Mathematische Statistik - Übung Section 1.2 Beispiel 1a

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From Bickel-Doksum Page 71

Find the posterior frequency function \pi(\theta|X) \,

\pi(\theta_1) = \pi(\theta_2) = \frac 1 2 \,


P(X|\theta) = 
\begin{array}{l||c|c}
\theta / X & 0 & 1 \\
\hline \hline
\theta_1 & 0.8 & 0.2 \\
\hline
\theta_2 & 0.4 & 0.6
\end{array}

Remember en:Bayes' theorem

P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}. 

P(A|B)=\frac{P(A \cap B)}{P(B)}.

\begin{align}
\pi(\theta_1|X=0) 
& = \frac{\pi(\theta_1 \cap X=0)}{\pi(X=0)} \\
& = \frac{\pi(X=0|\theta_1) \pi(\theta_1)}{\pi(X=0 \cap \theta_1) + \pi(X=1 \cap \theta_2)} \\
& = \frac{\pi(X=0|\theta_1) \pi(\theta_1)}{\pi(X=0 | \theta_1) \pi(\theta_1) + \pi(X=1 | \theta_2) \pi(\theta_2)} 
\end{align}\,


P(\theta|X) = 
\begin{array}{l||c|c}
\theta / X & 0 & 1 \\
\hline \hline
\theta_1 & 0.4/0.6 & 0.1/0.4 \\
\hline
\theta_2 & 0.2/0.6 & 0.3/0.4
\end{array}