target_Qbarbar_M1_star_times_M2_star_a: Function for targeting Qbarbar_M1_star_times_M2_star_a
In benkeser/intermed:

There are two interesting features of this targeting problem. First, we see that the nuisance parameter Qbarbar_M1_star_times_M2_star_a can be viewed in two ways: (1) the conditional mean of Qbarbar_M1_star_a given C with respect to the marginal of M_2 given A = a_star, C; (2) the conditional mean of Qbarbar_M2_star_a given C with respect to the marginal of M_1 given A = a_star, C. The natural inclination then is to use a sum loss function. Here it looks like we actually can use a sum loss approach, so long as the IPTW are incorporated into the loss function. We make two copies of each observation with A = a_star; assign Qbarbar_M1_star_a as outcome in half and Qbarbar_M2_star_a in the other half; then do one-shot targeting

target_Qbarbar_M1_star_times_M2_star_a(
  Qbarbar,
  Y,
  A,
  a,
  a_star,
  gn,
  tol = 1/(sqrt(length(Y)) * log(length(Y))),
  ...
)

`Qbarbar`	Iterated mean estimates
`Y`	A vector of continuous or binary outcomes.
`A`	A vector of binary treatment assignment (assumed to be equal to 0 or 1).
`a`	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of `a` versus `a_star`.
`a_star`	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of `a` versus `a_star`.
`gn`	Power users may wish to pass in their own properly formatted list of the propensity score so that nuisance parameters can be fitted outside of `intermed`.
`tol`	The tolerance for stopping the iterative targeting procedure.