Identification

Identification algorithms.

Identification

Algorithm	Implementation
ID [shpitser2006]	y0.algorithm.identify.id_std
Tian ID [tian03a]	y0.algorithm.tian_id
IDC [shpitser2008]	y0.algorithm.identify.idc
ID* [shpitser2012]	y0.algorithm.identify.id_star
IDC* [shpitser2012]	y0.algorithm.identify.idc_star
Cyclic ID [forre2019]	y0.algorithm.identify.cyclic_id
gID [correa2019]	Issue #72
gID* [correa2021]	Issue #121
Data ID [nabi2020]	Issue #73

This table partially draws from [tikka2021], which made a nice review of non-counterfactual identification methods.

Transport

Algorithm	Implementation
Surrogate Outcomes [tikka2018]	y0.algorithm.transport
Transportability [correa2020]	y0.algorithm.transport
Counterfactual Transportability	y0.algorithm.counterfactual_transport
[correa2022]

[shpitser2006] (1,2)

Identification of joint interventional distributions in recursive semi-Markovian causal models

[shpitser2008] (1,2)

Complete Identification Methods for the Causal Hierarchy

[shpitser2012] (1,2,3,4)

https://dl.acm.org/doi/10.5555/1597348.1597382

[tikka2018]

https://arxiv.org/abs/1806.07172

[correa2020]

General Transportability: Synthesis of Experiments from Heterogeneous Domains

[correa2022]

Counterfactual Transportability: A Formal Approach

[forre2019]

Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias

[nabi2020]

Full Law Identification In Graphical Models Of Missing Data: Completeness Results

[correa2019]

General Identifiability with Arbitrary Surrogate Experiments

[correa2021]

Nested Counterfactual Identification from Arbitrary Surrogate Experiments

[tikka2021]

Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approach

class Identification(query: Query, graph: NxMixedGraph, estimand: Expression | None = None)

A package of a query and resulting estimand from identification on a graph.

Instantiate an identification.

Parameters:

• query – The generalized identification query (outcomes/treatments/conditions)

• graph – The graph

• estimand – If none is given, will use the joint distribution over all variables in the graph.

classmethod from_parts(outcomes: Variable | Iterable[Variable], treatments: Variable | Iterable[Variable], graph: NxMixedGraph, estimand: Expression | None = None, conditions: Variable | Iterable[Variable] | None = None) → Identification

Instantiate an identification.

Parameters:

• outcomes – The outcomes in the query

• treatments – The treatments in the query (e.g., counterfactual variables)

• conditions – The conditions in the query (e.g., coming after the bar)

• graph – The graph

• estimand – If none is given, will use the joint distribution over all variables in the graph.

Returns:

An identification object

classmethod from_expression(*, query: Probability | Distribution, graph: NxMixedGraph, estimand: Expression | None = None) → Identification

Instantiate an identification.

Parameters:

• query – The query probability expression

• graph – The graph

• estimand – If none is given, will use the joint distribution over all variables in the graph.

Returns:

An identification object

property outcomes: set[Variable]

Return this identification object’s query’s outcomes.

property treatments: set[Variable]

Return this identification object’s query’s treatments.

property conditions: set[Variable]

Return this identification object’s query’s conditions.

exchange_observation_with_action(variables: Variable | Iterable[Variable]) → Identification

Move the condition variable(s) to the treatments.

exchange_action_with_observation(variables: Variable | Iterable[Variable]) → Identification

Move the treatment variable(s) to the conditions.

with_treatments(extra_treatments: Iterable[Variable]) → Identification

Create a new identification with additional treatments.

uncondition() → Identification

Move the conditions to outcomes.

class Query(outcomes: Variable | Iterable[Variable], treatments: Variable | Iterable[Variable], conditions: None | Variable | Iterable[Variable] = None)

An identification query.

Instantiate an identification.

Parameters:

• outcomes – The outcomes in the query

• treatments – The treatments in the query (e.g., counterfactual variables)

• conditions – The conditions in the query (e.g., coming after the bar)

classmethod from_str(outcomes: str | Iterable[str], treatments: str | Iterable[str], conditions: Iterable[str] | None = None) → Query

Construct a query from text variable names.

classmethod from_expression(query: Probability | Distribution) → Query

Instantiate an identification.

Parameters:

query – The query probability expression

Returns:

An identification tuple

Raises:

ValueError – If there are ragged counterfactual variables in the query

exchange_observation_with_action(variables: Variable | Iterable[Variable]) → Query

Move the condition variable(s) to the treatments.

exchange_action_with_observation(variables: Variable | Iterable[Variable]) → Query

Move the treatment variable(s) to the conditions.

with_treatments(extra_treatments: Iterable[Variable]) → Query

Create a new identification with additional treatments.

uncondition() → Query

Move the conditions to outcomes.

property expression: Expression

Return the query as a Probabilistic expression.

exception Unidentifiable

Raised on failure of the identification algorithm.

cyclic_id(graph: Annotated[NxMixedGraph, InPaperAs('G')], outcomes: Annotated[Variable | Iterable[Variable], InPaperAs('Y')], interventions: Annotated[Variable | Iterable[Variable], InPaperAs('W')], *, ordering: Sequence[Variable] | None = None) → Annotated[Expression, InPaperAs('P(Y \\mid do(W))')]

Identify causal effects in cyclic graphs.

