Relation of optimal local compression and local likelihood ratio

As Tenney and Sandell pointed out, the optimal local compression/decision rule has the form of a likelihood ratio when the local observations are not correlated. However, this does not hold in general for correlated observations. An interesting problem is to find the conditions under which the optimal local compression rule remains to have the form of a likelihood ratio even for correlated observations. In this paper, we prove that, with the model of Gaussian signal with independent Gaussian noise, the optimal local compression rule has the expected likelihood ratio form when the two conditional probability density functions are centrosymmetric. Computer simulation is provided in the paper for demonstration.