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"""
This module contains the implementation of ``BayesianNetwork``.
``BayesianNetwork`` is a class that represents a probabilistic, weighted, directed acyclic graph (DAG)
describing causal relationships between variables and their distribution in a factorised way.
"""
import copy
import re
from typing import Dict, Hashable, List, Optional, Set, Tuple, Union
import networkx as nx
import pandas as pd
from pgmpy.estimators import BayesianEstimator, MaximumLikelihoodEstimator
from pgmpy.factors.discrete.CPD import TabularCPD
from pgmpy.models import BayesianNetwork as BayesianModel
from causalnex.estimator.em import EMSingleLatentVariable
from causalnex.structure import StructureModel
from causalnex.utils.pgmpy_utils import pd_to_tabular_cpd
[docs]class BayesianNetwork:
"""
Base class for Bayesian Network (BN), a probabilistic weighted DAG where nodes represent variables,
edges represent the causal relationships between variables.
``BayesianNetwork`` stores nodes with their possible states, edges and
conditional probability distributions (CPDs) of each node.
``BayesianNetwork`` is built on top of the ``StructureModel``, which is an extension of ``networkx.DiGraph``
(see :func:`causalnex.structure.structuremodel.StructureModel`).
In order to define the ``BayesianNetwork``, users should provide a relevant ``StructureModel``.
Once ``BayesianNetwork`` is initialised, no changes to the ``StructureModel`` can be made
and CPDs can be learned from the data.
The learned CPDs can be then used for likelihood estimation and predictions.
Example:
::
>>> # Create a Bayesian Network with a manually defined DAG.
>>> from causalnex.structure import StructureModel
>>> from causalnex.network import BayesianNetwork
>>>
>>> sm = StructureModel()
>>> sm.add_edges_from([
>>> ('rush_hour', 'traffic'),
>>> ('weather', 'traffic')
>>> ])
>>> bn = BayesianNetwork(sm)
>>> # A created ``BayesianNetwork`` stores nodes and edges defined by the ``StructureModel``
>>> bn.nodes
['rush_hour', 'traffic', 'weather']
>>>
>>> bn.edges
[('rush_hour', 'traffic'), ('weather', 'traffic')]
>>> # A ``BayesianNetwork`` doesn't store any CPDs yet
>>> bn.cpds
>>> {}
>>>
>>> # Learn the nodes' states from the data
>>> import pandas as pd
>>> data = pd.DataFrame({
>>> 'rush_hour': [True, False, False, False, True, False, True],
>>> 'weather': ['Terrible', 'Good', 'Bad', 'Good', 'Bad', 'Bad', 'Good'],
>>> 'traffic': ['heavy', 'light', 'heavy', 'light', 'heavy', 'heavy', 'heavy']
>>> })
>>> bn = bn.fit_node_states(data)
>>> bn.node_states
{'rush_hour': {False, True}, 'weather': {'Bad', 'Good', 'Terrible'}, 'traffic': {'heavy', 'light'}}
>>> # Learn the CPDs from the data
>>> bn = bn.fit_cpds(data)
>>> # Use the learned CPDs to make predictions on the unseen data
>>> test_data = pd.DataFrame({
>>> 'rush_hour': [False, False, True, True],
>>> 'weather': ['Good', 'Bad', 'Good', 'Bad']
>>> })
>>> bn.predict(test_data, "traffic").to_dict()
>>> {'traffic_prediction': {0: 'light', 1: 'heavy', 2: 'heavy', 3: 'heavy'}}
>>> bn.predict_probability(test_data, "traffic").to_dict()
{'traffic_prediction': {0: 'light', 1: 'heavy', 2: 'heavy', 3: 'heavy'}}
{'traffic_light': {0: 0.75, 1: 0.25, 2: 0.3333333333333333, 3: 0.3333333333333333},
'traffic_heavy': {0: 0.25, 1: 0.75, 2: 0.6666666666666666, 3: 0.6666666666666666}}
"""
[docs] def __init__(self, structure: StructureModel):
"""
Create a ``BayesianNetwork`` with a DAG defined by ``StructureModel``.
