We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1-\alpha$ post-hoc confidence bounds on the number of positive and negative (or non-positive) parameters. The guarantee is simultaneous, for all subsets of parameters. Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the $p$-values of the one-sided hypotheses for the selection, and use the adjusted $p$-values for simultaneous inference on the selected $n$ one-sided hypotheses. The adjustment is straightforward assuming that the $p$-values of one-sided hypotheses have densities with monotone likelihood ratio, and are mutually independent. We show that the bounds we provide are tighter (often by a great margin) than existing alternatives, and that they can be obtained by at most a polynomial time. We demonstrate the usefulness of our simultaneous post-hoc bounds in the evaluation of treatment effects across studies or subgroups. Specifically, we provide a tight lower bound on the number of studies which are beneficial, as well as on the number of studies which are harmful (or non-beneficial), and in addition conclude on the effect direction of individual studies, while guaranteeing that the probability of at least one wrong inference is at most 0.05.

翻译：我们考虑对 $n>1$ 个参数的符号进行推断的问题。目标是为正参数和负（或非正）参数的数量提供 $1-\alpha$ 的事后置信界。该保证是同时的，适用于所有参数子集。我们的建议如下：首先利用数据为每个参数选择假设检验的方向；然后，针对选择过程调整单侧假设的 $p$ 值，并使用调整后的 $p$ 值对所选的 $n$ 个单侧假设进行同时推断。若单侧假设的 $p$ 值具有单调似然比的密度且相互独立，则调整过程直接明了。我们证明，所提供的界比现有替代方案更紧（通常大幅收紧），且可在多项式时间内获得。我们通过跨研究或亚组的治疗效果评估展示了同时事后界的有用性。具体而言，我们为有益研究的数量以及有害（或非有益）研究的数量提供了紧的下界，此外还推断出单个研究的效果方向，同时保证至少一次错误推断的概率不超过0.05。