from collections.abc import Iterable
from itertools import chain, product
import networkx as nx
import numpy as np
from tqdm.auto import tqdm
from pgmpy.estimators.LinearModel import LinearEstimator
from pgmpy.factors.discrete import DiscreteFactor
from pgmpy.global_vars import SHOW_PROGRESS
from pgmpy.models import BayesianNetwork
from pgmpy.utils.sets import _powerset, _variable_or_iterable_to_set
[docs]class CausalInference(object):
"""
This is an inference class for performing Causal Inference over Bayesian Networks or Structural Equation Models.
This class will accept queries of the form: P(Y | do(X)) and utilize its methods to provide an estimand which:
* Identifies adjustment variables
* Backdoor Adjustment
* Front Door Adjustment
* Instrumental Variable Adjustment
Parameters
----------
model: CausalGraph
The model that we'll perform inference over.
set_nodes: list[node:str] or None
A list (or set/tuple) of nodes in the Bayesian Network which have been set to a specific value per the
do-operator.
Examples
--------
Create a small Bayesian Network.
>>> from pgmpy.models import BayesianNetwork
>>> game = BayesianNetwork([('X', 'A'),
... ('A', 'Y'),
... ('A', 'B')])
Load the graph into the CausalInference object to make causal queries.
>>> from pgmpy.inference.CausalInference import CausalInference
>>> inference = CausalInference(game)
>>> inference.get_all_backdoor_adjustment_sets(X="X", Y="Y")
>>> inference.get_all_frontdoor_adjustment_sets(X="X", Y="Y")
References
----------
'Causality: Models, Reasoning, and Inference' - Judea Pearl (2000)
Many thanks to @ijmbarr for their implementation of Causal Graphical models available. It served as an invaluable
reference. Available on GitHub: https://github.com/ijmbarr/causalgraphicalmodels
"""
def __init__(self, model, set_nodes=None):
if not isinstance(model, BayesianNetwork):
raise NotImplementedError(
"Causal Inference is only implemented for BayesianNetworks at this time."
)
self.model = model
self.set_nodes = _variable_or_iterable_to_set(set_nodes)
self.observed_variables = frozenset(self.model.nodes()).difference(
model.latents
)
def __repr__(self):
variables = ", ".join(map(str, sorted(self.observed_variables)))
return f"{self.__class__.__name__}({variables})"
[docs] def is_valid_backdoor_adjustment_set(self, X, Y, Z=[]):
"""
Test whether Z is a valid backdoor adjustment set for estimating the causal impact of X on Y.
Parameters
----------
X: str
Intervention Variable
Y: str
Target Variable
Z: str or set[str]
Adjustment variables
Returns
-------
Is a valid backdoor adjustment set: bool
True if Z is a valid backdoor adjustment set else False
Examples
--------
>>> game1 = BayesianNetwork([('X', 'A'),
... ('A', 'Y'),
... ('A', 'B')])
>>> inference = CausalInference(game1)
>>> inference.is_valid_backdoor_adjustment_set("X", "Y")
True
"""
Z_ = _variable_or_iterable_to_set(Z)
observed = [X] + list(Z_)
parents_d_sep = []
for p in self.model.predecessors(X):
parents_d_sep.append(not self.model.is_dconnected(p, Y, observed=observed))
return all(parents_d_sep)
[docs] def get_all_backdoor_adjustment_sets(self, X, Y):
"""
Returns a list of all adjustment sets per the back-door criterion.
A set of variables Z satisfies the back-door criterion relative to an ordered pair of variabies (Xi, Xj) in a DAG G if:
(i) no node in Z is a descendant of Xi; and
(ii) Z blocks every path between Xi and Xj that contains an arrow into Xi.
TODO:
* Backdoors are great, but the most general things we could implement would be Ilya Shpitser's ID and
IDC algorithms. See [his Ph.D. thesis for a full explanation]
(https://ftp.cs.ucla.edu/pub/stat_ser/shpitser-thesis.pdf). After doing a little reading it is clear
that we do not need to immediatly implement this. However, in order for us to truly account for
unobserved variables, we will need not only these algorithms, but a more general implementation of a DAG.
Most DAGs do not allow for bidirected edges, but it is an important piece of notation which Pearl and
Shpitser use to denote graphs with latent variables.