Args:
structure: a graph representing a causal relationship between variables.
In the structure
- cycles are not allowed;
- multiple (parallel) edges are not allowed;
- isolated nodes and multiple components are not allowed.
Raises:
ValueError: If the structure is not a connected DAG.
"""
n_components = nx.number_weakly_connected_components(structure)
if n_components > 1:
raise ValueError(
f"The given structure has {n_components} separated graph components. "
"Please make sure it has only one."
)
if not nx.is_directed_acyclic_graph(structure):
cycle = nx.find_cycle(structure)
raise ValueError(
f"The given structure is not acyclic. Please review the following cycle: {cycle}"
)
# _node_states is a Dict in the form `dict: {node: dict: {state: index}}`.
# Underlying libraries expect all states to be integers from zero, and
# thus this dict is used to convert from state -> idx, and then back from idx -> state as required
self._node_states = {} # type: Dict[str: Dict[Hashable, int]]
self._structure = structure
# _model is a pgmpy Bayesian Model.
# It is used for:
# - probability fitting
# - predictions
self._model = BayesianModel()
self._model.add_edges_from(structure.edges)
@property
def structure(self) -> StructureModel:
"""
``StructureModel`` defining the DAG of the Bayesian Network.
Returns:
A ``StructureModel`` of the Bayesian Network.
"""
return self._structure
@property
def nodes(self) -> List[str]:
"""
List of all nodes contained within the Bayesian Network.
Returns:
A list of node names.
"""
return list(self._model.nodes)
@property
def node_states(self) -> Dict[str, Set[Hashable]]:
"""
Dictionary of all states that each node can take.
Returns:
A dictionary of node and its possible states, in format of `dict: {node: state}`.
"""
return {node: set(states.keys()) for node, states in self._node_states.items()}
@node_states.setter
def node_states(self, nodes: Dict[str, Set[Hashable]]):
"""
Set the list of nodes that are contained within the Bayesian Network.
The states of all nodes must be provided.
Args:
nodes: A dictionary of node and its possible states, in format of `dict: {node: state}`.
Raises:
ValueError: if a node contains a None state.
KeyError: if a node is missing.
"""
missing_feature = set(self.nodes).difference(set(nodes.keys()))
if missing_feature:
raise KeyError(
"The data does not cover all the features found in the Bayesian Network. "
f"Please check the following features: {missing_feature}"
)
self._node_states = {}
for node, states in nodes.items():
if any(pd.isnull(list(states))):
raise ValueError(f"node '{node}' contains None state")
self._node_states[node] = {v: k for k, v in enumerate(sorted(states))}
@property
def edges(self) -> List[Tuple[str, str]]:
"""
List of all edges contained within the Bayesian Network, as a Tuple(from_node, to_node).
Returns:
A list of all edges.
"""
return list(self._model.edges)
@property
def cpds(self) -> Dict[str, pd.DataFrame]:
"""
Conditional Probability Distributions of each node within the Bayesian Network.
The row-index of each dataframe is all possible states for the node.
The col-index of each dataframe is a MultiIndex that describes all possible permutations of parent states.
For example, for a node :math:`P(A | B, D)`, where
.. math::
- A \\in \\text{{"a", "b", "c", "d"}}
- B \\in \\text{{"x", "y", "z"}}
- C \\in \\text{{False, True}}
>>> b x y z
>>> d False True False True False True
>>> a
>>> a 0.265306 0.214286 0.066667 0.25 0.444444 0.000000
>>> b 0.183673 0.214286 0.200000 0.25 0.222222 0.666667
>>> c 0.285714 0.285714 0.400000 0.25 0.333333 0.333333
>>> d 0.265306 0.285714 0.333333 0.25 0.000000 0.000000
Returns:
Conditional Probability Distributions of each node within the Bayesian Network.
