# Quelle:Rh/Teh 2007

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 Autor Yee Whye Teh Titel Dirichlet Process Ort London Datum September 2007 Anmerkung Das verlinkte dejavu File wird für die Zeilenzählung herangezogen. Version von 2007 als djvu-Datei abrufbar unter http://web.archive.org/web/20071103075013/http://www.gatsby.ucl.ac.uk/~ywteh/research/projects.html#bayesian; der Preprint von 2007 wird in der Literatur explizit zitiert. URL http://web.archive.org/web/20071103075013/http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/dp.djvu Literaturverz. nein Fußnoten nein Fragmente 5

Fragmente der Quelle:
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Seite: 138, Zeilen: 17-21
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Seite(n): 7, Zeilen: 32-37
This slow growth of the number of clusters makes sense because of the rich-gets-richer phenomenon: we expect there to be large clusters thus the number of clusters l' to be far smaller than the number of observations L. Notice that α controls the number of clusters in a direct manner, with larger implying a larger number of clusters a priori. This intuition will help in the application of DPs to mixture models. This slow growth of the number of clusters makes sense because of the rich-gets-richer phenomenon: we expect there to be large clusters thus the number of clusters m to be far smaller than the number of observations n. Notice that α controls the number of clusters in a direct manner, with larger α implying a larger number of clusters a priori. This intuition will help in the application of DPs to mixture models.
 Anmerkungen keine Kennzeichnung der Übernahme; kein Quellenverweis. Man beachte das ausgelassene α in der Dissertation. Sichter (Graf Isolan), Hindemith

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Quelle: Teh 2007
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Formally, the DP is a stochastic process whose sample paths are probability measures with probability one. Stochastic processes are distributions over function spaces, with sample paths being random functions drawn from the distribution. In the case of the DP, it is a distribution over probability measures, which are functions with certain special properties which allow them to be interpreted [as distributions over some probability space.] Formally, the Dirichlet process (DP) is a stochastic process whose sample paths are probability measures with probability one. Stochastic processes are distributions over function spaces, with sample paths being random functions drawn from the distribution. In the case of the DP, it is a distribution over probability measures, which are functions with certain special properties which allow them to be interpreted as distributions over some probability space Θ.
 Anmerkungen kein Hinweis auf eine Übernahme, keine Quellenangabe. Die Übernahme wird hier fortgesetzt: Rh/Fragment_140_01 Sichter (Graf Isolan), Hindemith

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[In the case of the DP, it is a distribution over probability measures, which are functions with certain special properties which allow them to be interpreted] as distributions over some probability space. Thus, draws from a DP can be interpreted as random distributions. For a distribution over probability measures to be a DP, its marginal distributions have to take on a specific form which we shall give below. In the case of the DP, it is a distribution over probability measures, which are functions with certain special properties which allow them to be interpreted as distributions over some probability space Θ. Thus draws from a DP can be interpreted as random distributions. For a distribution over probability measures to be a DP, its marginal distributions have to take on a specific form which we shall give below.
 Anmerkungen kein Hinweis auf eine Übernahme, keine Quellenangabe. Die Übernahme beginnt schon auf der Vorseite: Rh/Fragment_139_15 Sichter (Graf Isolan), Hindemith

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Seite(n): 2-3, Zeilen: S.2,33-38 - S.3,1-4
Definition 5.3.1. We say G is DP distributed with base distribution G0 and concentration parameter α, written G~DP(α,G0), if

(G(A1),...,G(Ar))~Dirichlet(αG0(A1),...,αG0(Ar)) (5.3.1)

for every finite measurable partition A1,...,Ar over some probability space Θ.

The parameters G0 and α play intuitive roles in the definition of the DP. The base distribution is basically the mean of the DP for any measurable set A⊂Θ, that is,

E[G(A)] = G0(A).

On the other hand, the concentration parameter can be interpreted as an inverse variance

${\scriptstyle \mathbb{V}[G(A)] = \frac{G_0(A)(1-G_0(A))}{\alpha + 1}.}$

The larger α is the smaller the variance and the DP will concentrate more of its mass around the mean.

[Seite 2]

We say G is Dirichlet process distributed with base distribution H and concentration parameter α, written G~DP(α,H), if

(G(A1),...,G(Ar))~Dir(αH(A1),...,αH(Ar)) (1)

for every finite measurable partition A1,...,Ar of Θ.

