1 Queries

(Alan)

1.1 Estimands (causal questions)

Estimands are the things you want to know.

A few common causal estimands:

• Case-level causal effects
  • e.g., Did/does \(X\) have a positive effect on \(Y\) in this case?
  • e.g., Did \(X=1\) cause \(Y=1\) in this case with \(X=1, Y=1\)? (“causes of effects”)
• Causal pathways
  • e.g., Does \(X\) affect \(Y\) through \(M\)?
    • For this case?
    • For population as a whole?
• Average causal effects for a population
  • e.g., What is the average effect of \(X\) on \(Y\)? (“effects of causes”)

1.2 Case-level causal effects

• In qualitative research, we often see an outcome and want to know what caused it
  • Why does \(Y=1\) in this case?
  • Why did Argentina democratize in the 1980s?
• And we have suspected causes?
  • Did \(X=1\) cause \(Y=1\) in this case?
  • Did \(W=1\) cause \(Y=1\) in this case?
  • Did military defeat cause Argentina’s democratization?
• “Did \(X=1\) cause \(Y=1\) in this case?” means (in a binary framework)….
  • If \(X\) had been \(0\) in this case, would \(Y\) have been \(0\)?
• “Did military defeat cause Argentina’s democratization?” means….
  • If military defeat had not happened, would Argentina’s democratization have failed to happen?

1.3 The probability of a case-level causal effect

• Our estimates of case-level causal effects have uncertainty

  • Our inferences thus take the form of probabilities – as degrees of belief

• We call this probability of causation

• We always condition case-level queries on what we already know about the case

  • Given that \(X=1\) and \(Y=1\) in this case, what’s the probability that \(X=1\) caused \(Y=1\)?
  • Given that Argentina experienced military defeat and democratized, what’s the probability that the former caused the latter?

1.4 Case-level causal effects and types

• “Did \(X=1\) cause \(Y=1\) in this case?” means (in a binary framework)….
  • If \(X\) had been \(0\) in this case, would \(Y\) have been \(0\)?
• This is the same as asking: is the case a \(\theta_{01}\) type?
  • This is a type in which
    • \(Y=1\) if \(X=1\)
    • \(Y=0\) if \(X=0\)
• More precisely we ask, what’s the probability that this case is a \(\theta_{01}\) type?

1.5 Causal queries on a DAG

Causal queries can be expressed as nodes on a causal graph

1.6 Causal queries on a DAG: case-level causal effect

• Does \(X\) have a positive effect on \(Y\) in this case?
  • This is a question about the value of \(\theta^Y\)
• Does \(\theta^Y = \theta^Y_{01}\)?

1.7 Pathway questions

• How did \(X\) cause \(Y\) in this case?

  • What is the pathway or mechanism?

• We might think there’s a possible pathway through \(M\), but did that pathway actually operate in this case?

• There could be other possibilities

• Did military defeat cause democratization in Argentina through creating division within the army?

  • Or did military defeat cause democratization in Argentina through generating popular protest?

• Also a question about types

1.8 Causal queries on a DAG: pathway questions

• Does \(X\) have a positive effect on \(Y\) through \(M\) in this case?
• A question about \(\theta^M\) and \(\theta^Y\) in this case

1.9 Causal queries on a DAG: pathway questions

• A question about \(\theta^M\)
  • Does \(M\) respond to \(X\)?
  • E.g., is \(\theta^M = \theta_{01}^M\)?
• Questions about \(\theta^Y\)
  • When \(M\) changes the way it does when \(X\) goes from 0 to 1, does \(Y\) go from 0 to 1, even when \(X\) is held constant?
  • Does \(Y\) increase if \(X\) increases but \(M\) is held constant?
• We work through this in the “Causal Questions” chapter, but larger point:
  • We can define this causal question (like any causal question) as a question about combinations of nodal types (i.e., causal types)

1.10 Causal queries with conditioning

• We can ask: what is the effect of \(X\) on \(Y\) given \(M=1\)?
• Importantly, different from: what is the effect of \(X\) on \(Y\) if \(M\) is set to \(1\)?

1.11 Types of conditioning

We distinguish between:

• the effect of \(X\) on \(Y\) when \(M\) “happens to” take on a given value
• the effect of \(X\) on \(Y\) when \(M\) “is set to” or “fixed at” a given value

Say we think that school attendance (X) increases income (Y) via education (M).

• then we will be more convinced that \(X=1\) caused \(Y=1\) if we observe that \(M=1\), BUT
• we would not expect \(X\) to cause \(Y\) if we were to fix \(M\) to \(1\) – if we were to set education level to some value

2 Queries in CausalQueries

(Macartan)

2.1 Defining queries in CausalQueries

• We use [condition] to describe conditions that are set
• we use “given = condition” arguments to describe conditions that happen to hold

Examples:

• \(Y[X=1]\) is the value of \(Y\) when \(X\) is set to 1. (do\((X=1)\), or \(X \leftarrow 1\))
• \(Y[X=1] - Y[X=0]\) is the effect of \(X\) on \(Y\)
• \(Y[X=1, M = 1] - Y[X=0, M=1]\) is the effect of \(X\) on \(Y\) when \(M\) is set to 1
• \(Y[X=1] - Y[X=0]\) given \(M==1\) is the effect of \(X\) on \(Y\) when \(M\) happens to be 1

2.2 Conditioning in CausalQueries

You can query using multiple queries, and asking for queries based on parameters, priors, or posteriors.