"""
cpds = {}
for cpd in self._model.cpds:
names = cpd.variables[1:]
cols = [""]
if names:
cols = pd.MultiIndex.from_product(
[sorted(self._node_states[var].keys()) for var in names],
names=names,
)
cpds[cpd.variable] = pd.DataFrame(
cpd.values.reshape(
len(self._node_states[cpd.variable]), max(1, len(cols))
)
)
cpds[cpd.variable][cpd.variable] = sorted(
self._node_states[cpd.variable].keys()
)
cpds[cpd.variable].set_index([cpd.variable], inplace=True)
cpds[cpd.variable].columns = cols
return cpds
[docs] def set_cpd(self, node: str, df: pd.DataFrame) -> "BayesianNetwork":
"""
Provide self-defined CPD to Bayesian Network
Args:
node: the node to add self-defined cpd.
df: self-defined cpd in pandas DataFrame format.
Returns:
self
Raises:
IndexError: if the index names of the pandas DataFrame does not match the expected DataFrame.
ValueError: if node does not exist in Bayesian Network or a bad cpd table is provided.
"""
if node not in self.nodes:
raise ValueError(f'Non-existing node "{node}"')
# Check Table
true_parents = {
parent_node: self.node_states[parent_node]
for parent_node in self._structure.predecessors(node)
}
if isinstance(df.columns, pd.MultiIndex):
table_parents = {
name: set(df.columns.levels[i].values)
for i, name in enumerate(df.columns.names)
}
else:
table_parents = {}
if not (
set(df.index.values) == self.node_states[node]
and true_parents == table_parents
and df.index.name == node
):
raise IndexError("Wrong index values. Please check your indices")
sorted_df = df.reindex(sorted(df.columns), axis=1)
node_card = len(self.node_states[node])
if any(table_parents): # Check whether table parents is empty
evidence, evidence_card = zip(
*[
(key, len(table_parents[key]))
for key in sorted(table_parents.keys())
]
)
else:
evidence, evidence_card = (None, None)
tabular_cpd = TabularCPD(
node,
node_card,
sorted_df.values,
evidence=evidence,
evidence_card=evidence_card,
)
model_copy = copy.deepcopy(self._model)
model_copy.add_cpds(tabular_cpd)
model_copy.check_model()
self._model = model_copy
return self
[docs] def fit_node_states(self, df: pd.DataFrame) -> "BayesianNetwork":
"""
Fit all states of nodes that can appear in the data.
The dataframe provided should contain every possible state (values that can be taken) for every column.
Args:
df: data to fit node states from. Each column indicates a node and each row
an observed combination of states.
Returns:
self
Raises:
ValueError: if dataframe contains any missing data.
"""
self.node_states = {c: set(df[c].unique()) for c in df.columns}
return self
def _state_to_index(
self,
df: pd.DataFrame,
nodes: Optional[List[str]] = None,
) -> pd.DataFrame:
"""
Transforms all values in df to an integer, as defined by the mapping from fit_node_states.
Args:
df: data to transform
nodes: list of nodes to map to index. None means all.
Returns:
The transformed dataframe.
Raises:
ValueError: if nodes have not been fit, or if column names do not match node names.
"""
df.is_copy = False
cols = nodes if nodes else df.columns
for col in cols:
df[col] = df[col].map(self._node_states[col])
df.is_copy = True
return df
[docs] def fit_cpds(
self,
data: pd.DataFrame,
method: str = "MaximumLikelihoodEstimator",
bayes_prior: Optional[str] = None,
equivalent_sample_size: Optional[int] = None,
) -> "BayesianNetwork":
"""
Learn conditional probability distributions for all nodes in the Bayesian Network, conditioned on
their incoming edges (parents).
Args:
data: dataframe containing one column per node in the Bayesian Network.
method: how to fit probabilities. One of:
- "MaximumLikelihoodEstimator": fit probabilities using Maximum Likelihood Estimation;
- "BayesianEstimator": fit probabilities using Bayesian Parameter Estimation. Use bayes_prior.
bayes_prior: how to construct the Bayesian prior used by method="BayesianEstimator". One of:
- "K2": shorthand for dirichlet where all pseudo_counts are 1
regardless of variable cardinality;
- "BDeu": equivalent of using Dirichlet and using uniform 'pseudo_counts' of
`equivalent_sample_size / (node_cardinality * np.prod(parents_cardinalities))`
for each node. Use equivelant_sample_size.
equivalent_sample_size: used by BDeu bayes_prior to compute pseudo_counts.