Parameters:

• graph – Causal graph

• outcomes – Target variables \(Y\)

• interventions – Intervention variables \(W\)

• ordering – Ordering of variables in the graph. If not given, an apt-order is calculated with get_apt_order()

Returns:

Identified causal effect \(P(Y | do(W))\)

Raises:

• ValueError – If preconditions are violated.

• Unidentifiable – If the causal effect cannot be identified based on the query and graph.

id_star(graph: NxMixedGraph, event: dict[Variable, Intervention], *, _number_recursions: int = 0) → Expression

Apply the ID* algorithm to the graph from [shpitser2012].

idc(identification: Identification, *, ordering: Sequence[Variable] | None = None) → Expression

Run the IDC algorithm from [shpitser2008].

Parameters:

• identification – The identification tuple

• ordering – A topological ordering of the variables. If not passed, is calculated from the directed component of the mixed graph.

Returns:

An expression created by the identify() algorithm after simplifying the original query

Raises “Unidentifiable” if no appropriate identification can be found.

idc_star(graph: NxMixedGraph, outcomes: dict[Variable, Intervention], conditions: dict[Variable, Intervention], *, _number_recursions: int = 0) → Expression

Run the IDC* algorithm from [shpitser2012].

Parameters:

• graph – The causal graph

• outcomes – The outcome events corresponds to \(\gamma\)

• conditions – The condition events corresponds to \(\delta\)

• _number_recursions – The number of times the algorithm has recurred

Returns:

An expression created by the idc_star() algorithm after simplifying the original query

Raises:

ValueError – If ID* returns zero

identify(identification: Identification, *, ordering: Sequence[Variable] | None = None) → Expression

Run the ID algorithm from [shpitser2006].

Parameters:

• identification – The identification tuple

• ordering – A topological ordering of the variables. If not passed, is calculated from the directed component of the mixed graph.

Returns:

the expression corresponding to the identification

Raises:

Unidentifiable – If no appropriate identification can be found

See also identify_outcomes() for a more idiomatic way of running the ID algorithm given a graph, treatments, and outcomes.

identify_outcomes(graph: NxMixedGraph, treatments: Variable | Iterable[Variable], outcomes: Variable | Iterable[Variable], conditions: None | Variable | Iterable[Variable] = None, *, strict: Literal[True] = False, ordering: Sequence[Variable] | None = None) → Expression

identify_outcomes(graph: NxMixedGraph, treatments: Variable | Iterable[Variable], outcomes: Variable | Iterable[Variable], conditions: None | Variable | Iterable[Variable] = None, *, strict: Literal[False] = False, ordering: Sequence[Variable] | None = None) → Expression | None

Calculate the estimand for the treatment(s)m outcome(s), and optional condition(s).

Parameters:

• graph – An acyclic directed mixed graph

• treatments – The node or nodes that are treated

• outcomes – The node or nodes that are outcomes

• conditions – Optional condition or condition nodes. If given, uses the IDC algorithm via y0.algorithm.identify.idc(). Otherwise, uses the ID algorithm via y0.algorithm.identify.identify().

• strict – If true, raises an error identification fails rather than returning None

• ordering – A topological ordering of the variables. If not passed, is calculated from the directed component of the mixed graph.

Returns:

An expression representing the estimand if the query is identifiable. If the query is not identifiable, returns none.

Do Calculus.

rule_2_of_do_calculus_applies(graph: Annotated[NxMixedGraph, InPaperAs('G')], *, treatments: Annotated[set[Variable], InPaperAs('\\mathbf{X}')], outcomes: Annotated[set[Variable], InPaperAs('\\mathbf{Y}')], conditions: Annotated[set[Variable], InPaperAs('\\mathbf{Z}')], condition: Variable) → bool

Check if Rule 2 of the Do-Calculus applies to the conditioned variable.

Parameters:

• graph – The graph

• treatments – The treatments

• conditions – The conditions

• outcomes – The outcomes

• condition – The condition to check

Returns:

If rule 2 applies, see below.

If Rule 2 of the do calculus applies to the conditioned variable, then it can be converted to a do variable.

\[\begin{split}\newcommand\ci{\perp\!\!\!\perp} \newcommand{\ubar}[1]{\underset{\bar{}}{#1}} \newcommand{\obar}[1]{\overset{\bar{}}{#1}} \text{if } (\exists Z \in \mathbf{Z})(\mathbf{Y} \ci Z | \mathbf{X}, \mathbf{Z} - \{Z\})_{G_{\bar{\mathbf{X}}\ubar{Z}}} \\ \text{then } P(\mathbf{Y}|do(\mathbf{X}),\mathbf{Z}) = P(\mathbf Y|do(\mathbf X), do(Z), \mathbf{Z} - \{Z\})\end{split}\]