  The covariance matrix adaptation evolution strategy (CMA-ES) is a stochastic
search algorithm using a multivariate normal distribution for continuous
black-box optimization. In addition to strong empirical results, part of the
CMA-ES can be described by a stochastic natural gradient method and can be
derived from information geometric optimization (IGO) framework. However, there
are some components of the CMA-ES, such as the rank-one update, for which the
theoretical understanding is limited. While the rank-one update makes the
covariance matrix to increase the likelihood of generating a solution in the
direction of the evolution path, this idea has been difficult to formulate and
interpret as a natural gradient method unlike the rank-$\mu$ update. In this
work, we provide a new interpretation of the rank-one update in the CMA-ES from
the perspective of the natural gradient with prior distribution. First, we
propose maximum a posteriori IGO (MAP-IGO), which is the IGO framework extended
to incorporate a prior distribution. Then, we derive the rank-one update from
the MAP-IGO by setting the prior distribution based on the idea that the
promising mean vector should exist in the direction of the evolution path.
Moreover, the newly derived rank-one update is extensible, where an additional
term appears in the update for the mean vector. We empirically investigate the
properties of the additional term using various benchmark functions.
