The observed frequency of a particular outcome in data-based simulation, known as bootstrap probability (BP) of Felsenstein (1985), is very useful as a confidence level of data analysis with discrete outcomes such as estimating the phylogenetic tree from aligned DNA sequences or identifying the clusters from microarray expression profiles. We argue that the length of simulated data sets should be
(-1) times the original data length for avoiding false positives, i.e., bias of hypothesis testing, although such a negative data length cannot be realized. In another word, we perform the “m out of n”
bootstrap with m=-n. This turns out to be equivalent to the approximately unbiased (AU) confidence level computed by the multiscale bootstrap of Shimodaira (2002), but such a notion of negative data length has not been known until Shimodaira (2008). The method is illustrated in real data analysis of phylogenetic inference and hierarchical clustering. In the latter part of the talk, the mathematical justification is explained in terms of distance and curvature with connection to the geometrical theory of Efron and Tibshirani (1998) and the argument of Perlman and Wu (1999). BP is interpreted as Bayesian posterior probability and AU is the frequentist p-value, and thus changing the length of simulated data sets bridges the gap between these two confidence levels.