Parameters
----------
X: str
Intervention Variable
Returns
-------
frozenset: A frozenset of frozensets
Y: str
Target Variable
Examples
--------
>>> game1 = BayesianNetwork([('X', 'A'),
... ('A', 'Y'),
... ('A', 'B')])
>>> inference = CausalInference(game1)
>>> inference.get_all_backdoor_adjustment_sets("X", "Y")
frozenset()
References
----------
"Causality: Models, Reasoning, and Inference", Judea Pearl (2000). p.79.
"""
try:
assert X in self.observed_variables
assert Y in self.observed_variables
except AssertionError:
raise AssertionError("Make sure both X and Y are observed.")
if self.is_valid_backdoor_adjustment_set(X, Y, Z=frozenset()):
return frozenset()
possible_adjustment_variables = (
set(self.observed_variables)
- {X}
- {Y}
- set(nx.descendants(self.model, X))
)
valid_adjustment_sets = []
for s in _powerset(possible_adjustment_variables):
super_of_complete = []
for vs in valid_adjustment_sets:
super_of_complete.append(vs.intersection(set(s)) == vs)
if any(super_of_complete):
continue
if self.is_valid_backdoor_adjustment_set(X, Y, s):
valid_adjustment_sets.append(frozenset(s))
if len(valid_adjustment_sets) == 0:
raise ValueError(f"No valid adjustment set found for {X} -> {Y}")
return frozenset(valid_adjustment_sets)
[docs] def is_valid_frontdoor_adjustment_set(self, X, Y, Z=None):
"""
Test whether Z is a valid frontdoor adjustment set for estimating the causal impact of X on Y via the frontdoor
adjustment formula.
Parameters
----------
X: str
Intervention Variable
Y: str
Target Variable
Z: set
Adjustment variables
Returns
-------
Is valid frontdoor adjustment: bool
True if Z is a valid frontdoor adjustment set.
"""
Z = _variable_or_iterable_to_set(Z)
# 0. Get all directed paths from X to Y. Don't check further if there aren't any.
directed_paths = list(nx.all_simple_paths(self.model, X, Y))
if directed_paths == []:
return False
# 1. Z intercepts all directed paths from X to Y
unblocked_directed_paths = [
path for path in directed_paths if not any(zz in path for zz in Z)
]
if unblocked_directed_paths:
return False
# 2. there is no backdoor path from X to Z
unblocked_backdoor_paths_X_Z = [
zz for zz in Z if not self.is_valid_backdoor_adjustment_set(X, zz)
]
if unblocked_backdoor_paths_X_Z:
return False
# 3. All back-door paths from Z to Y are blocked by X
valid_backdoor_sets = []
for zz in Z:
valid_backdoor_sets.append(self.is_valid_backdoor_adjustment_set(zz, Y, X))
if not all(valid_backdoor_sets):
return False
return True
[docs] def get_all_frontdoor_adjustment_sets(self, X, Y):
"""
Identify possible sets of variables, Z, which satisfy the front-door criterion relative to given X and Y.
Z satisfies the front-door criterion if:
(i) Z intercepts all directed paths from X to Y
(ii) there is no backdoor path from X to Z
(iii) all back-door paths from Z to Y are blocked by X
Returns
-------
frozenset: a frozenset of frozensets
References
----------
Causality: Models, Reasoning, and Inference, Judea Pearl (2000). p.82.
"""
assert X in self.observed_variables
assert Y in self.observed_variables
possible_adjustment_variables = set(self.observed_variables) - {X} - {Y}
valid_adjustment_sets = frozenset(
[
frozenset(s)
for s in _powerset(possible_adjustment_variables)
if self.is_valid_frontdoor_adjustment_set(X, Y, s)
]
)
return valid_adjustment_sets
[docs] def get_distribution(self):
"""
Returns a string representing the factorized distribution implied by the CGM.
"""
products = []
for node in nx.topological_sort(self.model):
if node in self.set_nodes:
continue
parents = list(self.model.predecessors(node))
if not parents:
p = f"P({node})"
else:
parents = [
f"do({n})" if n in self.set_nodes else str(n) for n in parents
]
p = f"P({node}|{','.join(parents)})"
products.append(p)
return "".join(products)
[docs] def simple_decision(self, adjustment_sets=[]):
"""
Selects the smallest set from provided adjustment sets.
