165:
1751:, a Bayesian nonparametric model for sequences which has a multi-level hierarchy of Pitman-Yor processes. In addition, Bayesian Multi-Domain Learning (BMDL) model derives domain-dependent latent representations of overdispersed count data based on hierarchical negative binomial factorization for accurate cancer subtyping even if the number of samples for a specific cancer type is small.
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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
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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
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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.
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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
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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:
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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
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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
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and
Hierarchical Gamma process. The hierarchy can be deeper, with multiple levels of groups arranged in a hierarchy. Such an arrangement has been exploited in the
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items, with clusters shared across all groups, and each group, having its own mixing proportions, composed of different combinations of clusters.
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To understand how the HDP implements a clustering model, and how clusters become shared across groups, recall that draws from a
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980:{\displaystyle {\begin{aligned}\theta _{ji}|G_{j}&\sim G_{j}\\x_{ji}|\theta _{ji}&\sim F(\theta _{ji})\end{aligned}}}
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is the base distribution shared across all groups. In turn, the common base distribution is
Dirichlet process distributed:
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words are organized into documents, with each document formed by a bag (group) of words (data items). Indexing groups by
1965:, et al. "A sticky HDP-HMM with application to speaker diarization." The Annals of Applied Statistics (2011): 1020-1056.
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1634:{\displaystyle {\begin{aligned}x_{ji}|G_{j}&\sim \sum _{k=1}^{\infty }\pi _{jk}F(\theta _{k}^{*})\end{aligned}}}
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1509:{\displaystyle {\begin{aligned}G_{j}&=\sum _{k=1}^{\infty }\pi _{jk}\delta _{\theta _{k}^{*}}\end{aligned}}}
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603:{\displaystyle {\begin{aligned}G_{j}|G_{0}&\sim \operatorname {DP} (\alpha _{j},G_{0})\end{aligned}}}
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are atomic probability measures with probability one. This means that the common base distribution
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The HDP can be generalized in a number of directions. The
Dirichlet processes can be replaced by
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Hajiramezanali, E. & Dadaneh, S. Z. & Karbalayghareh, A. & Zhou, Z. & Qian, X.
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733:{\displaystyle {\begin{aligned}G_{0}&\sim \operatorname {DP} (\alpha _{0},H)\end{aligned}}}
1978:(PDF). 32nd Conference on Neural Information Processing Systems (NIPS 2018), Montréal, Canada.
1953:(PDF). Advances in Neural Information Processing Systems 14:577–585. Cambridge, MA: MIT Press.
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is itself the base distribution for the group specific
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The first line states that each parameter has a prior distribution given by
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allowing the number of states to be unbounded and learnt from data.
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play the role of the mixture component parameters, while the masses
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The HDP mixture model is a natural nonparametric generalization of
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is the concentration parameter associated with the group, and
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Hierarchical
Bayesian Nonparametric Models with Applications
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Teh, Y. W.; Jordan, M. I.; Beal, M. J.; Blei, D. M. (2006).
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has infinite support. Each atom is associated with a mass
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th group is associated with a random probability measure
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Beal, M.J., Ghahramani, Z. and
Rasmussen, C.E. (2002).
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in 2006, as a formalization and generalization of the
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which has distribution given by a
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50:Learn how and when to remove these messages
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187:adding citations to reliable sources
1951:"The infinite hidden Markov model"
1875:Teh, Y. W.; Jordan, M. I. (2010).
1819:"Hierarchical Dirichlet Processes"
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1999:Nonparametric Bayesian statistics
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438:{\displaystyle x_{j1},...x_{jn}}
198:"Hierarchical Dirichlet process"
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96:"Hierarchical Dirichlet process"
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239:February 2012
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200: –
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194:Find sources:
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172:This article
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137:February 2012
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92:Find sources:
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70:This article
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313:Yee Whye Teh
306:. It uses a
304:grouped data
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181:Please help
176:verification
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33:Please help
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1838:1566–1581.
347:topic model
1988:Categories
1767:References
325:David Blei
281:statistics
209:newspapers
107:newspapers
74:references
36:improve it
1932:ignored (
1922:cite book
1890:CiteSeerX
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42:talk page
1755:See also
300:Bayesian
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230:JSTOR
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1934:help
1908:ISBN
1739:and
323:and
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