from typing import Dict, List, Optional, Set, Union
import networkx as nx
import numpy as np
from dowhy.causal_graph import CausalGraph
from dowhy.utils.api import parse_state
from dowhy.utils.graph_operations import find_ancestor, find_c_components, induced_graph
from dowhy.utils.ordered_set import OrderedSet
class IDExpression:
"""
Class for storing a causal estimand, as a result of the identification step using the ID algorithm.
The object stores a list of estimators(self._product) whose porduct must be obtained and a list of variables (self._sum) over which the product must be marginalized.
"""
def __init__(self):
self._product = []
self._sum = []
def add_product(self, element: Union[Dict, "IDExpression"]):
"""
Add an estimator to the list of product.
:param element: Estimator to append to the product list.
"""
self._product.append(element)
def add_sum(self, element: Set):
"""
Add variables to the list.
:param element: Set of variables to append to the list self._sum.
"""
for el in element:
self._sum.append(el)
def get_val(self, return_type: str):
"""
Get either the list of estimators (for product) or list of variables (for the marginalization).
:param return_type: "prod" to return the list of estimators or "sum" to return the list of variables.
"""
if return_type == "prod":
return self._product
elif return_type == "sum":
return self._sum
else:
raise Exception("Provide correct return type.")
def _print_estimator(self, prefix, estimator: Union[Dict, "IDExpression"] = None, start: bool = False):
"""
Print the IDExpression object.
"""
if estimator is None:
return None
string = ""
if isinstance(estimator, IDExpression):
s = True if len(estimator.get_val(return_type="sum")) > 0 else False
if s:
sum_vars = "{" + ",".join(estimator.get_val(return_type="sum")) + "}"
string += prefix + "Sum over " + sum_vars + ":\n"
prefix += "\t"
for expression in estimator.get_val(return_type="prod"):
add_string = self._print_estimator(prefix, expression)
if add_string is None:
return None
else:
string += add_string
else:
outcome_vars = list(estimator["outcome_vars"])
condition_vars = list(estimator["condition_vars"])
string += prefix + "Predictor: P(" + ",".join(outcome_vars)
if len(condition_vars) > 0:
string += "|" + ",".join(condition_vars)
string += ")\n"
if start:
string = string[:-1]
return string
def __str__(self):
string = self._print_estimator(prefix="", estimator=self, start=True)
if string is None:
return "The graph is not identifiable."
else:
return string
[docs]class IDIdentifier:
"""
This class is for backwards compatibility with CausalModel
Will be deprecated in the future in favor of function call id_identify_effect()
"""
[docs] def identify_effect(
self,
graph: CausalGraph,
treatment_name: Union[str, List[str]],
outcome_name: Union[str, List[str]],
node_names: Optional[Union[str, List[str]]] = None,
**kwargs,
):
return identify_effect_id(graph, treatment_name, outcome_name, node_names, **kwargs)
[docs]def identify_effect_id(
graph: CausalGraph,
treatment_name: Union[str, List[str]],
outcome_name: Union[str, List[str]],
node_names: Optional[Union[str, List[str]]] = None,
**kwargs,
) -> IDExpression:
"""
Implementation of the ID algorithm.
Link - https://ftp.cs.ucla.edu/pub/stat_ser/shpitser-thesis.pdf
The pseudo code has been provided on Pg 40.
:param treatment_names: OrderedSet comprising names of treatment variables.
:param outcome_names:OrderedSet comprising names of outcome variables.
:param node_names: OrderedSet comprising names of all nodes in the graph
:returns: target estimand, an instance of the IDExpression class.
"""
if node_names is None:
node_names = OrderedSet(graph._graph.nodes)
adjacency_matrix = graph.get_adjacency_matrix()
try:
tsort_node_names = OrderedSet(list(nx.topological_sort(graph._graph))) # topological sorting of graph nodes
except:
raise Exception("The graph must be a directed acyclic graph (DAG).")
return __adjacency_matrix_identify_effect(
adjacency_matrix,
OrderedSet(parse_state(treatment_name)),
OrderedSet(parse_state(outcome_name)),
tsort_node_names,
node_names,
)
def __adjacency_matrix_identify_effect(
adjacency_matrix: np.matrix,
treatment_name: OrderedSet,
outcome_name: OrderedSet,
tsort_node_names: OrderedSet,
node_names: OrderedSet = None,
):
node2idx, idx2node = __idx_node_mapping(node_names)
# Estimators list for returning after identification
estimators = IDExpression()
# Line 1
# If no action has been taken, the effect on Y is just the marginal of the observational distribution P(v) on Y.
if len(treatment_name) == 0:
identifier = IDExpression()
estimator = {}
estimator["outcome_vars"] = node_names
estimator["condition_vars"] = OrderedSet()
identifier.add_product(estimator)
identifier.add_sum(node_names.difference(outcome_name))
estimators.add_product(identifier)
return estimators
# Line 2
# If we are interested in the effect on Y, it is sufficient to restrict our attention on the parts of the model ancestral to Y.
