  Simultaneous, post-hoc inference is desirable in large-scale hypotheses
testing as it allows for exploration of data while deciding on criteria for
proclaiming discoveries. It was recently proved that all admissible post-hoc
inference methods for the number of true discoveries must be based on closed
testing. In this paper we investigate tractable and efficient closed testing
with local tests of different properties, such as monotonicty, symmetry and
separability, meaning that the test thresholds a monotonic or symmetric
function or a function of sums of test scores for the individual hypotheses.
This class includes well-known global null tests by Fisher, Stouffer and
Ruschendorf, as well as newly proposed ones based on harmonic means and Cauchy
combinations. Under monotonicity, we propose a new linear time statistic
("coma") that quantifies the cost of multiplicity adjustments. If the tests are
also symmetric and separable, we develop several fast (mostly linear-time)
algorithms for post-hoc inference, making closed testing tractable. Paired with
recent advances in global null tests based on generalized means, our work
immediately instantiates a series of simultaneous inference methods that can
handle many complex dependence structures and signal compositions. We provide
guidance on choosing from these methods via theoretical investigation of the
conservativeness and sensitivity for different local tests, as well as
simulations that find analogous behavior for local tests and full closed
testing. One result of independent interest is the following: if
$P_1,\dots,P_d$ are $p$-values from a multivariate Gaussian with arbitrary
covariance, then their arithmetic average P satisfies $Pr(P \leq t) \leq t$ for
$t \leq \frac{1}{2d}$.
