On Rogers' Proof of Identifiability for the GTR plus Γ plus I Model
On Rogers' Proof of Identifiability for the GTR plus Γ plus I Model
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DOI：
10.1093/sysbio/syr023
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发表时间：
2011-10-01
影响因子：
6.5
通讯作者：
Housworth, Elizabeth A.
中科院分区：
文献类型：
--
作者：
Chai, Juanjuan;Housworth, Elizabeth A.
模型可识别性是任何统计一致性证明的关键组成部分。可识别性意味着在给定来自模型的无限数量的数据的情况下，可以推断出模型的所有参数。例如，对于F81 + Γ模型下的系统发育推断，参数是具有分支长度的无根系统发育树、特定的F81速率矩阵（Felsenstein 1981）和描述速率异质性的γ（Γ）分布的形状参数。F81速率矩阵特别简单，突变到状态x的速率仅取决于状态x的长期频率。该模型仅使用成对物种比较（即，联合成对DNA状态分布）是不可识别的（Steel 2009）。对于任何F81矩阵，任何两个不同的伽马分布，以及任何四个或更多物种的集合，Steel（2009）表明每个伽马分布都有不同的拓扑结构，这些拓扑结构给出了这些物种的相同联合成对DNA序列分布。任何仅使用成对比较的模型参数的统计估计，如基于距离的方法，将试图估计参数空间中的两个独立点，违反一致性的定义。为了这项工作的目的，我们定义的通用可识别性意味着一个模型是不可识别的参数集具有比整个参数空间更小的维度。Wu和Susko（2010）通过两两比较证明了一般时间可逆（GTR）+ Γ模型的通有可辨识性。模型参数是具有分支长度的遗传过程、描述GTR DNA沿着遗传过程进化的4 × 4瞬时速率矩阵以及伽马分布的形状参数。具体地说，他们证明了除了F81家族的矩阵和至少有两个不同种间距离的所有同源性，速率矩阵，伽马分布的形状，同源性及其分支长度都可以通过成对比较来识别。因为GTR马尔可夫矩阵是用三个非零特征值参数化的，而F81矩阵的子族是用一个非零特征值参数化的，所以F81模型是整个GTR类的低维子集。对于GTR + Γ + I下的系统发育推断，我们增加了一个参数，即不变（I）位点的比例（Gu et al. 1995）。Rogers（2001）认为，这种流行的模型一般可从成对比较中识别，Wald证明最大似然估计量是一致的所有其他方面都适用于系统发育推断。然而，他关于可识别性的论点有一个缺陷（Allman et al. 2008）。
Model identifiability is a key component of any proof of statistical consistency. Identifiability means that it is possible to infer all of the model’s parameters given an infinite amount of data from the model. For phylogenetic inference under the F81+ Γ model, for example, the parameters are the unrooted phylogenetic tree with branch lengths, the particular F81 rate matrix (Felsenstein 1981), and a shape parameter for the gamma (Γ) distribution describing the rate heterogeneity. The F81 rate matrix is particularly simple with the rate of mutating to state x depending only on the longrun frequency of state x. This model is not identifiable using only pairwise species comparisons, that is, the joint pairwise DNA state distributions (Steel 2009). For any F81 matrix, any two distinct gamma distributions, and any set of four or more species, Steel (2009) showed that there are distinct topologies for each gamma distribution that give the same joint pairwise DNA sequence distributions for those species. Any statistical estimator of these model parameters using only pairwise comparisons, such as distance-based methods, will be trying to estimate two separate points in parameter space, violating the definition of consistency. For the purposes of this work, we define generic identifiability to mean that the set of parameters for which a model is not identifiable has a smaller dimension than the whole parameter space. Wu and Susko (2010) proved generic identifiability for the general time reversible (GTR)+ Γ model from pairwise comparisons. The model parameters are the phylogeny with branch lengths, the 4× 4 instantaneous rate matrix describing GTR DNA evolution along the phylogeny, and a shape parameter for the gamma distribution. Specifically, they proved that for all but the F81 family of matrices and for all phylogenies with at least two distinct interspecies distances, the rate matrix, the shape of the gamma distribution, and the phylogeny and its branch lengths are identifiable from pairwise comparisons. Because a GTR Markov matrix is parameterized using three nonzero eigenvalues but the F81 subfamily of matrices is parameterized using only one nonzero eigenvalue, the F81 model is a lower dimensional subset of the whole GTR class. Similarly, given more than two taxa, phylogenies with only one value for all of their interspecies distances make up a lower dimensional subset of the whole of tree space.For phylogenetic inference under GTR+ Γ+ I, we add an additional parameter, which is the proportion of invariable (I) sites (Gu et al. 1995). Rogers (2001) argued that this popular model was generically identifiable from pairwise comparisons and that all other aspects of Wald’s proof that maximum likelihood estimators are consistent held for phylogenetic inference. His argument of identifiability, however, contained a flaw (Allman et al. 2008).