networks, contagion, causality faculty at MIT
![Figure 1: The complete set of potential outcome functions yi(z) for every unit i ∈ [n] is encoded in the unknown “science” table (in blue), where
each row is a unit i, and each column is a set of treated units {j ∈ [n] | Zj = 1}. We only observe one column (in red) of the table: Yi ≜ yi(Z) for
the realized treatment vector Z ∈ {0, 1}n, Z ∼ π, V ≜ {j ∈ [n] | Zj = 1}. We can “compress” this table into interpretable estimands by moving
in one of two directions: (a) integrating over the columns, i.e. marginalizing over the treatments under unit-specific policies πi (move right),
or (b) integrating over the rows, i.e. averaging the outcomes over treatment-specific node sets (move down). These averages can be executed
in succession — a form of “double averaging” — and the resulting estimands will be typically sensitive to their ordering. In other words, the
averaging operations generally do not commute; see Proposition 2. Here, the averaging is done with respect to the “exposure map” Di ≜ di(Z)
of the number of treated neighbors a unit has.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:zobujqws6rru6wvxtggnkwh7/bafkreic6jntenifpmpaneb3a2jisz2l5w4medx2micsykecuup4r62umha)
networks, contagion, causality faculty at MIT