Information Geometry of the Locally Most Powerful Test

The locally most powerful (LMP) test for weak signal detection is studied from a information-geometric perspective. In such a framework, the LMP test is identified as the norm of natural whitened gradient on the statistical manifold consisting of a family of parametric probability distributions, which indicates that the LMP test pursues the steepest learning directions from the null hypothesis to the empirical distribution of the observed data on the manifold. A concrete geometrical interpretation of the LMP test in the theory of information geometry is presented, which leads to an immediate extension of the LMP test to a vector valued parameter case. Example of multi-component sinusoidal signal detection under low SNR conditions confirms a practical importance of the extended test.