Optimize backdoor criterion identification for large graphs (#342)

* Used the d-separation algorithm from networkx
* Updated backdoor identification to be faster
* Refactor code for the new function
* Changed the logic for unobserved confounding to simply remove the unobserved confounder when returning a backdoor set. Simplifies the logic.

dowhy/causal_graph.py

@@ -159,8 +159,11 @@ class CausalGraph:
                 if node_name not in common_cause_names:
                     for outcome in self.outcome_name:
                         self._graph.add_node(node_name, observed="yes")
-                        self._graph.add_edge(node_name, outcome, style = "dotted", headport="s", tailport="n")
-                        self._graph.add_edge(outcome, node_name, style = "dotted", headport="n", tailport="s") # TODO make the ports more general so that they apply not just to top-bottom node configurations
+                        # Assuming the simple form of effect modifier
+                        # that directly causes the outcome.
+                        self._graph.add_edge(node_name, outcome)
+                        #self._graph.add_edge(node_name, outcome, style = "dotted", headport="s", tailport="n")
+                        #self._graph.add_edge(outcome, node_name, style = "dotted", headport="n", tailport="s") # TODO make the ports more general so that they apply not just to top-bottom node configurations
         if mediator_names is not None:
             for node_name in mediator_names:
                 for treatment, outcome in itertools.product(self.treatment_name, self.outcome_name):
@@ -209,6 +212,7 @@ class CausalGraph:
 
     def do_surgery(self, node_names, remove_outgoing_edges=False,
                    remove_incoming_edges=False):
+        node_names = parse_state(node_names)
         new_graph = self._graph.copy()
         for node_name in node_names:
             if remove_outgoing_edges:
@@ -236,15 +240,39 @@ class CausalGraph:
             causes = causes.union(self.get_ancestors(v, new_graph=new_graph))
         return causes
 
-    def check_valid_backdoor_set(self, nodes1, nodes2, nodes3, backdoor_paths=None):
+    def check_dseparation(self, nodes1, nodes2, nodes3, new_graph=None,
+            dseparation_algo="default"):
+        if dseparation_algo == "default":
+            if new_graph is None:
+                new_graph = self._graph
+            dseparated = nx.algorithms.d_separated(new_graph,
+                    set(nodes1), set(nodes2), set(nodes3))
+        else:
+            raise ValueError(f"{dseparation_algo} method for d-separation not supported.")
+        return dseparated
+
+    def check_valid_backdoor_set(self, nodes1, nodes2, nodes3,
+            backdoor_paths=None, new_graph=None, dseparation_algo="default"):
+        """ Assume that the first parameter (nodes1) is the treatment,
+        the second is the outcome, and the third is the candidate backdoor set
+        """
         # also return the number of backdoor paths blocked by observed nodes
-        if backdoor_paths is None:
-            backdoor_paths = self.get_backdoor_paths(nodes1, nodes2)
-        d_separated = all([self.is_blocked(path, nodes3) for path in backdoor_paths])
-        observed_nodes3 = self.filter_unobserved_variables(nodes3)
-        num_paths_blocked = sum([self.is_blocked(path, observed_nodes3) for path in backdoor_paths])
-        return {'is_dseparated': d_separated,
-                'num_paths_blocked_by_observed_nodes': num_paths_blocked}
+        if dseparation_algo == "default":
+            if new_graph is None:
+                # Assume that nodes1 is the treatment
+                new_graph = self.do_surgery(nodes1,
+                    remove_outgoing_edges=True)
+            dseparated = nx.algorithms.d_separated(new_graph,
+                    set(nodes1), set(nodes2), set(nodes3))
+        elif dseparation_algo == "naive":
+            # ignores new_graph parameter, always uses self._graph
+            if backdoor_paths is None:
+                backdoor_paths = self.get_backdoor_paths(nodes1, nodes2)
+            dseparated = all([self.is_blocked(path, nodes3) for path in backdoor_paths])
+        else:
+            raise ValueError(f"{dseparation_algo} method for d-separation not supported.")
+        return {'is_dseparated': dseparated}
+
 
     def get_backdoor_paths(self, nodes1, nodes2):
         paths = []
@@ -406,8 +434,9 @@ class CausalGraph:
 
         Currently only supports singleton sets.
         """
-        dpaths = self.get_all_directed_paths(nodes1, nodes2)
-        return len(dpaths) > 0
+        #dpaths = self.get_all_directed_paths(nodes1, nodes2)
+        #return len(dpaths) > 0
+        return nx.has_path(self._graph, nodes1[0], nodes2[0])
 
     def get_adjacency_matrix(self, *args, **kwargs):
         '''
@@ -416,14 +445,25 @@ class CausalGraph:
         '''
         return nx.convert_matrix.to_numpy_matrix(self._graph, *args, **kwargs)
 
