1. Hypothesis testing problem

$ H_0 : \theta \in \Theta_0 \quad\quad \text{vs} \quad\quad H_1: \theta \in \Theta_1 \quad\quad (\Theta_0 \uplus \Theta_1 = \Theta = \text{whole parameter space})$

2. Bayesian approach

1. Calculate the posterior prob. for each model ($\Theta_0, \Theta_1$): $ \pi(H_0 \mid x), \pi(H_1 \mid x) $
2. Support the hypothesis that has larger posterior prob.

3. Calculate posterior

1. Posterior $ \pi(H_i | x) $
   \(\begin{equation} \begin{aligned} \pi(H_i | x) &\propto p(H_i) \color{red}{f(x | H_i)} \\ &= \color{blue}{\pi_i} f(x | H_i) \quad\quad \cdots \quad\quad \color{blue}{\pi_i} \equiv p(H_i) \\ \color{red}{f(x | H_i)} &= \int_{\theta \in \Theta_i} f(x | \theta, H_i) \pi(\theta | H_i) d\theta \\ &= \int_{\theta \in \Theta_i} f(x | \theta) \pi(\theta | H_i) d\theta \quad\quad \cdots \quad\quad \text{Assume that } f(x | \theta) = f(x | \theta, H_i) \\ &= \int_{\theta \in \Theta_i} f(x | \theta) \color{blue}{g_i(\theta)} d\theta \quad\quad \cdots \quad\quad \color{blue}{g_i(\theta)} \equiv \pi(\theta | H_i) \\ \end{aligned} \end{equation}\)
2. Compare posterior prob.
   \(\pi(H_0 \mid x) = \frac{f(x, H_0)}{f(x, H_0) + f(x, H_1)} = \frac{\pi_0 \int_{\theta \in \Theta_0} f(x | \theta) g_0(\theta) d\theta}{\pi_0 \int_{\theta \in \Theta_0} f(x | \theta) g_0(\theta) d\theta + \pi_1 \int_{\theta \in \Theta_1} f(x | \theta) g_1(\theta) d\theta} \\ \pi(H_1 \mid x) = \frac{f(x, H_1)}{f(x, H_0) + f(x, H_1)} = \frac{\pi_1 \int_{\theta \in \Theta_1} f(x | \theta) g_1(\theta) d\theta}{\pi_0 \int_{\theta \in \Theta_0} f(x | \theta) g_0(\theta) d\theta + \pi_1 \int_{\theta \in \Theta_1} f(x | \theta) g_1(\theta) d\theta}\) Support $\text{argmax}_i \pi(H_i \mid x)$

4. Bayes factor

1. Prior odds
   $ \frac{\pi_0}{\pi_1} $

2. Posterior odds
   $ \frac{\pi(H_0 \mid x)}{\pi(H_1 \mid x)} $

3. Bayes factor
   $ B_{01}(x) = \frac{\pi(H_0 \mid x)}{\pi(H_1 \mid x)} \times \frac{\pi_1}{\pi_0} = \frac{f(x \mid H_0)}{f(x \mid H_1)} $ $ B_{10}(x) = \frac{1}{B_{01}(x)} \quad\quad \text{(symmetric)} $
   $ f(x \mid H_i) $ : the amount of evidence that the data $x$ supports hypothesis $H_i$