## FANDOM

32.681 Seiten

 Typus Verschleierung Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 140, Zeilen: 14-31
Quelle: Teh 2007
Seite(n): 4, Zeilen: 5-23
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