-    def check_valid_frontdoor_set(self, nodes1, nodes2, candidate_nodes, frontdoor_paths=None):
+    def check_valid_frontdoor_set(self, nodes1, nodes2, candidate_nodes,
+            frontdoor_paths=None, new_graph = None,
+            dseparation_algo="default"):
         """Check if valid the frontdoor variables for set of treatments, nodes1 to set of outcomes, nodes2.
         """
-        if frontdoor_paths is None:
-            frontdoor_paths = self.get_all_directed_paths(nodes1, nodes2)
+        # Condition 1: node 1 ---> node 2 is intercepted by candidate_nodes
+        if dseparation_algo == "default":
+            if new_graph is None:
+                new_graph = self._graph
+            dseparated = nx.algorithms.d_separated(new_graph,
+                    set(nodes1), set(nodes2), set(candidate_nodes))
+        elif dseparation_algo == "naive":
+            if frontdoor_paths is None:
+                frontdoor_paths = self.get_all_directed_paths(nodes1, nodes2)
 
-        d_separated = all([self.is_blocked(path, candidate_nodes) for path in frontdoor_paths])
-        return d_separated
+            dseparated = all([self.is_blocked(path, candidate_nodes) for path in frontdoor_paths])
+        else:
+            raise ValueError(f"{dseparation_algo} method for d-separation not supported.")
+        return dseparated
 
     def check_valid_mediation_set(self, nodes1, nodes2, candidate_nodes, mediation_paths=None):
         """Check if candidate nodes are valid mediators for set of treatments, nodes1 to set of outcomes, nodes2.

dowhy/causal_identifier.py

@@ -30,7 +30,7 @@ class CausalIdentifier:
     BACKDOOR_MIN="minimal-adjustment"
     BACKDOOR_MAX="maximal-adjustment"
     METHOD_NAMES = {BACKDOOR_DEFAULT, BACKDOOR_EXHAUSTIVE, BACKDOOR_MIN, BACKDOOR_MAX}
-    DEFAULT_BACKDOOR_METHOD = BACKDOOR_MAX
+    DEFAULT_BACKDOOR_METHOD = BACKDOOR_DEFAULT
 
     def __init__(self, graph, estimand_type,
             method_name = "default",
@@ -249,19 +249,28 @@ class CausalIdentifier:
         )
         return estimand
 
-    def identify_backdoor(self, treatment_name, outcome_name, include_unobserved=True):
+    def identify_backdoor(self, treatment_name, outcome_name,
+            include_unobserved=True, dseparation_algo="default"):
         backdoor_sets = []
-        backdoor_paths = self._graph.get_backdoor_paths(treatment_name, outcome_name)
+        backdoor_paths = None
+        bdoor_graph = None
+        if dseparation_algo == "naive":
+            backdoor_paths = self._graph.get_backdoor_paths(treatment_name, outcome_name)
+        elif dseparation_algo == "default":
+            bdoor_graph = self._graph.do_surgery(treatment_name,
+                    remove_outgoing_edges=True)
+        else:
+            raise ValueError(f"d-separation algorithm {dseparation_algo} is not supported")
         method_name = self.method_name if self.method_name != CausalIdentifier.BACKDOOR_DEFAULT else CausalIdentifier.DEFAULT_BACKDOOR_METHOD
 
