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19
Bayesian nonparametrics and the probabilistic approach to modelling
"... be thought of as a representation of possible data one could predict from a system. The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of pro ..."
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be thought of as a representation of possible data one could predict from a system. The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of probability theory in order to make predictions, compare alternative models, and learn model parameters and structure from data. This simple and elegant framework is most powerful when coupled with flexible probabilistic models. Flexibility is achieved through the use of Bayesian nonparametrics. This article provides an overview of probabilistic modelling and an accessible survey of some of the main tools in Bayesian nonparametrics. The survey covers the use of Bayesian nonparametrics for modelling unknown functions, density estimation, clustering, time series modelling, and representing sparsity, hierarchies, and covariance structure. More specifically it gives brief nontechnical overviews of Gaussian processes, Dirichlet processes, infinite hidden Markov models, Indian buffet processes, Kingman’s coalescent, Dirichlet diffusion trees, and Wishart processes. Key words: probabilistic modelling; Bayesian statistics; nonparametrics; machine learning. 1.
Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
, 2009
"... We introduce the notion of a restricted exchangeable partition of N and study natural classes of such partitions. We obtain integral representations, study associated coalescents and fragmentations, embeddings into continuum random trees and convergence to such limit trees. As an application, we ded ..."
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We introduce the notion of a restricted exchangeable partition of N and study natural classes of such partitions. We obtain integral representations, study associated coalescents and fragmentations, embeddings into continuum random trees and convergence to such limit trees. As an application, we deduce from the general theory developed here a particular limit result conjectured previously for Ford’s alpha model and its nonbinary extension, the alphagamma model, where restricted exchangeability arises naturally.
Nonparametric bayesian models of hierarchical structure in complex networks,” ArXiv
, 2012
"... Analyzing and understanding the structure of complex relational data is important in many applications including analysis of the connectivity in the human brain. Such networks can have prominent patterns on different scales, calling for a hierarchically structured model. We propose two nonparametr ..."
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Analyzing and understanding the structure of complex relational data is important in many applications including analysis of the connectivity in the human brain. Such networks can have prominent patterns on different scales, calling for a hierarchically structured model. We propose two nonparametric Bayesian hierarchical network models based on Gibbs fragmentation tree priors, and demonstrate their ability to capture nested patterns in simulated networks. On real networks we demonstrate detection of hierarchical structure and show predictive performance on par with the state of the art. We envision that our methods can be employed in exploratory analysis of large scale complex networks for example to model human brain connectivity. 1
Random permutations and partition models
, 2010
"... Set partitions For n ≥ 1, a partition B of the finite set [n] = {1,..., n} is • a collection B = {b1,...} of disjoint nonempty subsets, called blocks, whose union is [n]; • an equivalence relation or Boolean function B: [n] × [n] → {0, 1} that is reflexive, symmetric and transitive; • a symmetri ..."
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Set partitions For n ≥ 1, a partition B of the finite set [n] = {1,..., n} is • a collection B = {b1,...} of disjoint nonempty subsets, called blocks, whose union is [n]; • an equivalence relation or Boolean function B: [n] × [n] → {0, 1} that is reflexive, symmetric and transitive; • a symmetric Boolean matrix such that Bij = 1 if i, j belong to the same block. These equivalent representations are not distinguished in the notation, so B is a set of subsets, a matrix, a Boolean function, or a subset of [n] × [n], as the context demands. In practice, a partition is sometimes written in an abbreviated form, such as B = 213 for a partition of [3]. In this notation, the five partitions of [3] are 123, 123, 132, 231, 123. The blocks are unordered, so 213 is the same partition as 132 and 231. A partition B is a subpartition of B ∗ if each block of B is a subset of some block of B ∗ or, equivalently, if Bij = 1 implies B ∗ ij = 1. This relationship is a partial order denoted by B ≤ B ∗, which can be interpreted as B ⊂ B ∗ if each partition is regarded as a subset of [n] 2. The partition lattice En is the set of partitions of [n] with this partial order. To each pair of partitions B, B ′ there corresponds a greatest lower bound B ∧ B ′ , which is the set intersection or Hadamard componentwise matrix product. The least upper bound B ∨ B ′ is the least element that is greater than both, the transitive completion of B ∪ B ′. The least element of En is the partition 0n with n singleton blocks, and the greatest element is the singleblock partition denoted by 1n. A permutation σ: [n] → [n] induces an action B ↦ → B σ by composition such that B σ (i, j) = B(σ(i), σ(j)). In matrix notation, B σ = σBσ −1, so the action by conjugation permutes both the rows and columns of B in the same way. The block sizes are preserved and are maximally invariant under conjugation. In this way, the 15 partitions of [4] may be grouped into five orbits or equivalence classes as follows:
The TimeMarginalized Coalescent Prior for Hierarchical Clustering
"... We introduce a new prior for use in Nonparametric Bayesian Hierarchical Clustering. The prior is constructed by marginalizing out the time information of Kingman’s coalescent, providing a prior over tree structures which we call the TimeMarginalized Coalescent (TMC). This allows for models which fa ..."
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We introduce a new prior for use in Nonparametric Bayesian Hierarchical Clustering. The prior is constructed by marginalizing out the time information of Kingman’s coalescent, providing a prior over tree structures which we call the TimeMarginalized Coalescent (TMC). This allows for models which factorize the tree structure and times, providing two benefits: more flexible priors may be constructed and more efficient Gibbs type inference can be used. We demonstrate this on an example model for density estimation and show the TMC achieves competitive experimental results. 1
Bayesian nonparametrics and the
"... One contribution of 17 to a Discussion Meeting Issue ‘Signal processing and inference for the physical sciences’. Subject Areas: pattern recognition, computer modelling and simulation, statistics, mathematical modelling Keywords: probabilistic modelling, Bayesian statistics, nonparametrics, machine ..."
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One contribution of 17 to a Discussion Meeting Issue ‘Signal processing and inference for the physical sciences’. Subject Areas: pattern recognition, computer modelling and simulation, statistics, mathematical modelling Keywords: probabilistic modelling, Bayesian statistics, nonparametrics, machine learning Author for correspondence:
Supplementary data References
, 2013
"... Bayesian nonparametrics and the probabilistic ..."
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ELECTRONIC COMMUNICATIONS in PROBABILITY Consistent Markov branching trees
"... We study consistent collections of random fragmentation trees with random integervalued edge lengths. We prove several equivalent necessary and sufficient conditions under which Geometrically distributed edge lengths can be consistently assigned to a Markov branching tree. Among these conditions is ..."
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We study consistent collections of random fragmentation trees with random integervalued edge lengths. We prove several equivalent necessary and sufficient conditions under which Geometrically distributed edge lengths can be consistently assigned to a Markov branching tree. Among these conditions is a characterization by a unique probability measure, which plays a role similar to the dislocation measure for homogeneous fragmentation processes. We discuss this and other connections to previous work on Markov branching trees and homogeneous fragmentation processes.
Regenerative tree growth: structural results and convergence
, 2013
"... We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n ≥ 1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as nonexchangeablemodelssuch asthe a ..."
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We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n ≥ 1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as nonexchangeablemodelssuch asthe alphathetamodel, the alphagammamodeland all restricted exchangeable models previously studied. Our main structural result is a representation of the growthrulebyaσfinite dislocationmeasureκonthe setofpartitionsofNextendingBertoin’s notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish selfsimilar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under κ, in addition to a regular variation condition.