Returns:
self
Raises:
ValueError: if an invalid method or bayes_prior is specified.
"""
state_names = {k: list(v.values()) for k, v in self._node_states.items()}
transformed_data = data.copy(deep=True) # type: pd.DataFrame
transformed_data = self._state_to_index(transformed_data[self.nodes])
if method == "MaximumLikelihoodEstimator":
self._model.fit(
data=transformed_data,
estimator=MaximumLikelihoodEstimator,
state_names=state_names,
)
elif method == "BayesianEstimator":
valid_bayes_priors = ["BDeu", "K2"]
if bayes_prior not in valid_bayes_priors:
raise ValueError(
f"unrecognised bayes_prior, please use one of {valid_bayes_priors}"
)
self._model.fit(
data=transformed_data,
estimator=BayesianEstimator,
prior_type=bayes_prior,
equivalent_sample_size=equivalent_sample_size,
state_names=state_names,
)
else:
valid_methods = ["MaximumLikelihoodEstimator", "BayesianEstimator"]
raise ValueError(f"unrecognised method, please use one of {valid_methods}")
return self
[docs] def fit_node_states_and_cpds(
self,
data: pd.DataFrame,
method: str = "MaximumLikelihoodEstimator",
bayes_prior: Optional[str] = None,
equivalent_sample_size: Optional[int] = None,
) -> "BayesianNetwork":
"""
Call `fit_node_states` and then `fit_cpds`.
Args:
data: dataframe containing one column per node in the Bayesian Network.
method: how to fit probabilities. One of:
- "MaximumLikelihoodEstimator": fit probabilities using Maximum Likelihood Estimation;
- "BayesianEstimator": fit probabilities using Bayesian Parameter Estimation. Use bayes_prior.
bayes_prior: how to construct the Bayesian prior used by method="BayesianEstimator". One of:
- "K2": shorthand for dirichlet where all pseudo_counts are 1
regardless of variable cardinality;
- "BDeu": equivalent of using dirichlet and using uniform 'pseudo_counts' of
`equivalent_sample_size / (node_cardinality * np.prod(parents_cardinalities))`
for each node. Use equivelant_sample_size.
equivalent_sample_size: used by BDeu bayes_prior to compute pseudo_counts.
Returns:
self
"""
return self.fit_node_states(data).fit_cpds(
data, method, bayes_prior, equivalent_sample_size
)
[docs] def add_node(
self,
node: str,
edges_to_add: List[Tuple[str, str]],
edges_to_remove: List[Tuple[str, str]],
) -> "BayesianNetwork":
"""
Adding a latent variable to the structure model, as well as its corresponding edges
Args:
node: Name of the node
edges_to_add: which edges to add to the structure
edges_to_remove: which edges to remove from the structure
Returns:
self
Raises:
ValueError: If lv_name exists in the network or
if `edges_to_add` include edges NOT containing the latent variable or
if `edges_to_remove` include edges containing the latent variable
"""
if any(node not in edges for edges in edges_to_add):
raise ValueError(f"Should only add edges containing node '{node}'")
if any(node in edges for edges in edges_to_remove):
raise ValueError(f"Should only remove edges NOT containing node '{node}'")
self._structure.add_edges_from(edges_to_add)
self._structure.remove_edges_from(edges_to_remove)
self._model.add_edges_from(edges_to_add)
self._model.remove_edges_from(edges_to_remove)
return self
[docs] def fit_latent_cpds( # pylint: disable=too-many-arguments
self,
lv_name: str,
lv_states: List,
data: pd.DataFrame,
box_constraints: Optional[Dict[str, Tuple[pd.DataFrame, pd.DataFrame]]] = None,
priors: Optional[Dict[str, pd.DataFrame]] = None,
initial_params: Union[str, Dict[str, pd.DataFrame]] = "random",
non_missing_data_factor: int = 1,
n_runs: int = 20,
stopping_delta: float = 0.0,
) -> "BayesianNetwork":
"""
This runs the EM algorithm to estimate the CPDs of latent variables and their corresponding Markov blanket
Args:
lv_name: Latent variable name
lv_states: the states the LV can assume
data: dataframe, must contain all variables in the Markov Blanket of the latent variable. Include one column
with the latent variable name, filled with np.nan for missing info about LV.
