Michael Data

Binary Classification with Univariate Gaussian

For univariate Gaussian binary classification:
Optimal decision regions are defined by decision boundaries in $x$ where $P(c_1)P(x|\theta_1) = P(c_2)P(x|\theta_2)$.


\begin{align*}
    *    P(c_1)P(x|\theta_1) &= P(c_2)P(x|\theta_2)
\\  P(c_1)\frac{1}{\sqrt{2\pi}\sigma_1} e^{-\frac{1}{2\sigma_1^2}(\mu_1-x)^2} &= 
    *    P(c_2)\frac{1}{\sqrt{2\pi}\sigma_2} e^{-\frac{1}{2\sigma_2^2}(\mu_2-x)^2}
\end{align*}

Cancel common terms and take the log


\begin{align*}
\\  P(c_1)\sigma_2^2 e^{-\frac{1}{2\sigma_1^2}(\mu_1-x)^2} &= 
    *    P(c_2)\sigma_1^2 e^{-\frac{1}{2\sigma_2^2}(\mu_2-x)^2}
\\  \log P(c_1) + \log \sigma_2 - \frac{1}{2\sigma_1^2}(\mu_1-x)^2 &= 
    *    \log P(c_2) + \log \sigma_1 - \frac{1}{2\sigma_2^2}(\mu_2-x)^2
\\  \frac{1}{2\sigma_2^2}(\mu_2-x)^2 - \frac{1}{2\sigma_1^2}(\mu_1-x)^2&= 
    *    \log P(c_2) - \log P(c_1) + \log \sigma_1 - \log \sigma_2
\\  \frac{1}{\sigma_2^2}(\mu_2-x)^2 - \frac{1}{\sigma_1^2}(\mu_1-x)^2 &= 
    *    2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\\  \sigma_1^2(\mu_2-x)^2 - \sigma_2^2(\mu_1-x)^2 &= 
    *    2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\\  \sigma_1^2(\mu_2^2 - 2\mu_2x + x^2) - \sigma_2^2(\mu_1^2 - 2x\mu_1 + x^2) &= 
    *    2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\\  \sigma_1^2\mu_2^2 - 2\sigma_1^2\mu_2x + \sigma_1^2x^2 - \sigma_2^2\mu_1^2 + 2\sigma_2^2\mu_1x - \sigma_2^2x^2&= 
    *    2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\\  \sigma_1^2x^2 - \sigma_2^2x^2 + 2\sigma_2^2\mu_1x - 2\sigma_1^2\mu_2x - \sigma_2^2\mu_1^2 + \sigma_1^2\mu_2^2 &= 
    *    2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\end{align*}

Use the quadratic formula:


\begin{align*}
    *    0 &= (\sigma_1^2 - \sigma_2^2)x^2 + \left(2(\sigma_2^2\mu_1 - \sigma_1^2\mu_2) \right) x 
    *    + \sigma_1^2\mu_2^2 - \sigma_2^2\mu_1^2 - 2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 }
\\  x &= \frac{-b \pm \sqrt(b^2-4ac)}{2a}
\end{align*}
\begin{align*}
    *   a &= (\sigma_1^2 - \sigma_2^2)
\\ b &= 2(\sigma_2^2\mu_1 - \sigma_1^2\mu_2)
\\ c &= \sigma_1^2\mu_2^2 - \sigma_2^2\mu_1^2 - 2 \sigma_1^2 \sigma_2^2 \log \frac{P(c_2) \sigma_1 }{ P(c_1) \sigma_2 } 
\end{align*}