         # First, checking if empty set is a valid backdoor set
         empty_set = set()
-        check = self._graph.check_valid_backdoor_set(treatment_name, outcome_name, empty_set,
-                backdoor_paths=backdoor_paths)
+        check = self._graph.check_valid_backdoor_set(treatment_name,
+                outcome_name, empty_set,
+                backdoor_paths=backdoor_paths, new_graph=bdoor_graph,
+                dseparation_algo=dseparation_algo)
         if check["is_dseparated"]:
-            backdoor_sets.append({
-                'backdoor_set':empty_set,
-                'num_paths_blocked_by_observed_nodes': check["num_paths_blocked_by_observed_nodes"]})
+            backdoor_sets.append({'backdoor_set':empty_set})
             # If the method is `minimal-adjustment`, return the empty set right away.
             if method_name == CausalIdentifier.BACKDOOR_MIN:
                 return backdoor_sets
@@ -271,33 +280,75 @@ class CausalIdentifier:
             - set(treatment_name) \
             - set(outcome_name)
         eligible_variables -= self._graph.get_descendants(treatment_name)
-
-        num_iterations = 0
-        found_valid_adjustment_set = False
+        # If var is d-separated from both treatment or outcome, it cannot
+        # be a part of the backdoor set
+        filt_eligible_variables = set()
+        for var in eligible_variables:
+            dsep_treat_var = self._graph.check_dseparation(
+                    treatment_name, parse_state(var),
+                    set())
+            dsep_outcome_var = self._graph.check_dseparation(
+                    outcome_name, parse_state(var), set())
+            if not dsep_outcome_var or not dsep_treat_var:
+                filt_eligible_variables.add(var)
         if method_name in CausalIdentifier.METHOD_NAMES:
-            # If `minimal-adjustment` method is specified, start the search from the set with minimum size. Otherwise, start from the largest.
-            set_sizes = range(1, len(eligible_variables) + 1, 1) if method_name == CausalIdentifier.BACKDOOR_MIN else range(len(eligible_variables), 0, -1)
-            for size_candidate_set in set_sizes:
-                for candidate_set in itertools.combinations(eligible_variables, size_candidate_set):
-                    check = self._graph.check_valid_backdoor_set(treatment_name,
-                            outcome_name, candidate_set, backdoor_paths=backdoor_paths)
-                    self.logger.debug("Candidate backdoor set: {0}, is_dseparated: {1}, No. of paths blocked by observed_nodes: {2}".format(candidate_set, check["is_dseparated"], check["num_paths_blocked_by_observed_nodes"]))
-                    if check["is_dseparated"]:
-                        backdoor_sets.append({
-                            'backdoor_set': candidate_set,
-                            'num_paths_blocked_by_observed_nodes': check["num_paths_blocked_by_observed_nodes"]})
-                        found_valid_adjustment_set = True
-                    num_iterations += 1
-                    if method_name == CausalIdentifier.BACKDOOR_EXHAUSTIVE and num_iterations > CausalIdentifier.MAX_BACKDOOR_ITERATIONS:
-                        break
-                # If the backdoor method is `maximal-adjustment` or `minimal-adjustment`, return the first found adjustment set.
-                if method_name in {CausalIdentifier.BACKDOOR_MAX, CausalIdentifier.BACKDOOR_MIN} and found_valid_adjustment_set:
-                    break
+            backdoor_sets, found_valid_adjustment_set = self.find_valid_adjustment_sets(
+                    treatment_name, outcome_name,
+                    backdoor_paths, bdoor_graph,
+                    dseparation_algo,
+                    backdoor_sets, filt_eligible_variables,
+                    method_name=method_name,
+                    max_iterations= CausalIdentifier.MAX_BACKDOOR_ITERATIONS)
+            if method_name == CausalIdentifier.BACKDOOR_DEFAULT and found_valid_adjustment_set:
+                # repeat the above search with BACKDOOR_MIN
+                backdoor_sets, _ = self.find_valid_adjustment_sets(
+                        treatment_name, outcome_name,
+                        backdoor_paths, bdoor_graph,
+                        dseparation_algo,
+                        backdoor_sets, filt_eligible_variables,
+                        method_name=CausalIdentifier.BACKDOOR_MIN,
+                        max_iterations= CausalIdentifier.MAX_BACKDOOR_ITERATIONS)
         else:
             raise ValueError(f"Identifier method {method_name} not supported. Try one of the following: {CausalIdentifier.METHOD_NAMES}")
 