If some data is present about the LV, create complete columns.
n_runs: max number of EM alternations
stopping_delta: if max difference in current - last iteration CPDS < stopping_delta => convergence reached
initial_params: way to initialise parameters. Can be:
- "random": random values (default)
- "avg": uniform distributions everywhere. Not advised, as it may be the a stationary point on itself
- if provide a dictionary of dataframes, this will be used as the initialisation
box_constraints: minimum and maximum values for each model parameter. Specified with a dictionary mapping:
- Node
- two dataframes, in order: Min(P(Node|Par(Node))) and Max(P(Node|Par(Node)))
priors: priors, provided as a mapping Node -> dataframe with Dirichilet priors for P(Node|Par(Node))
non_missing_data_factor:
This is a weight added to the non-missing data samples. The effect is as if the amount of data provided
was bigger. Empirically, helps to set the factor to 10 if the non missing data is ~1% of the dataset
Returns:
self
Raises:
ValueError: if the latent variable is not a string or
if the latent variable cannot be found in the network or
if the latent variable is present/observed in the data
if the latent variable states are empty
if additional non-lv nodes are added to the subgraph without being fit
"""
if not isinstance(lv_name, str):
raise ValueError(f"Invalid latent variable name '{lv_name}'")
if lv_name not in self._structure:
raise ValueError(f"Latent variable '{lv_name}' not added to the network")
if not isinstance(lv_states, list) or len(lv_states) == 0:
raise ValueError(f"Latent variable '{lv_name}' contains no states")
# Unaccounted nodes that have not been fit will result in infinite loop during
# generation of InferenceEngine
unaccounted_nodes = []
for node in self.nodes:
if (node not in self.cpds) and (node != lv_name):
unaccounted_nodes.append(node)
if len(unaccounted_nodes) > 0:
raise ValueError(
f"Node(s) {unaccounted_nodes} have not had their states and cpds fit. "
"Before fitting latent variable cpds, add the additional nodes and"
"edges to the subgraph and fit with .fit_node_states_and_cpds() first."
)
# Register states for the latent variable
self._node_states[lv_name] = {v: k for k, v in enumerate(sorted(lv_states))}
# Run EM algorithm
estimator = EMSingleLatentVariable(
sm=self.structure,
data=data,
lv_name=lv_name,
node_states={n: sorted(s) for n, s in self.node_states.items()},
initial_params=initial_params,
box_constraints=box_constraints,
priors=priors,
non_missing_data_factor=non_missing_data_factor,
)
estimator.run(n_runs=n_runs, stopping_delta=stopping_delta)
# Add CPDs into the model
tab_cpds = [pd_to_tabular_cpd(el) for el in estimator.cpds.values()]
self._model.add_cpds(*tab_cpds)
return self
[docs] def predict(self, data: pd.DataFrame, node: str) -> pd.DataFrame:
"""
Predict the state of a node based on some input data, using the Bayesian Network.
Args:
data: data to make prediction.
node: the node to predict.
Returns:
A dataframe of predictions, containing a single column name {node}_prediction.
"""
if all(parent in data.columns for parent in self._model.get_parents(node)):
return self._predict_from_complete_data(data, node)
return self._predict_from_incomplete_data(data, node)
def _predict_from_complete_data(
self,
data: pd.DataFrame,
node: str,
) -> pd.DataFrame:
"""
Predict state of node given all parents of node exist within data.
This method inspects the CPD of node directly, since all parent states are known.
This avoids traversing the full network to compute marginals.
This method is fast.
Args:
data: data to make prediction.
node: the node to predict.
Returns:
A dataframe of predictions, containing a single column named {node}_prediction.
