Abstract Reject inference has an established role in the development of scorecards for credit applications. The performance of the rejects, had they been accepted, is inferred to be good or bad in order to obtain a complete picture of the population applying for credit. Once this is done, the scorecard to assess this population can then be developed. But consider the following problem. A company mails its customer base with an offer of additional finance facilities. The problem is: who should it mail to maximize response while minimizing risk, while also minimizing the number of responders who are rejected at application time to avoid jeopardizing the existing customer relationship? 1bis problem has three inferences required to tackle it completely. First, there is the classic inference at the point of application to infer which rejects, had they been accepted, would have been good (or bad). Second, there is the inference at the point of mailing to infer which customers, had they been previously mailed, would have responded-a response inference. But third, and most interesting, is the double inference of inferring which of the inferred responders would have subsequently been good, bad, or rejected-an inference on an inference!