  Selective inference is considered for testing trees and edges in phylogenetic
tree selection from molecular sequences. This improves the previously proposed
approximately unbiased test by adjusting the selection bias when testing many
trees and edges at the same time. The newly proposed selective inference
$p$-value is useful for testing selected edges to claim that they are
significantly supported if $p>1-\alpha$, whereas the non-selective $p$-value is
still useful for testing candidate trees to claim that they are rejected if
$p<\alpha$. The selective $p$-value controls the type-I error conditioned on
the selection event, whereas the non-selective $p$-value controls it
unconditionally. The selective and non-selective approximately unbiased
$p$-values are computed from two geometric quantities called signed distance
and mean curvature of the region representing tree or edge of interest in the
space of probability distributions. These two geometric quantities are
estimated by fitting a model of scaling-law to the non-parametric multiscale
bootstrap probabilities. Our general method is applicable to a wider class of
problems; phylogenetic tree selection is an example of model selection, and it
is interpreted as the variable selection of multiple regression, where each
edge corresponds to each predictor. Our method is illustrated in a previously
controversial phylogenetic analysis of human, rabbit and mouse.

Applications;Populations and Evolution