What are each of these?

model <- make_model("X -> M -> Y")

model |> query_model("Y[X=1] - Y[X=0]")

model |> query_model("Y[X=1, M=1] - Y[X=0, M=1]")

model |> query_model("Y[X=1] - Y[X=0]", given = "M==1")

model |> query_model("Y[X=1] - Y[X=0]", using = "posteriors")

2.3 From statements to types

In general even complex queries can be written as sets of causal types. CausalQueries translates from causal statements to causal types.

make_model("X->Y") |>
  get_query_types("Y[X=1] ==1 ")

Causal types satisfying query's condition(s)  

 query =  Y[X=1]==1 

X0.Y01  X1.Y01
X0.Y11  X1.Y11


 Number of causal types that meet condition(s) =  4
 Total number of causal types in model =  8

2.4 From statements to types

In general even complex queries can be written as sets of causal types. CausalQueries translates from causal statements to causal types.

make_model("X->Y") |>
  get_query_types("Y[X=1] == Y[X=0]")

Causal types satisfying query's condition(s)  

 query =  Y[X=1]==Y[X=0] 

X0.Y00  X1.Y00
X0.Y11  X1.Y11


 Number of causal types that meet condition(s) =  4
 Total number of causal types in model =  8

3 Bayesian updating

(Macartan)

Updating on causal quantities

3.1 Outline

1. Bayesian reasoning
2. Bayesian calculations by hand

3.2 Bayes reasoning

• Bayesian methods are just sets of procedures to figure out how to update beliefs in light of new information.

• We begin with a prior belief about the probability that a hypothesis is true.

• New data then allow us to form a posterior belief about the probability of the hypothesis.

3.3 Continuous distributions

For continuous distributions and parameter vector \(\theta\):

\[p(\theta|\mathcal{D})=\frac{p(\mathcal{D}|\theta)p(\theta)}{\int_{\theta'}p(\mathcal{D|\theta'})p(\theta')d\theta}\]

4 Conditional independence

(Alan)

Figuring out when A tells you something about B

4.1 Key insight linking Causal Models to Bayes

The key insight that connects a causal model to Bayesian updating is this:

• a causal model can tell you how to update on one feature given information on another

4.3 Theorem

Key insight for process tracing:

• Informativeness of a node hinges on conditional independence
• If \(K\) is not conditionally independent of \(\theta\) then \(K\) is (possibly) informative for \(\theta\).
• If \(K\) is conditionally independent of \(\theta\) then \(K\) is not informative for \(\theta\).

5 Updating in CausalQueries

(Macartan)

CausalQueries brings these elements together

5.1 Big picture

CausalQueries brings these elements together by allowing users to:

1. Make model: Specify a DAG: CausalQueries figures out all causal types and places probabilities on these (parameters)
2. Query model: CausalQueries figures out which types correspond to a given causal query given case data and figures out the probability of these (given some parameters)
3. Update model: CausalQueries figures out which parameters are more likely given data, allowing you to query using these parameters

5.2 Illustration of 1 and 2

When we pose queries given different data we are in fact using Bayesian updating.

Below we update on our beliefs about a single case given different data we might see. This, essentially, is what process tracing is (in our view).

make_model('X -> M -> Y') |>
  set_restrictions(decreasing("X", "M"))|>
  set_restrictions(decreasing("M", "Y"))|>
  query_model(
    query = "Y[X=1] != Y[X=0]",
    given = c("All",
              "X==1 & Y ==1", 
              "M==0", 
              "M==1",
              "X==1 & Y ==1 & M==0",
              "X==1 & Y ==1 & M==1")) 

Causal queries generated by query_model (all at population level)

|label                                      |query            |given               |using      |  mean|
|:------------------------------------------|:----------------|:-------------------|:----------|-----:|
|Y[X=1] != Y[X=0]                           |Y[X=1] != Y[X=0] |-                   |parameters | 0.111|
|Y[X=1] != Y[X=0] given X==1 & Y ==1        |Y[X=1] != Y[X=0] |X==1 & Y ==1        |parameters | 0.200|
|Y[X=1] != Y[X=0] given M==0                |Y[X=1] != Y[X=0] |M==0                |parameters | 0.111|
|Y[X=1] != Y[X=0] given M==1                |Y[X=1] != Y[X=0] |M==1                |parameters | 0.111|
|Y[X=1] != Y[X=0] given X==1 & Y ==1 & M==0 |Y[X=1] != Y[X=0] |X==1 & Y ==1 & M==0 |parameters | 0.000|
|Y[X=1] != Y[X=0] given X==1 & Y ==1 & M==1 |Y[X=1] != Y[X=0] |X==1 & Y ==1 & M==1 |parameters | 0.250|

5.3 Illustration of 3

We will spend more time on 3 later but in brief:

• update_model() lets you update over population parameters not just case types

model <- make_model("X->Y") 
data  <- data.frame(X = c(0,0,1,1), Y = c(0,0,1,1))

model <- update_model(model, data) |>
  query_model("Y[X=1] - Y[X=0]", using = "posteriors")

Causal queries generated by query_model (all at population level)

|label           |using      |  mean|    sd| cred.low| cred.high|
|:---------------|:----------|-----:|-----:|--------:|---------:|
|Y[X=1] - Y[X=0] |posteriors | 0.349| 0.259|   -0.168|     0.817|