Consider a binary classification problem to be solved using Bayes decision rule. It is given that $P(x | y_0) = \sigma(0.9x_1 + 0.3x_2 + \theta)$ and $P(x | y_1) = \sigma(0.3x_1 - 0.1x_2 + 0.7)$, and prior probabilities of class $y_0$ and $y_1$ are 0.2 and 0.8, respectively. $\sigma$ represents the sigmoid (logistic) function. Find the value of $\theta$ for which the predicted class $x = (-1, 1)$ is $y_0$.
This is the problem I was trying to solve. Can you please tell me how to solve it. Below is how my friend told me to solve it. But I do not know how the ratio is greater than 1.
$P(x | y_0) = \sigma(-0.6 + \theta)$
$P(x | y_1) = \sigma(0.3)$
$P(y_0) = 0.2$
$P(y_1) = 0.8$
$\frac{P(x | y_1)P(y_1)}{P(x | y_0)P(y_0)} > 1$