Selective inference on multiple families of hypotheses

In many complex multiple-testing problems the hypotheses are divided into families. Given the data, families with evidence for true discoveries are selected, and hypotheses within them are tested. Neither controlling the error rate in each family separately nor controlling the error rate over all hypotheses together can assure some level of confidence about the filtration of errors within the selected families. We formulate this concern about selective inference in its generality, for a very wide class of error rates and for any selection criterion, and present an adjustment of the testing level inside the selected families that retains control of the expected average error over the selected families.