Parameters
----------
adjustment_sets: iterable
A frozenset or list of valid adjustment sets
Returns
-------
frozenset
"""
adjustment_list = list(adjustment_sets)
if adjustment_list == []:
return frozenset([])
return adjustment_list[np.argmin(adjustment_list)]
[docs] def estimate_ate(
self,
X,
Y,
data,
estimand_strategy="smallest",
estimator_type="linear",
**kwargs,
):
"""
Estimate the average treatment effect (ATE) of X on Y.
Parameters
----------
X: str
Intervention Variable
Y: str
Target Variable
data: pandas.DataFrame
All observed data for this Bayesian Network.
estimand_strategy: str or frozenset
Either specify a specific backdoor adjustment set or a strategy.
The available options are:
smallest:
Use the smallest estimand of observed variables
all:
Estimate the ATE from each identified estimand
estimator_type: str
The type of model to be used to estimate the ATE.
All of the linear regression classes in statsmodels are available including:
* GLS: generalized least squares for arbitrary covariance
* OLS: ordinary least square of i.i.d. errors
* WLS: weighted least squares for heteroskedastic error
Specify them with their acronym (e.g. "OLS") or simple "linear" as an alias for OLS.
**kwargs: dict
Keyward arguments specific to the selected estimator.
linear:
missing: str
Available options are "none", "drop", or "raise"
Returns
-------
The average treatment effect: float
Examples
--------
>>> import pandas as pd
>>> game1 = BayesianNetwork([('X', 'A'),
... ('A', 'Y'),
... ('A', 'B')])
>>> data = pd.DataFrame(np.random.randint(2, size=(1000, 4)), columns=['X', 'A', 'B', 'Y'])
>>> inference = CausalInference(model=game1)
>>> inference.estimate_ate("X", "Y", data=data, estimator_type="linear")
"""
valid_estimators = ["linear"]
try:
assert estimator_type in valid_estimators
except AssertionError:
print(
f"{estimator_type} if not a valid estimator_type. Please select from {valid_estimators}"
)
all_simple_paths = nx.all_simple_paths(self.model, X, Y)
all_path_effects = []
for path in all_simple_paths:
causal_effect = []
for (x1, x2) in zip(path, path[1:]):
if isinstance(estimand_strategy, frozenset):
adjustment_set = frozenset({estimand_strategy})
assert self.is_valid_backdoor_adjustment_set(
x1, x2, Z=adjustment_set
)
elif estimand_strategy in ["smallest", "all"]:
adjustment_sets = self.get_all_backdoor_adjustment_sets(x1, x2)
if estimand_strategy == "smallest":
adjustment_sets = frozenset(
{self.simple_decision(adjustment_sets)}
)
if estimator_type == "linear":
self.estimator = LinearEstimator(self.model)
ate = [
self.estimator.fit(X=x1, Y=x2, Z=s, data=data, **kwargs)._get_ate()
for s in adjustment_sets
]
causal_effect.append(np.mean(ate))
all_path_effects.append(np.prod(causal_effect))
return np.sum(all_path_effects)
[docs] def get_proper_backdoor_graph(self, X, Y, inplace=False):
"""
Returns a proper backdoor graph for the exposure `X` and outcome `Y`.
A proper backdoor graph is a graph which remove the first edge of every
proper causal path from `X` to `Y`.
Parameters
----------
X: list (array-like)
A list of exposure variables.
Y: list (array-like)
A list of outcome variables
inplace: boolean
If inplace is True, modifies the object itself. Otherwise retuns
a modified copy of self.
Examples
--------
>>> from pgmpy.models import BayesianNetwork
>>> from pgmpy.inference import CausalInference
>>> model = BayesianNetwork([("x1", "y1"), ("x1", "z1"), ("z1", "z2"),
... ("z2", "x2"), ("y2", "z2")])
>>> c_infer = CausalInference(model)
>>> c_infer.get_proper_backdoor_graph(X=["x1", "x2"], Y=["y1", "y2"])
<pgmpy.models.BayesianNetwork.BayesianNetwork at 0x7fba501ad940>
References
----------
[1] Perkovic, Emilija, et al. "Complete graphical characterization and construction of adjustment sets in Markov equivalence classes of ancestral graphs." The Journal of Machine Learning Research 18.1 (2017): 8132-8193.
