Gauß Rechenregeln

Produkt zweier Gaußdichten

Gegeben

$$ \begin{aligned} f_1(x) &= \frac{1}{\sqrt{2\pi} \sigma_1} \exp\left\{-\frac{1}{2} \frac{(x - m_1)^2}{\sigma_1^2}\right\} \\ f_2(x) &= \frac{1}{\sqrt{2\pi} \sigma_2} \exp\left\{-\frac{1}{2} \frac{(x - m_2)^2}{\sigma_2^2}\right\} \end{aligned} $$

Gesucht:

$$ \begin{aligned} f(x) &= \frac{1}{\sqrt{2\pi} \sigma} \exp\left\{-\frac{1}{2} \frac{(x - m)^2}{\sigma_1^2}\right\} \\\\ &\propto f_1(x) \cdot f_2(x) = \frac{1}{\sqrt{2\pi} \sigma_1}\frac{1}{\sqrt{2\pi} \sigma_2} \cdot e^{-\frac{1}{2} \frac{(x - m_1)^2}{\sigma_1^2}} e^{-\frac{1}{2} \frac{(x - m_2)^2}{\sigma_2^2}} \end{aligned} $$

Exponent:

$$ \begin{aligned} &\frac{\left(x-m_{1}\right)^{2}}{\sigma_{1}^{2}}+\frac{\left(x-m_{2}\right)^{2}}{\sigma_{2}^{2}} \overset{!}{=} \frac{(x-m)^{2}}{\sigma^{2}}+2C \\\\ &\frac{x^{2}-2 m_{1} x+m_{1}^{2}}{\sigma_{1}^{2}}+\frac{x-2 m_{2} x+m_{2}^{2}}{\sigma_{2}{ }^{2}} \stackrel{!}{=} \frac{x^{2}-2 mx+m^{2}}{\sigma^{2}}+2 C \\\\ &x^{2}\left(\frac{1}{\sigma_{1}^{2}}+\frac{1}{\sigma_{2}^{2}}-\frac{1}{\sigma^{2}}\right)-2\left(\frac{m_{1}}{\sigma_{1}^{2}}+\frac{m_{2}}{\sigma_{2}^{2}}-\frac{m}{\sigma^{2}}\right) \cdot x +\frac{m_{1}^{2}}{\sigma_{1}^{2}}+\frac{m_{2}^{2}}{\sigma_{2}^{2}}-\frac{m^{2}}{\sigma^{2}}-2 C \stackrel{!}{=} 0 \end{aligned} $$

Ergebnis:

$$ \begin{aligned} \sigma^{2}&=\frac{1}{\frac{1}{\sigma_{1}^{2}}+\frac{1}{\sigma_{2}^{2}}}=\frac{\sigma_{1}^{2} \sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}} \\\\ m &= \sigma^2 \left(\frac{m_1}{\sigma_1^2} + \frac{m_2}{\sigma_2^2} \right)\\\\ 2C &= \frac{m_1^2}{\sigma_1^2} + \frac{m_2^2}{\sigma_2^2} - \frac{m^2}{\sigma^2} = \frac{(m_1 - m_2)^2}{\sigma_1^2 + \sigma_2^2} \end{aligned} $$

(See also: Product of Two Gaussian PDFs)

Andere Darstellung:

$$ \begin{aligned} f(x) &\propto \frac{1}{\sqrt{2\pi} \sigma_1}\frac{1}{\sqrt{2\pi} \sigma_2} \cdot e^{-\frac{1}{2} \frac{(m_1 - m_2)^2}{\sigma_1^2 + \sigma_2^2}} e^{-\frac{1}{2} \frac{(x - m)^2}{\sigma^2}} \\\\ &= \underbrace{\frac{1}{\sqrt{2\pi} \sqrt{\sigma_1^2 + \sigma_2^2}} e^{-\frac{1}{2} \frac{(m_1 - m_2)^2}{\sigma_1^2 + \sigma_2^2}}}_{\text{Gewicht (norm.)}} \cdot \underbrace{\frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{1}{2} \frac{(x - m)^2}{\sigma^2}}}_{\text{Ergebnisdichte (n orm.)}} \end{aligned} $$

Dekomposition einer Gaußdichten

Gegeben: Gaußdichte mit $m, \sigma$

Gesucht: Dekomposition, d.h. mögliche Werte für $m_1, m_2, \sigma_1, \sigma_2$

$$ \begin{aligned} \frac{1}{\sigma^2} &= \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} \\\\ \Rightarrow \kappa^2 &= \kappa_1^2 + \kappa_2^2 \\\\ \Rightarrow \kappa_1^2 &= (1 - \gamma)\kappa^2, \kappa_2^2 = \gamma \cdot \kappa^2 \qquad \gamma \in [0, 1] \end{aligned} $$ $$ m=\frac{1}{\kappa^{2}}\left((1-\gamma) \kappa^{2} \cdot m_{1}+\gamma \kappa^{2} \cdot m_{2}\right)=(1-\gamma) \cdot m_{1}+\gamma \cdot m_{2} \tag{*} $$

Gilt offennsichtlich für $m_1 = m_2 = m$ , aber auch Wahl von $m_1, m_2$ nach $(*)$ möglich