ancestors = find_ancestor(outcome_name, node_names, adjacency_matrix, node2idx, idx2node)
if (
len(node_names.difference(ancestors)) != 0
): # If there are elements which are not the ancestor of the outcome variables
# Modify list of valid nodes
treatment_name = treatment_name.intersection(ancestors)
node_names = node_names.intersection(ancestors)
adjacency_matrix = induced_graph(node_set=node_names, adjacency_matrix=adjacency_matrix, node2idx=node2idx)
return __adjacency_matrix_identify_effect(
treatment_name=treatment_name,
outcome_name=outcome_name,
adjacency_matrix=adjacency_matrix,
tsort_node_names=tsort_node_names,
node_names=node_names,
)
# Line 3 - forces an action on any node where such an action would have no effect on Y – assuming we already acted on X.
# Modify adjacency matrix to obtain that corresponding to do(X)
adjacency_matrix_do_x = adjacency_matrix.copy()
for x in treatment_name:
x_idx = node2idx[x]
for i in range(len(node_names)):
adjacency_matrix_do_x[i, x_idx] = 0
ancestors = find_ancestor(outcome_name, node_names, adjacency_matrix_do_x, node2idx, idx2node)
W = node_names.difference(treatment_name).difference(ancestors)
if len(W) != 0:
return __adjacency_matrix_identify_effect(
treatment_name=treatment_name.union(W),
outcome_name=outcome_name,
adjacency_matrix=adjacency_matrix,
tsort_node_names=tsort_node_names,
node_names=node_names,
)
# Line 4 - Decomposes the problem into a set of smaller problems using the key property of C-component factorization of causal models.
# If the entire graph is a single C-component already, further problem decomposition is impossible, and we must provide base cases.
# Modify adjacency matrix to remove treatment variables
node_names_minus_x = node_names.difference(treatment_name)
node2idx_minus_x, idx2node_minus_x = __idx_node_mapping(node_names_minus_x)
adjacency_matrix_minus_x = induced_graph(
node_set=node_names_minus_x, adjacency_matrix=adjacency_matrix, node2idx=node2idx
)
c_components = find_c_components(
adjacency_matrix=adjacency_matrix_minus_x, node_set=node_names_minus_x, idx2node=idx2node_minus_x
)
if len(c_components) > 1:
identifier = IDExpression()
sum_over_set = node_names.difference(outcome_name.union(treatment_name))
for component in c_components:
expressions = __adjacency_matrix_identify_effect(
treatment_name=node_names.difference(component),
outcome_name=OrderedSet(list(component)),
adjacency_matrix=adjacency_matrix,
tsort_node_names=tsort_node_names,
node_names=node_names,
)
for expression in expressions.get_val(return_type="prod"):
identifier.add_product(expression)
identifier.add_sum(sum_over_set)
estimators.add_product(identifier)
return estimators
# Line 5 - The algorithms fails due to the presence of a hedge - the graph G, and a subgraph S that does not contain any X nodes.
S = c_components[0]
c_components_G = find_c_components(adjacency_matrix=adjacency_matrix, node_set=node_names, idx2node=idx2node)
if len(c_components_G) == 1 and c_components_G[0] == node_names:
return None
# Line 6 - If there are no bidirected arcs from X to the other nodes in the current subproblem under consideration, then we can replace acting on X by conditioning, and thus solve the subproblem.
if S in c_components_G:
sum_over_set = S.difference(outcome_name)
prev_nodes = []
for node in tsort_node_names:
if node in S:
identifier = IDExpression()
estimator = {}
estimator["outcome_vars"] = OrderedSet([node])
estimator["condition_vars"] = OrderedSet(prev_nodes)
identifier.add_product(estimator)
identifier.add_sum(sum_over_set)
estimators.add_product(identifier)
prev_nodes.append(node)
return estimators
# Line 7 - This is the most complicated case in the algorithm. Explain in the second last paragraph on Pg 41 of the link provided in the docstring above.
for component in c_components_G:
C = S.difference(component)
if C.is_empty() is None:
return __adjacency_matrix_identify_effect(
treatment_name=treatment_name.intersection(component),
outcome_name=outcome_name,
adjacency_matrix=induced_graph(
node_set=component, adjacency_matrix=adjacency_matrix, node2idx=node2idx
),
tsort_node_names=tsort_node_names,
node_names=node_names,
)
def __idx_node_mapping(node_names: OrderedSet):
"""
Obtain the node name to index and index to node name mappings.
:param node_names: Name of all nodes in the graph.
:return: node to index and index to node mappings.
"""
node2idx = {}
idx2node = {}
for i, node in enumerate(node_names.get_all()):
node2idx[node] = i
idx2node[i] = node
return node2idx, idx2node