Causal Model

CausalModel is a core object for performing Identification and finding Instrumental Variables.

Before introducing the causal model, we should clarify the definition of interventions first. Interventions would be to take the whole population and give every one some operation. [Pearl] defined the \(do\)-operator to describe such operations. Probabilistic models can not serve to predict the effect of interventions which leads to the need for causal model.

The formal definition of causal model is due to [Pearl]. A causal model is a triple

\[M = \left< U, V, F\right>\]

where

• \(U\) are exogenous (variables that are determined by factors outside the model);

• \(V\) are endogenous that are determined by \(U \cup V\), and \(F\) is a set of functions such that

\[V_i = F_i(pa_i, U_i)\]

with \(pa_i \subset V \backslash V_i\).

For example, \(M = \left< U, V, F\right>\) is a causal model where

\[ \begin{align}\begin{aligned}V = \{V_1, V_2\},\\U = \{ U_1, U_2, I, J\},\\F = \{F_1, F_2 \}\end{aligned}\end{align} \]

such that

\[\begin{split}V_1 = F_1(I, U_1) = \theta_1 I + U_1\\ V_2 = F_2(V_1, J, U_2, ) = \phi V_1 + \theta_2 J + U_2.\end{split}\]

Note that every causal model can be associated with a DAG and encodes necessary information of the causal relationships between variables. YLearn uses CausalModel to represent a causal model and support many operations related to the causal model such as Identification.

Identification

To characterize the effect of the intervention, one needs to consider the causal effect which is a causal estimand including the \(do\)-operator. The action that converts the causal effect into corresponding statistical estimands is called Identification and is implemented in CausalModel in YLearn. Note that not all causal effects can be converted to statistical estimands. We refer to such causal effects as not identifiable. We list several identification methods supported by CausalModel.

Backdoor adjustment

The causal effect of \(X\) on \(Y\) is given by

\[P(y|do(x)) = \sum_w P(y|x, w)P(w)\]

if the set of variables \(W\) satisfies the back-door criterion relative to \((X, Y)\).

Frontdoor adjustment

The causal effect of \(X\) on \(Y\) is given by

\[P(y|do(x)) = \sum_w P(w|x) \sum_{x'}P(y|x', w)P(x')\]

if the set of variables \(W\) satisfies the front-door criterion relative to \((X, Y)\) and if \(P(x, w) > 0\).

General identification

[Shpitser2006] gives a necessary and sufficient graphical condition such that the causal effect of an arbitrary set of variables on another arbitrary set can be identified uniquely whenever its identifiable. We call the corresponding action of verifying this condition as general identification.

Finding Instrumental Variables

Instrumental variables are useful to identify and estimate the causal effect of \(X\) on \(Y\) when there are unobserved confoundings of \(X\) and \(Y\). A set of variables \(Z\) is said to be a set of instrumental variables if for any \(z\) in \(Z\):

1. \(z\) has a causal effect on \(X\).

2. The causal effect of \(z\) on \(Y\) is fully mediated by \(X\).

3. There are no back-door paths from \(z\) to \(Y\).

Example 1: Identify the causal effect with the general identification method

Causal structures where all unobserved variables are removed and their related causations are replaced by the confounding arcs (black doted lines with two arrows).

For the causal structure in the figure, we want to identify the causal effect of \(X\) on \(Y\) using the general identification method. The first step is to represent the causal structure with CausalModel.

from ylearn.causal_model.graph import CausalGraph

causation = {
    'X': ['Z2'],
    'Z1': ['X', 'Z2'],
    'Y': ['Z1', 'Z3'],
    'Z3': ['Z2'],
    'Z2': [],
}
arcs = [('X', 'Z2'), ('X', 'Z3'), ('X', 'Y'), ('Z2', 'Y')]
cg = CausalGraph(causation=causation, latent_confounding_arcs=arcs)

Then we need to define an instance of CausalModel for the causal structure encoded in cg to perform the identification.

from ylearn.causal_model.model import CausalModel
cm = CausalModel(causal_model=cg)
stat_estimand = cm.id(y={'Y'}, x={'X'})
stat_estimand.show_latex_expression()

>>> :math:`\sum_{Z3, Z1, Z2}[P(Z2)P(Y|Z3, Z2)][P(Z1|Z2, X)][P(Z3|Z2)]`

The result is the desired identified causal effect of \(X\) on \(Y\) in the given causal structure.

