  We consider statistical inference problems under uncertain equality
constraints, and provide asymptotically valid uncertainty estimates for
inferred parameters. The proposed approach leverages the implicit function
theorem and primal-dual optimality conditions for a particular problem class.
The motivating application is multi-dimensional raking, where observations are
adjusted to match marginals; for example, adjusting estimated deaths across
race, county, and cause in order to match state all-race all-cause totals. We
review raking from a convex optimization perspective, providing explicit
primal-dual formulations, algorithms, and optimality conditions for a wide
array of raking applications, which are then leveraged to obtain the
uncertainty estimates. Empirical results show that the approach obtains, at the
cost of a single solve, nearly the same uncertainty estimates as
computationally intensive Monte Carlo techniques that pass thousands of
observed and of marginal draws through the entire raking process.