         return backdoor_sets
 
+    def find_valid_adjustment_sets(self, treatment_name, outcome_name,
+            backdoor_paths, bdoor_graph, dseparation_algo,
+            backdoor_sets, filt_eligible_variables,
+            method_name, max_iterations):
+        num_iterations = 0
+        found_valid_adjustment_set = False
+        # If `minimal-adjustment` method is specified, start the search from the set with minimum size. Otherwise, start from the largest.
+        set_sizes = range(1, len(filt_eligible_variables) + 1, 1) if method_name == CausalIdentifier.BACKDOOR_MIN else range(len(filt_eligible_variables), 0, -1)
+        for size_candidate_set in set_sizes:
+            for candidate_set in itertools.combinations(filt_eligible_variables, size_candidate_set):
+                check = self._graph.check_valid_backdoor_set(treatment_name,
+                        outcome_name, candidate_set,
+                        backdoor_paths=backdoor_paths,
+                        new_graph = bdoor_graph,
+                        dseparation_algo = dseparation_algo)
+                self.logger.debug("Candidate backdoor set: {0}, is_dseparated: {1}".format(candidate_set, check["is_dseparated"]))
+                if check["is_dseparated"]:
+                    backdoor_sets.append({'backdoor_set': candidate_set})
+                    found_valid_adjustment_set = True
+                num_iterations += 1
+                if method_name == CausalIdentifier.BACKDOOR_EXHAUSTIVE and num_iterations > max_iterations:
+                    self.logger.warning(f"Max number of iterations {max_iterations} reached.")
+                    break
+            # If the backdoor method is `maximal-adjustment` or `minimal-adjustment`, return the first found adjustment set.
+            if method_name in {CausalIdentifier.BACKDOOR_DEFAULT, CausalIdentifier.BACKDOOR_MAX, CausalIdentifier.BACKDOOR_MIN} and found_valid_adjustment_set:
+                break
+            # If all variables are observed, and the biggest eligible set
+            # does not satisfy backdoor, then none of its subsets will.
+            if method_name in {CausalIdentifier.BACKDOOR_DEFAULT, CausalIdentifier.BACKDOOR_MAX} and self._graph.all_observed(filt_eligible_variables):
+                break
+            if num_iterations > max_iterations:
+                self.logger.warning(f"Max number of iterations {max_iterations} reached. Could not find a valid backdoor set.")
+                break
+        return backdoor_sets, found_valid_adjustment_set
+
+
     def get_default_backdoor_set_id(self, backdoor_sets_dict):
         # Adding a None estimand if no backdoor set found
         if len(backdoor_sets_dict) == 0:
@@ -310,12 +361,12 @@ class CausalIdentifier:
         min_iv_keys = {key for key, iv_count in iv_count_dict.items() if iv_count == min_iv_count}
         min_iv_backdoor_sets_dict = {key: backdoor_sets_dict[key] for key in min_iv_keys}
 
-        # Default set is the one with the most number of adjustment variables (optimizing for minimum (unknown) bias not for efficiency)
-        max_set_length = -1
+        # Default set is the one with the least number of adjustment variables (optimizing for efficiency)
+        min_set_length = 1000000
         default_key = None
         for key, bdoor_set in min_iv_backdoor_sets_dict.items():
-            if len(bdoor_set) > max_set_length:
-                max_set_length = len(bdoor_set)
+            if len(bdoor_set) < min_set_length:
+                min_set_length = len(bdoor_set)
                 default_key = key
         return default_key
 
@@ -328,11 +379,12 @@ class CausalIdentifier:
             proceed_when_unidentifiable = self._proceed_when_unidentifiable
         is_identified = [ self._graph.all_observed(bset["backdoor_set"]) for bset in backdoor_sets ]
 
-        if all(is_identified):
-            self.logger.info("All common causes are observed. Causal effect can be identified.")
+        if any(is_identified):
+            self.logger.info("Causal effect can be identified.")
             backdoor_sets_arr = [list(
                 bset["backdoor_set"])
-                for bset in backdoor_sets]
+                for bset in backdoor_sets
+                if self._graph.all_observed(bset["backdoor_set"]) ]
         else: # there is unobserved confounding
             self.logger.warning("If this is observed data (not from a randomized experiment), there might always be missing confounders. Causal effect cannot be identified perfectly.")
             response = False # user response
@@ -349,11 +401,11 @@ class CausalIdentifier:
                     self.logger.warn("Identification failed due to unobserved variables.")
                     backdoor_sets_arr = []
             if proceed_when_unidentifiable or response is True:
-                max_paths_blocked = max( bset['num_paths_blocked_by_observed_nodes'] for bset in backdoor_sets)
-                backdoor_sets_arr = [list(
-                    self._graph.filter_unobserved_variables(bset["backdoor_set"]))
-                    for bset in backdoor_sets
-                    if bset["num_paths_blocked_by_observed_nodes"]==max_paths_blocked]
+                # Just removing the unobserved variable
+                backdoor_sets_arr = []
+                for bset in backdoor_sets:
+                    curr_set = list(self._graph.filter_unobserved_variables(bset["backdoor_set"]))
+                    backdoor_sets_arr.append(curr_set)
 
         for i in range(len(backdoor_sets_arr)):
             backdoor_estimand_expr = self.construct_backdoor_estimand(
@@ -364,20 +416,59 @@ class CausalIdentifier:
             backdoor_variables_dict["backdoor"+str(i+1)] = backdoor_sets_arr[i]
         return estimands_dict, backdoor_variables_dict
 
-    def identify_frontdoor(self):
+    def identify_frontdoor(self, dseparation_algo="default"):
         """ Find a valid frontdoor variable if it exists.
 
