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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hierarchical Dirichlet process ： ウィキペディア英語版 Hierarchical Dirichlet process In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. It uses a Dirichlet process for each group of data, with the Dirichlet processes for all groups sharing a base distribution which is itself drawn from a Dirichlet process. This method allows groups to share statistical strength via sharing of clusters across groups. The base distribution being drawn from a Dirichlet process is important, because draws from a Dirichlet process are atomic probability measures, and the atoms will appear in all group-level Dirichlet processes. Since each atom corresponds to a cluster, clusters are shared across all groups. It was developed by Yee Whye Teh, Michael I. Jordan, Matthew J. Beal and David Blei and published in the ''Journal of the American Statistical Association'' in 2006.〔 ==Model== This model description is sourced from.〔 The HDP is a model for grouped data. What this means is that the data items come in multiple distinct groups. For example, in a topic model words are organized into documents, with each document formed by a bag (group) of words (data items). Indexing groups by $j=1,...J$, suppose each group consist of data items $x\_,...x\_$. The HDP is parameterized by a base distribution $H$ that governs the a priori distribution over data items, and a number of concentration parameters that govern the a priori number of clusters and amount of sharing across groups. The $j$th group is associated with a random probability measure $G\_j$ which has distribution given by a Dirichlet process: $$ \begin G_j|G_0 &\sim \operatorname(\alpha_j,G_0) \end where $\backslash alpha\_j$ is the concentration parameter associated with the group, and $G\_0$ is the base distribution shared across all groups. In turn, the common base distribution is Dirichlet process distributed: $$ \begin G_0 &\sim \operatorname(\alpha_0,H) \end with concentration parameter $\backslash alpha\_0$ and base distribution $H$. Finally, to relate the Dirichlet processes back with the observed data, each data item $x\_$ is associated with a latent parameter $\backslash theta\_$: $$ \begin \theta_|G_j &\sim G_j \\ x_|\theta_ &\sim F(\theta_) \end The first line states that each parameter has a prior distribution given by $G\_j$, while the second line states that each data item has a distribution $F(\backslash theta\_)$ parameterized by its associated parameter. The resulting model above is called a HDP mixture model, with the HDP referring to the hierarchically linked set of Dirichlet processes, and the mixture model referring to the way the Dirichlet processes are related to the data items. To understand how the HDP implements a clustering model, and how clusters become shared across groups, recall that draws from a Dirichlet process are atomic probability measures with probability one. This means that the common base distribution $G\_0$ has a form which can be written as: $$ \begin G_0 &= \sum_^\infty \pi_\delta_ \end where there are an infinite number of atoms, $\backslash theta^$ *_k, k=1,2,..., assuming that the overall base distribution $H$ has infinite support. Each atom is associated with a mass $\backslash pi\_$. The masses have to sum to one since $G\_0$ is a probability measure. Since $G\_0$ is itself the base distribution for the group specific Dirichlet processes, each $G\_j$ will have atoms given by the atoms of $G\_0$, and can itself be written in the form: $$ \begin G_j &= \sum_^\infty \pi_\delta_ \end Thus the set of atoms is shared across all groups, with each group having its own group-specific atom masses. Relating this representation back to the observed data, we see that each data item is described by a mixture model: $$ \begin x_|G_j &\sim \sum_^\infty \pi_ F(\theta^ *_k) \end where the atoms $\backslash theta^$ *_k play the role of the mixture component parameters, while the masses $\backslash pi\_$ play the role of the mixing proportions. In conclusion, each group of data is modeled using a mixture model, with mixture components shared across all groups but mixing proportions being group-specific. In clustering terms, we can interpret each mixture component as modeling a cluster of data items, with clusters shared across all groups, and each group, having its own mixing proportions, composed of different combinations of clusters. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hierarchical Dirichlet process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.