The parameters H and α play intuitive roles in the definition of the DP. The base distribution is basically the mean of the DP: for any measurable set A⊂Θ,

[Seite 3]

we have E[G(A)]=H(A). On the other hand, the concentration parameter can be understood as an inverse variance: V[G(A)]=H(A)(1-H(A))/(α + 1). The larger α is, the smaller the variance, and the DP will concentrate more of its mass around the mean.

 Anmerkungen Hier wurde nicht nur eine Definition übernommen, sondern auch große Teile der übrigen Darstellung (insbesondere die kommentierenden/erläuternden Zwischentexte). Dies wird sich im nächsten Fragment nahtlos fortsetzen, sodass sich Seite 140 im wesentlichen als Komplettübernahme von Material aus Teh (2007) erweist. Ein Hinweis darauf unterbleibt ebenso wie eine Kennzeichnung übernommener Formulierungen. Sichter (Graf Isolan), Hindemith

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Now we are interested in the posterior distribution of G given some observed values. Let π1, ..., πn be a sequence of independent draws from G. Note that the πi's take values in Θ since G is a distribution over Θ. Let A1,...,Ar be a finite measurable partition of Θ, and let nk be the number of observed values in Ak. Then by the conjugancy [sic!] between the Dirichlet and the multinomial distributions, we have
(G(A1),...,G(Ar))│π1, ..., πn ~ Dir(αG0(A1)+n1, ..., αG0(Ar)+nr). (5.3.2)

Since the above is true for all finite measurable partitions, the posterior distribution over G must be a DP as well.

In fact, the posterior DP is

${\scriptstyle G\mid \pi_1,\dots,\pi_n\sim DP \left( \alpha+n,\frac{\alpha G_0 + \sum^n_{i=1}\delta_{\pi_i}}{\alpha+n}\right).}$

Notice that the DP has updated concentration parameter α+n and base distribution ${\scriptstyle\frac{\alpha G_0 + \sum^n_{i=1}\delta_{\pi_i}}{\alpha+n},}$ where δi is a point mass located at πi and ${\scriptstyle n_k = \sum^n_{i=1}\delta_{\pi_i}(A_k).}$ In other words, the DP provides a conjugate family of priors over distributions that are closed under posterior updates given observations.

Furthermore, notice that the posterior base distribution is weighted average between the prior base G0 and the empirical distribution ${\scriptstyle\frac{\sum^n_{i=1}\delta_{\pi_i}}{n}.}$ Indeed, the weight associated with the prior base distribution is proportional to α, while the empirical distribution has weight proportional to the number of observations n.

Let θ1, ..., θn be a sequence of independent draws from G. Note that the θi's take values in Θ since G is a distribution over Θ. We are interested in the posterior distribution of G given observed values of θ1, ..., θn. Let A1,...,Ar be a finite measurable partition of Θ, and let nk=#{i: θi ∈ Ak} be the number of observed values in Ak. By (1) and the conjugacy between the Dirichlet and the multinomial distributions, we have
(G(A1),...,G(Ar))│θ1, ..., θn ~ Dir(αH(A1)+n1, ..., αH(Ar)+nr). (2)

Since the above is true for all finite measurable partitions, the posterior distribution over G must be a DP as well. A little algebra shows that the posterior DP has updated concentration parameter α+n and base distribution ${\scriptstyle\frac{\alpha H + \sum^n_{i=1}\delta_{\theta_i}}{\alpha+n},}$ where δi is a point mass located at θi and ${\scriptstyle n_k = \sum^n_{i=1}\delta_i(A_k).}$ In other words, the DP provides a conjugate family of priors over distributions that is closed under posterior updates given observations. Rewriting the posterior DP, we have

${\scriptstyle G\mid \theta_1,\dots,\theta_n\sim DP\left(\alpha+n,\frac{\alpha}{\alpha+n}H + \frac{n}{\alpha+n} \frac{\sum^n_{i=1}\delta_{\pi_i}}{n}\right).}$ (3).

Notice that the posterior base distribution is a weighted average between the prior base distribution H and the empirical distribution ${\scriptstyle\frac{\sum^n_{i=1}\delta_{\theta_i}}{n}.}$ The weight associated with the prior base distribution is proportional to α, while the empirical distribution has weight proportional to the number of observations n.

 Anmerkungen keine Kennzeichnung der übernommenen Passagen, keine Quellenangabe; Sichter (Graf Isolan), Hindemith