         Currently only supports a single variable frontdoor set.
         """
         frontdoor_var = None
-        frontdoor_paths = self._graph.get_all_directed_paths(self.treatment_name, self.outcome_name)
+        frontdoor_paths = None
+        fdoor_graph = None
+        if dseparation_algo == "default":
+            cond1_graph = self._graph.do_surgery(self.treatment_name,
+                    remove_incoming_edges=True)
+            bdoor_graph1 = self._graph.do_surgery(self.treatment_name,
+                    remove_outgoing_edges=True)
+        elif dseparation_algo == "naive":
+            frontdoor_paths = self._graph.get_all_directed_paths(self.treatment_name, self.outcome_name)
+        else:
+            raise ValueError(f"d-separation algorithm {dseparation_algo} is not supported")
+
+
         eligible_variables = self._graph.get_descendants(self.treatment_name) \
-            - set(self.outcome_name)
+            - set(self.outcome_name) \
+            - set(self._graph.get_descendants(self.outcome_name))
         # For simplicity, assuming a one-variable frontdoor set
         for candidate_var in eligible_variables:
-            is_valid_frontdoor = self._graph.check_valid_frontdoor_set(self.treatment_name,
-                    self.outcome_name, parse_state(candidate_var), frontdoor_paths=frontdoor_paths)
-            self.logger.debug("Candidate frontdoor set: {0}, is_dseparated: {1}".format(candidate_var, is_valid_frontdoor))
+            # Cond 1: All directed paths intercepted by candidate_var
+            cond1 = self._graph.check_valid_frontdoor_set(
+                self.treatment_name, self.outcome_name,
+                parse_state(candidate_var),
+                frontdoor_paths=frontdoor_paths,
+                new_graph=cond1_graph,
+                dseparation_algo=dseparation_algo)
+            self.logger.debug("Candidate frontdoor set: {0}, is_dseparated: {1}".format(candidate_var, cond1))
+            if not cond1:
+                continue
+            # Cond 2: No confounding between treatment and candidate var
+            cond2 = self._graph.check_valid_backdoor_set(
+                self.treatment_name, parse_state(candidate_var),
+                set(),
+                backdoor_paths=None,
+                new_graph= bdoor_graph1,
+                dseparation_algo=dseparation_algo)
+            if not cond2:
+                continue
+            # Cond 3: treatment blocks all confounding between candidate_var and outcome
+            bdoor_graph2 = self._graph.do_surgery(candidate_var,
+                    remove_outgoing_edges=True)
+            cond3 = self._graph.check_valid_backdoor_set(
+                parse_state(candidate_var), self.outcome_name,
+                self.treatment_name,
+                backdoor_paths=None,
+                new_graph= bdoor_graph2,
+                dseparation_algo=dseparation_algo)
+            is_valid_frontdoor = cond1 and cond2 and cond3
             if is_valid_frontdoor:
                 frontdoor_var = candidate_var
                 break

tests/causal_identifiers/base.py

@@ -8,6 +8,7 @@ class IdentificationTestGraphSolution(object):
 
     def __init__(self, graph_str, observed_variables, biased_sets, minimal_adjustment_sets, maximal_adjustment_sets):
         self.graph = CausalGraph("X", "Y", graph_str, observed_node_names=observed_variables)
+        self.graph_str = graph_str
         self.observed_variables = observed_variables
         self.biased_sets = biased_sets
         self.minimal_adjustment_sets = minimal_adjustment_sets