"""
transformed_data = data.copy(deep=True) # type: pd.DataFrame
parents = sorted(self._model.get_parents(node))
cpd = self.cpds[node]
transformed_data[f"{node}_prediction"] = transformed_data.apply(
lambda row: cpd[tuple(row[parent] for parent in parents)].idxmax()
if parents
else cpd[""].idxmax(),
axis=1,
)
return transformed_data[[node + "_prediction"]]
def _predict_from_incomplete_data(
self,
data: pd.DataFrame,
node: str,
) -> pd.DataFrame:
"""
Predict state of node when some parents of node do not exist within data.
This method uses the pgmpy predict function, which predicts the most likely state for every node
that is not contained within data.
With incomplete data, pgmpy goes beyond parents in the network to determine the most likely predictions.
This method is slow.
Args:
data: data to make prediction.
node: the node to predict.
Returns:
A dataframe of predictions, containing a single column name {node}_prediction.
"""
transformed_data = data.copy(deep=True) # type: pd.DataFrame
self._state_to_index(transformed_data)
# pgmpy will predict all missing data, so drop column we want to predict
transformed_data = transformed_data.drop(columns=[node])
predictions = self._model.predict(transformed_data)[[node]]
return predictions.rename(columns={node: node + "_prediction"})
[docs] def predict_probability(self, data: pd.DataFrame, node: str) -> pd.DataFrame:
"""
Predict the probability of each possible state of a node, based on some input data.
Args:
data: data to make prediction.
node: the node to predict probabilities.
Returns:
A dataframe of predicted probabilities, contained one column per possible state, named {node}_{state}.
"""
if all(parent in data.columns for parent in self._model.get_parents(node)):
return self._predict_probability_from_complete_data(data, node)
return self._predict_probability_from_incomplete_data(data, node)
def _predict_probability_from_complete_data(
self,
data: pd.DataFrame,
node: str,
) -> pd.DataFrame:
"""
Predict the probability of each possible state of a node, based on some input data.
This method inspects the CPD of node directly, since all parent states are known.
This avoids traversing the full network to compute marginals.
This method is fast.
Args:
data: data to make prediction.
node: the node to predict probabilities.
Returns:
A dataframe of predicted probabilities, contained one column per possible state, named {node}_{state}.
"""
transformed_data = data.copy(deep=True) # type: pd.DataFrame
parents = sorted(self._model.get_parents(node))
cpd = self.cpds[node]
def lookup_probability(row, s):
"""Retrieve probability from CPD"""
if parents:
return cpd[tuple(row[parent] for parent in parents)].loc[s]
return cpd.at[s, ""]
for state in self.node_states[node]:
transformed_data[f"{node}_{state}"] = transformed_data.apply(
lambda row, st=state: lookup_probability(row, st), axis=1
)
return transformed_data[[f"{node}_{state}" for state in self.node_states[node]]]
def _predict_probability_from_incomplete_data(
self,
data: pd.DataFrame,
node: str,
) -> pd.DataFrame:
"""
Predict the probability of each possible state of a node, based on some input data.
This method uses the pgmpy predict_probability function, which predicts the probability
of every state for every node that is not contained within data.
With incomplete data, pgmpy goes beyond parents in the network to determine the most likely predictions.
This method is slow.
Args:
data: data to make prediction.
node: the node to predict probabilities.
Returns:
A dataframe of predicted probabilities, contained one column per possible state, named {node}_{state}.
"""
transformed_data = data.copy(deep=True) # type: pd.DataFrame
self._state_to_index(transformed_data)
# pgmpy will predict all missing data, so drop column we want to predict
transformed_data = transformed_data.drop(columns=[node])
probability = self._model.predict_probability(
transformed_data
) # type: pd.DataFrame
# keep only probabilities for the node we are interested in
cols = []
pattern = re.compile(f"^{node}_[0-9]+$")
# disabled open pylint issue (https://github.com/PyCQA/pylint/issues/2962)
for col in probability.columns:
if pattern.match(col):
cols.append(col)
probability = probability[cols]
probability.columns = cols
return probability