The Fisher-Rao Metric for Projective Transformations of the Line

A conditional probability density function is defined for measurements arising from a projective transformation of the line. The conditional density is a member of a parameterised family of densities in which the parameter takes values in the three dimensional manifold of projective transformations of the line. The Fisher information of the family defines on the manifold a Riemannian metric known as the Fisher-Rao metric. The Fisher-Rao metric has an approximation which is accurate if the variance of the measurement errors is small. It is shown that the manifold of parameter values has a finite volume under the approximating metric.