tests/causal_identifiers/example_graphs.py

@@ -6,7 +6,7 @@ Each example graph is contained of the following values:
     * graph_str - The graph string in GML format.
     * observed_variables - A list of observed variables in the graph. This will be used to test no unobserved variables are offered in the solution.
     * biased_sets - The sets that we shouldn't get in the output as they incur biased estimates of the causal effect.
-    * minimal_adjustment_sets - Sets of observed variables that should be returned when 'minimal-adjustment' is specified as the backdoor method. 
+    * minimal_adjustment_sets - Sets of observed variables that should be returned when 'minimal-adjustment' is specified as the backdoor method.
         If no adjustment is necessary given the graph, minimal adjustment set should be the empty set.
     * maximal_adjustment_sets - Sets of observed variables that should be returned when 'maximal-adjustment' is specified as the backdoor method.
 """
@@ -14,14 +14,14 @@ Each example graph is contained of the following values:
 TEST_GRAPH_SOLUTIONS = {
     # Example is selected from Pearl J. "Causality" 2nd Edition, from chapter 3.3.1 on backoor criterion.
     "pearl_backdoor_example_graph": dict(
-        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]  
+        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]
                         node[id "Z2" label "Z2"]
                         node[id "Z3" label "Z3"]
                         node[id "Z4" label "Z4"]
                         node[id "Z5" label "Z5"]
                         node[id "Z6" label "Z6"]
                         node[id "X" label "X"]
-                        node[id "Y" label "Y"]      
+                        node[id "Y" label "Y"]
                         edge[source "Z1" target "Z3"]
                         edge[source "Z1" target "Z4"]
                         edge[source "Z2" target "Z4"]
@@ -31,7 +31,7 @@ TEST_GRAPH_SOLUTIONS = {
                         edge[source "Z4" target "Y"]
                         edge[source "Z5" target "Y"]
                         edge[source "Z6" target "Y"]
-                        edge[source "X" target "Z6"]]    
+                        edge[source "X" target "Z6"]]
                     """,
         observed_variables = ["Z1", "Z2", "Z3", "Z4", "Z5", "Z6", "X", "Y"],
         biased_sets = [{"Z4"}, {"Z6"}, {"Z5"}, {"Z2"}, {"Z1"}, {"Z3"}, {"Z1", "Z3"}, {"Z2", "Z5"}, {"Z1", "Z2"}],
@@ -39,22 +39,22 @@ TEST_GRAPH_SOLUTIONS = {
         maximal_adjustment_sets = [{"Z1", "Z2", "Z3", "Z4", "Z5"}]
     ),
     "simple_selection_bias_graph": dict(
-        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]  
+        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]
                     node[id "X" label "X"]
-                    node[id "Y" label "Y"]      
+                    node[id "Y" label "Y"]
                     edge[source "X" target "Y"]
                     edge[source "X" target "Z1"]
                     edge[source "Y" target "Z1"]]
                     """,
         observed_variables = ["Z1", "X", "Y"],
-        biased_sets = [{"Z1",}], 
+        biased_sets = [{"Z1",}],
         minimal_adjustment_sets = [{}],
         maximal_adjustment_sets = [{}]
     ),
     "simple_no_confounder_graph": dict(
-        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]  
+        graph_str = """graph[directed 1 node[id "Z1" label "Z1"]
                 node[id "X" label "X"]
-                node[id "Y" label "Y"]      
+                node[id "Y" label "Y"]
                 edge[source "X" target "Y"]
                 edge[source "Z1" target "X"]]
                 """,
@@ -65,11 +65,11 @@ TEST_GRAPH_SOLUTIONS = {
     ),
     # The following simpsons paradox examples are taken from Pearl, J {2013}. "Understanding Simpson’s Paradox" - http://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf
     "pearl_simpsons_paradox_1c": dict(
-        graph_str = """graph[directed 1 node[id "Z" label "Z"]  
+        graph_str = """graph[directed 1 node[id "Z" label "Z"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
                 node[id "L1" label "L1"]
-                node[id "L2" label "L2"]      
+                node[id "L2" label "L2"]
                 edge[source "X" target "Y"]
                 edge[source "L1" target "X"]
                 edge[source "L1" target "Z"]
@@ -82,7 +82,7 @@ TEST_GRAPH_SOLUTIONS = {
         maximal_adjustment_sets = [{}]
     ),
     "pearl_simpsons_paradox_1d": dict(
-        graph_str = """graph[directed 1 node[id "Z" label "Z"]  
+        graph_str = """graph[directed 1 node[id "Z" label "Z"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
                 node[id "L1" label "L1"]
@@ -97,10 +97,10 @@ TEST_GRAPH_SOLUTIONS = {
         maximal_adjustment_sets = [{"Z",}]
     ),
     "pearl_simpsons_paradox_2a": dict(
-        graph_str = """graph[directed 1 node[id "Z" label "Z"]  
+        graph_str = """graph[directed 1 node[id "Z" label "Z"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
-                node[id "L" label "L"]      
+                node[id "L" label "L"]
                 edge[source "X" target "Y"]
                 edge[source "X" target "Z"]
                 edge[source "L" target "Z"]
@@ -112,30 +112,30 @@ TEST_GRAPH_SOLUTIONS = {
         maximal_adjustment_sets = [{}]
     ),
     "pearl_simpsons_paradox_2b": dict(
-        graph_str = """graph[directed 1 node[id "Z" label "Z"]  
+        graph_str = """graph[directed 1 node[id "Z" label "Z"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
-                node[id "L" label "L"]      
+                node[id "L" label "L"]
                 edge[source "X" target "Y"]
                 edge[source "Z" target "X"]
                 edge[source "L" target "X"]
                 edge[source "L" target "Y"]]""",
         observed_variables = ["Z", "X", "Y"],
-        biased_sets = [], 
+        biased_sets = [],
         minimal_adjustment_sets = [],
         maximal_adjustment_sets = [] # Should this be {"Z"}?
     ),
     "pearl_simpsons_paradox_2b_L_observed": dict(
-        graph_str = """graph[directed 1 node[id "Z" label "Z"]  
+        graph_str = """graph[directed 1 node[id "Z" label "Z"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
-                node[id "L" label "L"]      
+                node[id "L" label "L"]
                 edge[source "X" target "Y"]
                 edge[source "Z" target "X"]
                 edge[source "L" target "X"]
                 edge[source "L" target "Y"]]""",
         observed_variables = ["Z", "X", "Y", "L"],
-        biased_sets = [], 
+        biased_sets = [],
         minimal_adjustment_sets = [{"L"}],
         maximal_adjustment_sets = [{"L", "Z"}]
     ),
@@ -143,7 +143,7 @@ TEST_GRAPH_SOLUTIONS = {
         graph_str = """graph[directed 1 node[id "Z1" label "Z1"]
                 node[id "Z2" label "Z2"]
                 node[id "Z3" label "Z3"]
-                node[id "L" label "L"]  
+                node[id "L" label "L"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
                 edge[source "X" target "Y"]
@@ -163,7 +163,7 @@ TEST_GRAPH_SOLUTIONS = {
         graph_str = """graph[directed 1 node[id "A" label "A"]
                 node[id "B" label "B"]
                 node[id "C" label "C"]
-                node[id "D" label "D"]  
+                node[id "D" label "D"]
                 node[id "E" label "E"]
                 node[id "X" label "X"]
                 node[id "Y" label "Y"]
@@ -220,7 +220,7 @@ TEST_GRAPH_SOLUTIONS = {
         graph_str = """graph[directed 1 node[id "X" label "X"]
                 node[id "Y" label "Y"]
                 node[id "Z1" label "Z1"]
-                node[id "Z2" label "Z2"]      
+                node[id "Z2" label "Z2"]
                 edge[source "X" target "Y"]
                 edge[source "X" target "Z1"]
                 edge[source "Z1" target "Y"]

