Effective Sampling Rates for Signal Detection; or Can the Gaussian Model be Salvaged?

The usual justification for talking about signal-to-noise ratio is in terms of a Gaussian model. This model can be treated either by means of information theory or by means of (earlier) methods of statistical inference. In either case the justification is often achieved by assuming that sampling is done at the Nyquist rate. This justification collapses if we are given a record of finite duration, since the sampling theorem is then inapplicable. In fact the Gaussian model itself collapses since it leads to the absurd conclusion that an infinite amount of information can be obtained in a finite time. But the mathematical convenience of the Gaussian model cannot be lightly brushed aside. The intention of this paper is primarily to try to salvage the Gaussian model by assuming that there is an effective sampling rate that cannot be exceeded. This rate could be slower or faster than the Nyquist rate. If an inefficient, but pleasantly simple statistic (the “power statistic”) is used, then there is less point in sampling faster than the Nyquist rate. For the reader's convenience, some material on spectral analysis and other matters is collected together in the Appendices. Most of it could be found, explicitly or implicitly, in previous literature. The notion of interaction for weight of evidence, and its relationship to spectral analysis, is explained in Appendix 6, and does not seem to have been previously published.