms implemented in R package pcalg on CRAN: Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 24 / 26
36 Implementation Algorithms implemented in R package pcalg on CRAN: addbgknowledge(g, X, Y) Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 24 / 26
37 Implementation Algorithms implemented in R package pcalg on CRAN: addbgknowledge(g, X, Y) isvalidgraph(g,graph.type) Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 24 / 26
38 Implementation Algorithms implemented in R package pcalg on CRAN: addbgknowledge(g, X, Y) isvalidgraph(g,graph.type) gac(g,x,y,s,graph.type) adjustment(g,graph.type,x,y,set.type) Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 24 / 26
39 Implementation Algorithms implemented in R package pcalg on CRAN: addbgknowledge(g, X, Y) isvalidgraph(g,graph.type) gac(g,x,y,s,graph.type) adjustment(g,graph.type,x,y,set.type) ida(x, Y,cov.mat,G,graph.type) jointida(x, Y,cov.mat,technique, G, graph.type) Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 24 / 26
40 Our contribution Observational data Background knowledge Learn the causal structure with background knowledge PC, GES, PC LINGAM, GIES, IGSP, AGES Causal graph: DAG, maximal PDAG, CPDAG Graphically find X There is no X Estimate total causal e ect E ciently estimate the set of possible total causal e ects A necessary and su cient graphical adjustment criterion for maximal PDAGs. E cient IDA and joint-ida algorithms for maximal PDAGs. Implemented new algorithms and extended the existing algorithms in pcalg. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 25 / 26
41 Thanks! See you at the poster session! Joint work with Marloes Maathuis, Markus Kalisch References: Perkovi, Kalisch and Maathuis (2017b). Interpreting and using CPDAGs with background knowledge UAI Perkovi, Textor, Kalisch and Maathuis (2015). A complete generalized adjustment criterion. UAI Perkovi, Textor, Kalisch and Maathuis (2017a). Complete graphical characterization and construction of adjustment sets in Markov equivalence classes of ancestral graphs. arxiv: , to appear in JMLR. Emilija Perkovi, ETH Zurich Using CPDAGs with background knowledge 26 / 26
Interpreting and using CPDAGs with background knowledge
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