"""
for var in chain(X, Y):
if var not in self.model.nodes():
raise ValueError(f"{var} not found in the model.")
model = self.model if inplace else self.model.copy()
edges_to_remove = []
for source in X:
paths = nx.all_simple_edge_paths(model, source, Y)
for path in paths:
edges_to_remove.append(path[0])
model.remove_edges_from(edges_to_remove)
return model
[docs] def is_valid_adjustment_set(self, X, Y, adjustment_set):
"""
Method to test whether `adjustment_set` is a valid adjustment set for
identifying the causal effect of `X` on `Y`.
Parameters
----------
X: list (array-like)
The set of cause variables.
Y: list (array-like)
The set of predictor variables.
adjustment_set: list (array-like)
The set of variables for which to test whether they satisfy the
adjustment set criteria.
Returns
-------
Is valid adjustment set: bool
Returns True if `adjustment_set` is a valid adjustment set for
identifying the effect of `X` on `Y`. Else returns False.
Examples
--------
>>> from pgmpy.models import BayesianNetwork
>>> from pgmpy.inference import CausalInference
>>> model = BayesianNetwork([("x1", "y1"), ("x1", "z1"), ("z1", "z2"),
... ("z2", "x2"), ("y2", "z2")])
>>> c_infer = CausalInference(model)
>>> c_infer.is_valid_adjustment_set(X=['x1', 'x2'], Y=['y1', 'y2'], adjustment_set=['z1', 'z2'])
True
References
----------
[1] Perkovic, Emilija, et al. "Complete graphical characterization and construction of adjustment sets in Markov equivalence classes of ancestral graphs." The Journal of Machine Learning Research 18.1 (2017): 8132-8193.
"""
backdoor_graph = self.get_proper_backdoor_graph(X, Y, inplace=False)
for (x, y) in zip(X, Y):
if backdoor_graph.is_dconnected(start=x, end=y, observed=adjustment_set):
return False
return True
[docs] def get_minimal_adjustment_set(self, X, Y):
"""
Method to test whether `adjustment_set` is a valid adjustment set for
identifying the causal effect of `X` on `Y`.
Parameters
----------
X: str (variable name)
The cause/exposure variables.
Y: str (variable name)
The outcome variable
Returns
-------
Minimal adjustment set: set or None
A set of variables which are the minimal possible adjustment set. If
None, no adjustment set is possible.
Examples
--------
References
----------
[1] Perkovic, Emilija, et al. "Complete graphical characterization and construction of adjustment sets in Markov equivalence classes of ancestral graphs." The Journal of Machine Learning Research 18.1 (2017): 8132-8193.
"""
backdoor_graph = self.get_proper_backdoor_graph(X, Y, inplace=False)
return backdoor_graph.minimal_dseparator(X, Y)
[docs] def query(
self,
variables,
do=None,
evidence=None,
adjustment_set=None,
inference_algo="ve",
show_progress=True,
**kwargs,
):
"""
Performs a query on the model of the form :math:`P(X | do(Y), Z)` where :math:`X`
is `variables`, :math:`Y` is `do` and `Z` is the `evidence`.
Parameters
----------
variables: list
list of variables in the query i.e. `X` in :math:`P(X | do(Y), Z)`.
do: dict (default: None)
Dictionary of the form {variable_name: variable_state} representing
the variables on which to apply the do operation i.e. `Y` in
:math:`P(X | do(Y), Z)`.
evidence: dict (default: None)
Dictionary of the form {variable_name: variable_state} repesenting
the conditional variables in the query i.e. `Z` in :math:`P(X |
do(Y), Z)`.
adjustment_set: str or list (default=None)
Specifies the adjustment set to use. If None, uses the parents of the
do variables as the adjustment set.
inference_algo: str or pgmpy.inference.Inference instance
The inference algorithm to use to compute the probability values.
String options are: 1) ve: Variable Elimination 2) bp: Belief
Propagation.
kwargs: Any
Additional paramters which needs to be passed to inference
algorithms. Please refer to the pgmpy.inference.Inference for
details.
Returns
-------
Queried distribution: pgmpy.factor.discrete.DiscreteFactor
A factor object representing the joint distribution over the variables in `variables`.