Example 2: Identify the causal effect with the back-door adjustment

All nodes are observed variables.

For the causal structure in the figure, we want to identify the causal effect of \(X\) on \(Y\) using the back-door adjustment method.

from ylearn.causal_model.graph import CausalGraph
from ylearn.causal_model.model import CausalModel

causation = {
    'X1': [],
    'X2': [],
    'X3': ['X1'],
    'X4': ['X1', 'X2'],
    'X5': ['X2'],
    'X6': ['X'],
    'X': ['X3', 'X4'],
    'Y': ['X6', 'X4', 'X5', 'X'],
}

cg = CausalGraph(causation=causation)
cm = CausalModel(causal_graph=cg)
backdoor_set, prob = cm3.identify(treatment={'X'}, outcome={'Y'}, identify_method=('backdoor', 'simple'))['backdoor']

print(backdoor_set)

>>> ['X3', 'X4']

Example 3: Find the valid instrumental variables

Causal structure for the variables \(p, t, l, g\)

We want to find the valid instrumental variables for the causal effect of \(t\) on \(g\).

causation = {
    'p':[],
    't': ['p'],
    'l': ['p'],
    'g': ['t', 'l']
}
arc = [('t', 'g')]
cg = CausalGraph(causation=causation, latent_confounding_arcs=arc)
cm = CausalModel(causal_graph=cg)

cm.get_iv('t', 'g')

>>> No valid instrument variable has been found.

Another causal structure for the variables \(p, t, l, g\)

We still want to find the valid instrumental variables for the causal effect of \(t\) on \(g\) in this new causal structure.

causation = {
    'p':[],
    't': ['p', 'l'],
    'l': [],
    'g': ['t', 'l']
}
arc = [('t', 'g')]
cg = CausalGraph(causation=causation, latent_confounding_arcs=arc)
cm = CausalModel(causal_graph=cg)

cm.get_iv('t', 'g')

>>> {'p'}

Class Structures

class ylearn.causal_model.CausalModel(causal_graph=None, data=None)

Parameters:

• causal_graph (CausalGraph, optional, default=None) – An instance of CausalGraph which encodes the causal structures.

• data (pandas.DataFrame, optional, default=None) – The data used to discover the causal structures if causal_graph is not provided.

id(y, x, prob=None, graph=None)

Identify the causal quantity \(P(y|do(x))\) if identifiable else return raise IdentificationError. Note that here we only consider semi-Markovian causal model, where each unobserved variable is a parent of exactly two nodes. This is because any causal model with unobserved variables can be converted to a semi-Markovian causal model encoding the same set of conditional independences.

Parameters:

• y (set of str) – Set of names of outcomes.

• x (set of str) – Set of names of treatments.

• prob (Prob, optional, default=None) – Probability distribution encoded in the graph.

• graph (CausalGraph) – CausalGraph encodes the information of corresponding causal structures.

Returns:

The probability distribution of the converted casual effect.

Return type:

Prob

Raises:

IdentificationError – If the interested causal effect is not identifiable, then raise IdentificationError.

is_valid_backdoor_set(set_, treatment, outcome)

Determine if a given set is a valid backdoor adjustment set for causal effect of treatments on the outcomes.

Parameters:

• set (set) – The adjustment set.

• treatment (set or list of str) – Names of the treatment. str is also acceptable for single treatment.

• outcome (set or list of str) – Names of the outcome. str is also acceptable for single outcome.

