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zAPuvDhD2;!2S7it8+9M7Av6>_L$CGufb`Lbe3%9~pe;1QXSDI7 Date: Tue, 19 Apr 2016 17:27:00 +0200 Subject: [PATCH 2/2] update on likelihood definition --- likelihood/lecture/likelihood.tex | 27 ++++++++++++--------------- 1 file changed, 12 insertions(+), 15 deletions(-) diff --git a/likelihood/lecture/likelihood.tex b/likelihood/lecture/likelihood.tex index 6c98f37..07d9e71 100644 --- a/likelihood/lecture/likelihood.tex +++ b/likelihood/lecture/likelihood.tex @@ -22,28 +22,25 @@ Parameter(n) $\theta$. Das k\"onnte die Normalverteilung p(x|\theta) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \end{equation} sein mit dem Mittelwert $\mu$ und der Standardabweichung $\sigma$ als -Parameter $\theta$. +den Parametern $\theta$. Wenn nun den $n$ unabh\"angigen Beobachtungen $x_1, x_2, \ldots x_n$ -die Wahrscheinlichkeitsverteilung $p(x|\theta)$ zugrundeliegt, dann -ist die Verbundwahrscheinlichkeit $p(x_1,x_2, \ldots x_n|\theta)$ des -Auftretens der Werte $x_1, x_2, \ldots x_n$ gegeben ein bestimmtes $\theta$ +die gleiche Wahrscheinlichkeitsverteilung $p(x|\theta)$ zugrundeliegt +(\enterm{i.i.d.} idependent and identically distributed), dann ist die +Verbundwahrscheinlichkeit $p(x_1,x_2, \ldots x_n|\theta)$ des +Auftretens der Werte $x_1, x_2, \ldots x_n$, gegeben ein bestimmtes +$\theta$, \begin{equation} p(x_1,x_2, \ldots x_n|\theta) = p(x_1|\theta) \cdot p(x_2|\theta) \ldots p(x_n|\theta) = \prod_{i=1}^n p(x_i|\theta) \; . \end{equation} Andersherum gesehen ist das die \determ{Likelihood} -(\enterm{likelihood}, deutsch immer noch ``Wahrscheinlichleit'') den -Parameter $\theta$ zu haben, gegeben die Me{\ss}werte $x_1, x_2, -\ldots x_n$, +(\enterm{likelihood}) den Parameter $\theta$ zu haben, gegeben die +Me{\ss}werte $x_1, x_2, \ldots x_n$, \begin{equation} - {\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta) + {\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta) \; . \end{equation} -Hinter dieser Umformung steht eigentlich der Satz von Bayes. Bei der -einfachen Gleichsetzung von ${\cal L}$ mit $p$ fehlen -Normierungsfaktoren, so dass ${\cal L}$ sich nicht auf Eins -aufintegriert, und ${\cal L}$ deshalb keine -Wahrscheinlichkeit(sdichte) ist. +Beachte, dass die Likelihood ${\cal L}$ keine Wahrscheinlichkeit im engeren Sinne ist, da sie sich nicht zu Eins aufintegriert ($\int {\cal L}(\theta|x_1,x_2, \ldots x_n) \, d\theta \ne 1$). Wir sind nun an dem Wert des Parameters $\theta_{mle}$ interessiert, der die Likelihood maximiert (Maximum-Likelihood Estimate ``mle''): @@ -55,8 +52,8 @@ dem $f(x)$ ihr globales Maximum annimmt. Wir suchen also den Wert von $\theta$ bei dem die Likelihood ${\cal L}(\theta)$ ihr Maximum hat. An der Stelle eines Maximums einer Funktion \"andert sich nichts, wenn -man die Funktionswerte mit einer streng monoton steigenden Funktion -transformiert. Aus numerischen und gleich ersichtlichen mathematischen +die Funktionswerte mit einer streng monoton steigenden Funktion +transformiert werden. Aus numerischen und gleich ersichtlichen mathematischen Gr\"unden wird meistens das Maximum der logarithmierten Likelihood (\determ{log-Likelihood}, \enterm{log-likelihood}) gesucht: \begin{eqnarray}