Before revision:  The geometry of wald spaceis that of the Fisher information metric of character distributions , either from a discrete Bernoulli or from a continuous Gaussian model. The latter can be viewed as the trace metric of the affine-invariant metric for covariance matrices, the former is that of the Hellinger divergence, or  , as we show, equivalent to any metric obtained from an f -divergence , such as the Jensen-Shannon metric .
After revision:  The geometry of wald spaceis that of the Fisher information metric of character distributions induced by the two-state symmetric Markov substitution process on each tree. Infinitesimally, the metric is proportional to the Kullback-Leibler divergence, or , as we show, equivalent to any metric obtained from an f -divergence , such as the Jensen-Shannon metric .