Mathematische Statistik - Übung Ergänzungsaufgabe 2 Beispiel 4

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Let X_i|\theta, i =1,\ldots,n \, be bernoulli trails with parameter \theta \in [0,1] \,. Find a conjugate family of prior distributions for the joint sampling density. 

Density of a Beta-Distribution  \pi(\theta) = \frac{ \theta^{p-1} (1-\theta)^{q-1}}{ \int_0^1 u^{p-1} (1-u)^{q-1} \, du } \propto \theta^{p-1} (1-\theta)^{q-1}  \,

 
\begin{align}
f(\theta|x) 
& \propto \pi(\theta) f(x|\theta) \\
& \propto \binom{n}{k} \theta^k (1-\theta)^{n-k} \pi(\theta) \\
& \propto \binom{n}{k} \theta^k (1-\theta)^{n-k} \theta^{p-1} (1-\theta)^{q-1} \\
& \propto \theta^{k + p -1} (1-\theta)^{n-k+q-1} \\
& = \frac{\theta^{k + p -1} (1-\theta)^{n-k+q-1}}{B(k+p, n-k+q)} \\
\end{align}
 \,

 \Rightarrow f(\theta|x) \, ist Beta verteilt mit k+p und n-k+q
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