A relationship between the likelihood ratio and a generalized causal conditional mean estimator is presented for the doubly composite hypotheses problem. The observation statistics are arbitrary and need not be Gaussian. The relationship parallelsthe well-known relationship for the Gaussian noise and composite signal hypothesis case, and the likelihood ratio can still be viewed as an estimator-correlator operation.

A new philosophy for designing detection devices, which embodies some of the better features of the classical generalized likelihood ratio test and the classical Bayes test, is presented. A feature of the new Bayes generalized likelihood ratio test is that it provides a unified procedure for utilizing parameter estimates in detector design. Its use depends on the existence of an asymptotically unbiased estimator for the unknown parameters. If such an estimator is easily implemented, then the new test should be particularly attractive.

A fundamental spatial property is noted that relates the likelihood ratio for random signals in Gaussian noise to conditional-mean parameter estimation. This allows a geometric interpretation of the estimation-correlation operation. Some additional properties exhibited by the likelihood ratio are also presented.