Synge's world function: Difference between revisions

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<math>\sigma</math> is half the geodesic length squared, <math>\sigma = 1/2 (\Delta \tau)^2</math>, not half the geodesic length
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In [[statistics]] and [[machine learning]], the '''hierarchical Dirichlet process''' (HDP) is a [[Non-parametric statistics|nonparametric]] [[Bayesian probability|Bayesian]] approach to clustering grouped data.<ref name="teh2006">{{cite journal
| last1 = Teh | first1 = Y. W.
| last2 = Jordan |first2 = M. I.
| last3 = Beal | first3 = M. J. | last4 = Blei | first4 = D. M. | title = Hierarchical Dirichlet Processes
| journal = [[Journal of the American Statistical Association]]
| year = 2006
| volume = 101
| pages = ''pp.'' 1566&ndash;1581
| url = http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/jasa2006.pdf
}}</ref><ref name="tehjor2010">{{cite journal
| last1 = Teh | first1 = Y. W.
| last2 = Jordan | first2 = M. I.
| title = Hierarchical Bayesian Nonparametric Models with Applications
| journal = Bayesian Nonparametrics
| year = 2010
| url = http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/TehJor2010a.pdf
| publisher = [[Cambridge University Press]]
}}</ref> 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]], [[David Blei]] and [[Matthew Beal]] and published in the [[Journal of the American Statistical Association]] in 2006.<ref name="teh2006" />


==Model==
The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.<br><br>Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.<br>This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford,  [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.<br><br>This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great<br>
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and Celine) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun<br><br>
Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul<br>
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an elegant collection of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte<br><br>
Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co<br>.
The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa<br><br>


This model description is sourced from.<ref name="teh2006" /> 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 <math>j=1,...J</math>, suppose each group consist of data items <math>x_{j1},...x_{jn}</math>. 
JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect<br><br>


The HDP is parameterized by a base distribution <math>H</math> which governs the a priori distribution over data items, and a number of concentration parameters which govern the a priori number of clusters and amount of sharing across groups.  The <math>j</math>th group is associated with a random probability measure <math>G_j</math> which has distribution given by a Dirichlet process:
It was all change at Mulberry last season too, as Emma Hill departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu<br><br>


<math>
This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp<br><br>
\begin{align}
G_j|G_0 &\sim \operatorname{DP}(\alpha_j,G_0)
\end{align}
</math>


where <math>\alpha_j</math> is the concentration parameter associated with the group, and <math>G_0</math> is the base distribution shared across all groups.  In turn, the common base distribution is Dirichlet process distributed:
Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr<br><br><br>


<math>
While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y<br>r.
\begin{align}
Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp<br><br>
G_0 &\sim \operatorname{DP}(\alpha_0,H)
\end{align}
</math>


with concentration parameter <math>\alpha_0</math> and base distribution <math>H</math>.  Finally, to relate the Dirichlet processes back with the observed data, each data item <math>x_{ji}</math> is associated with a latent parameter <math>\theta_{ji}</math>:
So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on [http://mondediplo.com/spip.php?page=recherche&recherche=Mount+Street Mount Street] this year.
 
<math>
\begin{align}
\theta_{ji}|G_j &\sim G_j \\
x_{ji}|\theta_{ji} &\sim F(\theta_{ji})
\end{align}
</math>
 
The first line states that each parameter has a prior distribution given by <math>G_j</math>, while the second line states that each data item has a distribution <math>F(\theta_{ji})</math> 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 <math>G_0</math> has a form which can be written as:
 
<math>
\begin{align}
G_0 &= \sum_{k=1}^\infty \pi_{0k}\delta_{\theta^*_k}
\end{align}
</math>
 
where there are an infinite number of atoms, <math>\theta^*_k, k=1,2,...</math>, assuming that the overall base distribution <math>H</math> has infinite support. Each atom is associated with a mass <math>\pi_{0k}</math>. The masses have to sum to one since <math>G_0</math> is a probability measure.  Since <math>G_0</math> is itself the base distribution for the group specific Dirichlet processes, each <math>G_j</math> will have atoms given by the atoms of <math>G_0</math>, and can itself be written in the form:
 
<math>
\begin{align}
G_j &= \sum_{k=1}^\infty \pi_{jk}\delta_{\theta^*_k}
\end{align}
</math>
 
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:
 
<math>
\begin{align}
x_{ji}|G_j &\sim \sum_{k=1}^\infty \pi_{jk} F(\theta^*_k)
\end{align}
</math>
 
where the atoms <math>\theta^*_k</math> play the role of the mixture component parameters, while the masses <math>\pi_{jk}</math> 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.
 
==Applications==
 
The HDP mixture model is a natural nonparametric generalization of [[Latent Dirichlet allocation]], where the number of topics can be unbounded and learnt from data.<ref name="teh2006" />  Here each group is a document consisting of a bag of words, each cluster is a topic, and each document is a mixture of topics.  The HDP is also a core component of the [[infinite hidden Markov model]],<ref name="teh2006" /> which is a nonparametric generalization of the [[hidden Markov model]] allowing the number of states to be unbounded and learnt from data.
 
==Generalizations==
 
The HDP can be generalized in a number of directions. The Dirichlet processes can be replaced by [[Pitman-Yor process]]es, resulting in the [[Hierarchical Pitman-Yor process]].  The hierarchy can be deeper, with multiple levels of groups arranged in a hierarchy.  Such an arrangement has been exploited in the [[sequence memoizer]], a Bayesian nonparametric model for sequences which has a multi-level hierarchy of Pitman-Yor processes.
 
==References==
<references/>
 
[[Category:Stochastic processes]]
[[Category:Non-parametric Bayesian methods]]

Latest revision as of 21:32, 29 March 2014


The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.

Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.
This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford, http://www.pcs-systems.co.uk/Images/celinebag.aspx which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.

This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and Celine) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun

Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an elegant collection of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte

Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co
. The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa

JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect

It was all change at Mulberry last season too, as Emma Hill departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu

This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp

Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr


While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y
r. Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp

So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on Mount Street this year.

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