53,876 research outputs found
The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance
For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this article,
this statistic is studied under the Yule and uniform model of phylogenetic
trees. The main tool of analysis is a coupling argument with another well-known
index called the Sackin statistic. Asymptotics for the mean, variance and
covariance of these two statistics are obtained, as well as their limiting
joint distribution for large phylogenies. Under the Yule model, the limiting
distribution arises as a solution of a functional fixed point equation. Under
the uniform model, the limiting distribution is the Airy distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and the
Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Evidence: Admission of Mathematical Probability Statistics Held Erroneous for Want of Demonstration of Validity
In State v. Sneed the New Mexico Supreme Court limited its disapproval of evidence of probability statistics to the particular facts presented but failed to articulate specific safeguards for subsequent use of such evidence. This note explores the nature of probability statistics, their potential utility in a legal context, and criteria by which their admissibility might be determined
Local Exchangeability
Exchangeability---in which the distribution of an infinite sequence is
invariant to reorderings of its elements---implies the existence of a simple
conditional independence structure that may be leveraged in the design of
probabilistic models, efficient inference algorithms, and randomization-based
testing procedures. In practice, however, this assumption is too strong an
idealization; the distribution typically fails to be exactly invariant to
permutations and de Finetti's representation theory does not apply. Thus there
is the need for a distributional assumption that is both weak enough to hold in
practice, and strong enough to guarantee a useful underlying representation. We
introduce a relaxed notion of local exchangeability---where swapping data
associated with nearby covariates causes a bounded change in the distribution.
We prove that locally exchangeable processes correspond to independent
observations from an underlying measure-valued stochastic process. We thereby
show that de Finetti's theorem is robust to perturbation and provide further
justification for the Bayesian modelling approach. Using this probabilistic
result, we develop three novel statistical procedures for (1) estimating the
underlying process via local empirical measures, (2) testing via local
randomization, and (3) estimating the canonical premetric of local
exchangeability. These three procedures extend the applicability of previous
exchangeability-based methods without sacrificing rigorous statistical
guarantees. The paper concludes with examples of popular statistical models
that exhibit local exchangeability
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