backwash_second_step: Second step of the backwash procedure.
In dcgerard/vicar: Various Ideas for Confounder Adjustment in Regression

Second step of the backwash procedure.

backwash_second_step(
  betahat_ols,
  S_diag,
  alpha_tilde,
  tau2_seq,
  lambda_seq,
  pi_init_type = c("zero_conc", "uniform", "random"),
  scale_var = TRUE,
  sprop = 0,
  var_inflate_pen = 0,
  verbose = TRUE
)

`betahat_ols`	A vector of numerics. The ordinary least squares estimates of the regression coefficients.
`S_diag`	A vector of positive numerics. The standard errors of `betahat_ols`.
`alpha_tilde`	A matrix of numerics. The estimated coefficients of the confounders.
`tau2_seq`	A vector of positive numerics. The known grid of prior mixing variances.
`lambda_seq`	A vector of penalties for the estimate of the mixing proportions.
`pi_init_type`	How should we initialize the mixing proportions? By concentrating on zero (`"zero_conc"`), by equal weights on all mixing distributions (`"uniform"`), or by sampling uniformly on the simplex (`"random"`)?
`scale_var`	A logical. Should we estimate a variance inflation parameter (`TRUE`) or not (`FALSE`)?
`sprop`	If b is an effect and s is an estimated standard error, then we model b/s^{sprop} as exchangeable. The default is 0. When `sprop = 1`, for identifiability reasons it must be the case that `scale_var = FALSE`.
`var_inflate_pen`	The penalty to apply on the variance inflation parameter. Defaults to 0, but should be something non-zero when `alpha = 1` and `scale_var = TRUE`.
`verbose`	If `verbose = TRUE`, print progress of the algorithm to the console.