Causal models for qualitative and mixed methods inference

Exercise 2: Bayes

Author

Macartan Humphreys and Alan Jacobs

1 Part 1: Queries (30 mins)

Return to the model that you created in Exercise 1. Now write down at least 3 queries that can be defined for this model and that might be of substantive interest.

You should write down each query in two ways:

  1. In plain, but maximally precise, plain language.

  2. Formally using CausalQueries. See the queries handout for examples. Be sure to distinguish between controlled interventions (using [X=...]) and simple conditioning (using “given…”).

2 Part II: A causal chain (30 mins)

Imagine a causal chain: \(A\) causes \(B\) and \(B\) causes \(C\)

Say you knew for the \(A\), \(B\) relationship:

  • there is a 50% chance that \(A\) causes \(B\) for a given unit.
  • there is a 25% chance that \(B=0\) regardless of \(A\)
  • there is a 25% chance that \(B=1\) regardless of \(A\)

Say you knew for the \(B\), \(C\) relationship:

  • there is a 50% chance that \(B\) causes \(C\) for a given unit.
  • there is a 25% chance that \(C=0\) regardless of \(B\)
  • there is a 25% chance that \(C=1\) regardless of \(B\)

Now:

  • What is the probability that \(A\) causes \(C\)?
  • What is the probability that \(A\) causes \(C\) for a case in which \(A=1\) and \(C=1\), when \(B\) is not observed?
  • What is the probability that \(A\) causes \(C\) for a case in which \(A=1\) and \(B=1\) and \(C=1\)?
  • What is the probability that \(A\) causes \(C\) for a case in which \(A=1\) and \(B=0\) and \(C=1\)?

Summarize: when is \(B\) informative for the \(A \rightarrow C\) relationship.

3 Group discussion (30 mins)

4 Additional (if you have time!): Bayesian sequencing and posterior variance (30 mins)

Say you knew the probability of \(Y\) given \(X_1\) and \(X_2\):

X_1 = 0 X1 = 1
X_2 = 0 .2 .5
X_2 = 1 .8 .9

In addition you know that \(X_1 = 1\) with probability 0.5, and \(X_2 = 1\) with probability 0.5, independently.

  1. Say you do not know \(X_1\) or \(X_2\). What is your prior that \(Y = 1\)?
  2. How uncertain are you? (You can use variance as a measure, where the variance in beliefs about a bernoulli event that arises with probability \(p\) is just \(p(1-p)\))
  3. Say you now learn \(X_1 = 0\): what is your posterior belief now?
  4. What is your uncertainty now? (Again use variance, but now you should look at the posterior variance). Has it gone up or down with the additional information?
  5. Say in addition you now learn \(X_2 = 1\): what now is your posterior? What now is your uncertainty?
  6. Ex ante you cannot tell what you would observe if you looked for \(X_1\) or \(X_2\). Thinking through all the patterns you might see, what is the expected estimate you would get if you just looked for \(X_1\) or just looked for \(X_2\)? What is the expected posterior variance? If you had to choose just one of \(X_1\) or \(X_2\) to look for, which would be more informative?

Bonus: Think through how things might change if X1 and X2 were not independent?