When analyzing a causal question, start with the probability of every possible combination of variable outcomes. A joint probability distribution maps each combination of outcomes for variables such as X and Y to a probability value. It describes how variables occur together, but it is not yet a data-generating process (DGP): causal modeling requires an account of how those outcomes are produced. section 2.5 section 2.3.2
Observed data provide an empirical joint distribution rather than the actual distribution. For discrete variables, count every observed outcome combination and divide by the number of observations. A zero count means an outcome was unseen in the sample, not necessarily impossible. With continuous variables, histograms, density curves, or related summaries represent the same empirical distribution. section 2.3.2
Use the joint distribution to keep several modeling distinctions explicit:
- Separate observational probabilities from causal mechanisms: the observational joint distribution describes what is seen, while the DGP describes how variables are generated. section 2.3.2
- Include latent variables when they help specify the mechanism. Marginalizing the full joint distribution over latent variables recovers the observational joint distribution. section 2.3.2
- Sample from a joint distribution to simulate complete combinations of variable values rather than sampling each variable in isolation. section 2.2.2
Rock throws and a broken window
Represent Jenny's throw, Brian's throw, and whether the window breaks as three binary variables. Five observations produce empirical proportions for all eight combinations, including several zero-proportion combinations. A fuller model can add latent impact strength; summing over that unobserved variable yields the observational joint distribution. A DGP can then generate the throw and window outcomes from underlying inclinations and window strength. section 2.3.2
Table 2.1 A simple data set with five examples
|
jenny_throws_rock
|
brian_throws_rock
|
window_breaks
|
|
|---|---|---|---|
|
1
|
False
|
True
|
False
|
|
2
|
True
|
False
|
True
|
|
3
|
False
|
False
|
False
|
|
4
|
False
|
False
|
False
|
|
5
|
True
|
True
|
True
|
Inspect the five-row dataset before comparing outcome counts, empirical proportions, and a hypothesized true observational distribution. section 2.3.2
