  In this article, we first establish the theory of optimal scores for speaker
recognition. Our analysis shows that the minimum Bayes risk (MBR) decisions for
both the speaker identification and speaker verification tasks can be based on
a normalized likelihood (NL). When the underlying generative model is a linear
Gaussian, the NL score is mathematically equivalent to the PLDA likelihood
ratio, and the empirical scores based on cosine distance and Euclidean distance
can be seen as approximations of this linear Gaussian NL score under some
conditions. We discuss a number of properties of the NL score and perform a
simple simulation experiment to demonstrate the properties of the NL score.