tests/causal_identifiers/test_backdoor_identifier.py

@@ -11,19 +11,19 @@ class TestBackdoorIdentification(object):
         graph = example_graph_solution.graph
         biased_sets = example_graph_solution.biased_sets
         identifier = CausalIdentifier(graph, "nonparametric-ate", method_name="exhaustive-search")
-        
+
         backdoor_results = identifier.identify_backdoor("X", "Y", include_unobserved=False)
         backdoor_sets = [
-            set(backdoor_result_dict["backdoor_set"]) 
+            set(backdoor_result_dict["backdoor_set"])
             for backdoor_result_dict in backdoor_results
             if len(backdoor_result_dict["backdoor_set"]) > 0
         ]
 
         assert (
             (len(backdoor_sets) == 0 and len(biased_sets) == 0) # No biased sets exist and that's expected.
-            or  
+            or
             all([
-                set(biased_backdoor_set) not in backdoor_sets 
+                set(biased_backdoor_set) not in backdoor_sets
                 for biased_backdoor_set in biased_sets
             ]) # No sets that would induce biased results are present in the solution.
         )
@@ -35,7 +35,7 @@ class TestBackdoorIdentification(object):
 
         backdoor_results = identifier.identify_backdoor("X", "Y", include_unobserved=False)
         backdoor_sets = [
-            set(backdoor_result_dict["backdoor_set"]) 
+            set(backdoor_result_dict["backdoor_set"])
             for backdoor_result_dict in backdoor_results
             if len(backdoor_result_dict["backdoor_set"]) > 0
         ]
@@ -46,18 +46,18 @@ class TestBackdoorIdentification(object):
         graph = example_graph_solution.graph
         expected_sets = example_graph_solution.minimal_adjustment_sets
         identifier = CausalIdentifier(graph, "nonparametric-ate", method_name="minimal-adjustment", proceed_when_unidentifiable=False)
-        
+
         backdoor_results = identifier.identify_backdoor("X", "Y", include_unobserved=False)
         backdoor_sets = [
-            set(backdoor_result_dict["backdoor_set"]) 
+            set(backdoor_result_dict["backdoor_set"])
             for backdoor_result_dict in backdoor_results
         ]
-        
+
         assert (
             ((len(backdoor_sets) == 0) and (len(expected_sets) == 0)) # No adjustments exist and that's expected.
             or
             all([
-                set(expected_set) in backdoor_sets 
+                set(expected_set) in backdoor_sets
                 for expected_set in expected_sets
             ])
         )
@@ -66,19 +66,19 @@ class TestBackdoorIdentification(object):
         graph = example_graph_solution.graph
         expected_sets = example_graph_solution.maximal_adjustment_sets
         identifier = CausalIdentifier(graph, "nonparametric-ate", method_name="maximal-adjustment", proceed_when_unidentifiable=False)
-        
+
         backdoor_results = identifier.identify_backdoor("X", "Y", include_unobserved=False)
-        
+
         backdoor_sets = [
-            set(backdoor_result_dict["backdoor_set"]) 
+            set(backdoor_result_dict["backdoor_set"])
             for backdoor_result_dict in backdoor_results
         ]
-        
+        print(backdoor_sets, expected_sets, example_graph_solution.graph_str)
         assert (
             ((len(backdoor_sets) == 0) and (len(expected_sets) == 0)) # No adjustments exist and that's expected.
             or
             all([
-                set(expected_set) in backdoor_sets 
+                set(expected_set) in backdoor_sets
                 for expected_set in expected_sets
             ])
         )