Examples
--------
>>> from pgmpy.utils import get_example_model
>>> model = get_example_model('alarm')
>>> infer = CausalInference(model)
>>> infer.query(['HISTORY'], do={'CVP': 'LOW'}, evidence={'HR': 'LOW'})
<DiscreteFactor representing phi(HISTORY:2) at 0x7f4e0874c2e0>
"""
# Step 1: Check if all the arguments are valid and get them to uniform types.
if (not isinstance(variables, Iterable)) or (isinstance(variables, str)):
raise ValueError(
f"variables much be a list (array-like). Got type: {type(variables)}."
)
elif not all([node in self.model.nodes() for node in variables]):
raise ValueError(
f"Some of the variables in `variables` are not in the model."
)
else:
variables = list(variables)
if do is None:
do = {}
elif not isinstance(do, dict):
raise ValueError(
"`do` must be a dict of the form: {variable_name: variable_state}"
)
if evidence is None:
evidence = {}
elif not isinstance(evidence, dict):
raise ValueError(
"`evidence` must be a dict of the form: {variable_name: variable_state}"
)
from pgmpy.inference import Inference
if inference_algo == "ve":
from pgmpy.inference import VariableElimination
inference_algo = VariableElimination
elif inference_algo == "bp":
from pgmpy.inference import BeliefPropagation
inference_algo = BeliefPropagation
elif not isinstance(inference_algo, Inference):
raise ValueError(
f"inference_algo must be one of: 've', 'bp', or an instance of pgmpy.inference.Inference. Got: {inference_algo}"
)
# Step 2: Check if adjustment set is provided, otherwise try calculating it.
if adjustment_set is None:
do_vars = [var for var, state in do.items()]
adjustment_set = set(
chain(*[self.model.predecessors(var) for var in do_vars])
)
if len(adjustment_set.intersection(self.model.latents)) != 0:
raise ValueError(
"Not all parents of do variables are observed. Please specify an adjustment set."
)
infer = inference_algo(self.model)
# Step 3.1: If no do variable specified, do a normal probabilistic inference.
if do == {}:
return infer.query(variables, evidence, show_progress=False)
# Step 3.2: If no adjustment is required, do a normal probabilistic
# inference with do variables as the evidence.
elif len(adjustment_set) == 0:
evidence = {**evidence, **do}
return infer.query(variables, evidence, show_progress=False)
# Step 4: For other cases, compute \sum_{z} p(variables | do, z) p(z)
values = []
# Step 4.1: Compute p_z and states of z to iterate over.
# For computing p_z, if evidence variables also in adjustment set,
# manually do reduce else inference will throw error.
evidence_adj_inter = {
var: state
for var, state in evidence.items()
if var in adjustment_set.intersection(evidence.keys())
}
if len(evidence_adj_inter) != 0:
p_z = infer.query(adjustment_set, show_progress=False).reduce(
[(key, value) for key, value in evidence_adj_inter.items()],
inplace=False,
)
# Since we are doing reduce over some of the variables, they are
# going to be removed from the factor but would be required to get
# values later. A hackish solution to reintroduce those variables in p_z
if set(p_z.variables) != adjustment_set:
p_z = DiscreteFactor(
p_z.variables + list(evidence_adj_inter.keys()),
list(p_z.cardinality) + [1] * len(evidence_adj_inter),
p_z.values,
state_names={
**p_z.state_names,
**{var: [state] for var, state in evidence_adj_inter.items()},
},
)
else:
p_z = infer.query(adjustment_set, evidence=evidence, show_progress=False)
adj_states = []
for var in adjustment_set:
if var in evidence_adj_inter.keys():
adj_states.append([evidence_adj_inter[var]])
else:
adj_states.append(self.model.get_cpds(var).state_names[var])
# Step 4.2: Iterate over states of adjustment set and compute values.
if show_progress and SHOW_PROGRESS:
pbar = tqdm(total=np.prod([len(states) for states in adj_states]))
for state_comb in product(*adj_states):
adj_evidence = {
var: state for var, state in zip(adjustment_set, state_comb)
}
evidence = {**do, **adj_evidence}
values.append(
infer.query(variables, evidence=evidence, show_progress=False)
* p_z.get_value(**adj_evidence)
)
if show_progress and SHOW_PROGRESS:
pbar.update(1)
return sum(values).normalize(inplace=False)