Remarks on statistical inference for statistical decisions

The Wald development of statistical decision theory addresses decision making with sample data. Wald's concept of a statistical decision function (SDF) embraces all mappings of the form [data => decision]. An SDF need not perform statistical inference; that is, it need not use data to draw conclusions about the true state of nature. Inference-based SDFs have the sequential form [data => inference => decision]. This paper offers remarks on the use of statistical inference in statistical decisions. Concern for tractability may provide a practical reason for study of inference-based SDFs. Another practical reason may be necessity. There often is an institutional separation between research and decision making, with researchers reporting inferences to the public. Then planners can perform the mapping [inference => decision], but they cannot perform the more basic mapping [data => decision]. The paper first addresses binary choice problems, where all SDFs may be viewed as hypothesis tests. It next considers as-if optimization, where one uses a point estimate of the true state as if the estimate is accurate. It then extend this idea to as-if decisions using set estimates of the true state, such as confidence sets.