tests/causal_identifiers/test_optimize_backdoor.py

@@ -11,7 +11,7 @@ class TestOptimizeBackdoorIdentifier(object):
         outcome = "Y"
         variables = ["X1", "X2"]
         causal_graph = "digraph{X1->T;X2->T;X1->X2;X2->Y;T->Y}"
-        
+
         vars = list(treatment) + list(outcome) + list(variables)
         df = pd.DataFrame(columns=vars)
 
@@ -30,12 +30,15 @@ class TestOptimizeBackdoorIdentifier(object):
         # Obtain backdoor sets
         path = Backdoor(identifier._graph._graph, treatment_name, outcome_name)
         backdoor_sets = path.get_backdoor_vars()
-
+        print(backdoor_sets)
         # Check if backdoor sets are valid i.e. if they block all paths between the treatment and the outcome
         backdoor_paths = identifier._graph.get_backdoor_paths(treatment_name, outcome_name)
         check_set = set(backdoor_sets[0]['backdoor_set'])
-        check = identifier._graph.check_valid_backdoor_set(treatment_name, outcome_name, check_set, backdoor_paths=backdoor_paths)
-        
+        check = identifier._graph.check_valid_backdoor_set(
+                        treatment_name, outcome_name, check_set,
+                        backdoor_paths=backdoor_paths,
+                        dseparation_algo="naive")
+        print(check)
         assert check["is_dseparated"]
 
     def test_2(self):
@@ -94,7 +97,7 @@ class TestOptimizeBackdoorIdentifier(object):
         backdoor_paths = identifier._graph.get_backdoor_paths(treatment_name, outcome_name)
         check_set = set(backdoor_sets[0]['backdoor_set'])
         check = identifier._graph.check_valid_backdoor_set(treatment_name, outcome_name, check_set, backdoor_paths=backdoor_paths)
-        
+
         assert check["is_dseparated"]
 
     def test_4(self):
@@ -102,7 +105,7 @@ class TestOptimizeBackdoorIdentifier(object):
         outcome = "Y"
         variables = ["X1", "X2"]
         causal_graph = "digraph{T->Y;X1->T;X1->Y;X2->T;}"
-        
+
         vars = list(treatment) + list(outcome) + list(variables)
         df = pd.DataFrame(columns=vars)
 
@@ -126,7 +129,7 @@ class TestOptimizeBackdoorIdentifier(object):
         backdoor_paths = identifier._graph.get_backdoor_paths(treatment_name, outcome_name)
         check_set = set(backdoor_sets[0]['backdoor_set'])
         check = identifier._graph.check_valid_backdoor_set(treatment_name, outcome_name, check_set, backdoor_paths=backdoor_paths)
-        
+
         assert check["is_dseparated"]
 
     def test_5(self):
@@ -158,7 +161,7 @@ class TestOptimizeBackdoorIdentifier(object):
         backdoor_paths = identifier._graph.get_backdoor_paths(treatment_name, outcome_name)
         check_set = set(backdoor_sets[0]['backdoor_set'])
         check = identifier._graph.check_valid_backdoor_set(treatment_name, outcome_name, check_set, backdoor_paths=backdoor_paths)
-        
+
         assert check["is_dseparated"]
 
     def test_6(self):
@@ -166,7 +169,7 @@ class TestOptimizeBackdoorIdentifier(object):
         outcome = "Y"
         variables = ["X1", "X2", "X3", "X4"]
         causal_graph = "digraph{X1->T;X1->X2;Y->X2;X3->T;X3->X4;X4->Y;T->Y}"
-        
+
         vars = list(treatment) + list(outcome) + list(variables)
         df = pd.DataFrame(columns=vars)
 
@@ -190,5 +193,5 @@ class TestOptimizeBackdoorIdentifier(object):
         backdoor_paths = identifier._graph.get_backdoor_paths(treatment_name, outcome_name)
         check_set = set(backdoor_sets[0]['backdoor_set'])
         check = identifier._graph.check_valid_backdoor_set(treatment_name, outcome_name, check_set, backdoor_paths=backdoor_paths)
-        
+
         assert check["is_dseparated"]