Returns:

True if the given set is a valid backdoor adjustment set for the causal effect of treatment on outcome in the current causal graph.

Return type:

bool

get_backdoor_set(treatment, outcome, adjust='simple', print_info=False)

Return the backdoor adjustment set for the given treatment and outcome.

Parameters:

• treatment (set or list of str) – Names of the treatment. str is also acceptable for single treatment.

• outcome (set or list of str) – Names of the outcome. str is also acceptable for single outcome.

• adjust (str) –

  Set style of the backdoor set. Available options are

  simple: directly return the parent set of treatment

  minimal: return the minimal backdoor adjustment set

  all: return all valid backdoor adjustment set.

• print_info (bool, default=False) – If True, print the identified results.

Returns:

The first element is the adjustment list, while the second is the encoded Prob.

Return type:

tuple of two element

Raises:

IdentificationError – Raise error if the style is not in simple, minimal or all or no set can satisfy the backdoor criterion.

get_backdoor_path(treatment, outcome)

Return all backdoor paths connecting treatment and outcome.

Parameters:

• treatment (str) – Name of the treatment.

• outcome (str) – Name of the outcome

Returns:

A list containing all valid backdoor paths between the treatment and outcome in the graph.

Return type:

list

has_collider(path, backdoor_path=True)

If the path in the current graph has a collider, return True, else return False.

Parameters:

• path (list of str) – A list containing nodes in the path.

• backdoor_path (bool, default=True) – Whether the path is a backdoor path.

Returns:

True if the path has a collider.

Return type:

bool

is_connected_backdoor_path(path)

Test whether a backdoor path is connected.

Parameters:

path (list of str) – A list describing the path.

Returns:

True if path is a d-connected backdoor path and False otherwise.

Return type:

bool

is_frontdoor_set(set_, treatment, outcome)

Determine if the given set is a valid frontdoor adjustment set for the causal effect of treatment on outcome.

Parameters:

• set (set) – The set waited to be determined as a valid front-door adjustment set.

• treatment (str) – Name of the treatment.

• outcome (str) – Name of the outcome.

Returns:

True if the given set is a valid frontdoor adjustment set for causal effects of treatments on outcomes.

Return type:

bool

get_frontdoor_set(treatment, outcome, adjust='simple')

Return the frontdoor set for adjusting the causal effect between treatment and outcome.

Parameters:

• treatment (set of str or str) – Name of the treatment. Should contain only one element.

• outcome (set of str or str) – Name of the outcome. Should contain only one element.

• adjust (str, default='simple') –

  Available options include ‘simple’: Return the frontdoor set with minimal number of elements.

  ’minimal’: Return the frontdoor set with minimal number of elements.

  ’all’: Return all possible frontdoor sets.

Returns:

2 elements (adjustment_set, Prob)

Return type:

tuple

Raises:

IdentificationError – Raise error if the style is not in simple, minimal or all or no set can satisfy the frontdoor criterion.

get_iv(treatment, outcome)

Find the instrumental variables for the causal effect of the treatment on the outcome.

Parameters:

• treatment (iterable) – Name(s) of the treatment.

• outcome (iterable) – Name(s) of the outcome.

Returns:

A valid instrumental variable set that will be an empty one if there is no such set.

Return type:

set

is_valid_iv(treatment, outcome, set_)

Determine whether a given set is a valid instrumental variable set.

Parameters:

• treatment (iterable) – Name(s) of the treatment.

• outcome (iterable) – Name(s) of the outcome.

• set (set) – The set waited to be tested.

Returns:

True if the set is a valid instrumental variable set and False otherwise.

Return type:

bool

identify(treatment, outcome, identify_method='auto')

Identify the causal effect expression. Identification is an operation that converts any causal effect quantity, e.g., quantities with the do operator, into the corresponding statistical quantity such that it is then possible to estimate the causal effect in some given data. However, note that not all causal quantities are identifiable, in which case an IdentificationError will be raised.

Parameters:

• treatment (set or list of str) – Set of names of treatments.

• outcome (set or list of str) – Set of names of outcomes.

• identify_method (tuple of str or str, optional, default='auto') –

  If the passed value is a tuple or list, then it should have two elements where the first one is for the identification methods and the second is for the returned set style.

  Available options:

  ’auto’ : Perform identification with all possible methods

  ’general’: The general identification method, see id()

  (‘backdoor’, ‘simple’): Return the set of all direct confounders of both treatments and outcomes as a backdoor adjustment set.

  (‘backdoor’, ‘minimal’): Return all possible backdoor adjustment sets with minimal number of elements.

  (‘backdoor’, ‘all’): Return all possible backdoor adjustment sets.

  (‘frontdoor’, ‘simple’): Return all possible frontdoor adjustment sets with minimal number of elements.

  (‘frontdoor’, ‘minimal’): Return all possible frontdoor adjustment sets with minimal number of elements.

  (‘frontdoor’, ‘all’): Return all possible frontdoor adjustment sets.

Returns:

A python dict where keys of the dict are identify methods while the values are the corresponding results.

Return type:

dict

Raises:

IdentificationError – If the causal effect is not identifiable or if the identify_method was not given properly.

estimate(estimator_model, data=None, *, treatment=None, outcome=None, adjustment=None, covariate=None, quantity=None, **kwargs)

Estimate the identified causal effect in a new dataset.

Parameters:

• estimator_model (EstimatorModel) – Any suitable estimator models implemented in the EstimatorModel can be applied here.

• data (pandas.DataFrame, optional, default=None) – The data set for causal effect to be estimated. If None, use the data which is used for discovering causal graph.

• treatment (set or list, optional, default=None) – Names of the treatment. If None, the treatment used for backdoor adjustment will be taken as the treatment.

• outcome (set or list, optional, default=None) – Names of the outcome. If None, the outcome used for backdoor adjustment will be taken as the outcome.

• adjustment (set or list, optional, default=None) – Names of the adjustment set. If None, the adjustment set is given by the simplest backdoor set found by CausalModel.

• covariate (set or list, optional, default=None) – Names of covariate set. Ignored if set as None.

• quantity (str, optional, default=None) – The interested quantity when evaluating causal effects.

Returns:

The estimated causal effect in data.

Return type:

np.ndarray or float

identify_estimate(data, outcome, treatment, estimator_model=None, quantity=None, identify_method='auto', **kwargs)

Combination of the identify method and the estimate method. However, since current implemented estimator models assume (conditionally) unconfoundness automatically (except for methods related to iv), we may only consider using backdoor set adjustment to fulfill the unconfoundness condition.

Parameters:

• treatment (set or list of str, optional) – Set of names of treatments.

• outcome (set or list of str, optional) – Set of names of outcome.

• identify_method (tuple of str or str, optional, default='auto') –

  If the passed value is a tuple or list, then it should have two elements where the first one is for the identification methods and the second is for the returned set style.

  Available options:

  ’auto’ : Perform identification with all possible methods

  ’general’: The general identification method, see id()

  (‘backdoor’, ‘simple’): Return the set of all direct confounders of both treatments and outcomes as a backdoor adjustment set.

  (‘backdoor’, ‘minimal’): Return all possible backdoor adjustment sets with minimal number of elements.

  (‘backdoor’, ‘all’): Return all possible backdoor adjustment sets.

  (‘frontdoor’, ‘simple’): Return all possible frontdoor adjustment sets with minimal number of elements.

  (‘frontdoor’, ‘minimal’): Return all possible frontdoor adjustment sets with minimal number of elements.

  (‘frontdoor’, ‘all’): Return all possible frontdoor adjustment sets.

• quantity (str, optional, default=None) – The interested quantity when evaluating causal effects.

Returns:

The estimated causal effect in data.

Return type:

np.ndarray or float