In the frequentist approach \(\theta\) is considered a fixed number that does not depend on the sample \(x\). Consequently

if \(\theta \in \Theta_0\) then \(P(\theta \in \Theta_0 | x) = P(H_0 |x) = 1\) and \(P(\theta \in \Theta_1 | x) = P(H_1 |x) = 0\).

if \(\theta \in \Theta_1\) then \(P(\theta \in \Theta_0 | x) = P(H_0 |x) = 0\) and \(P(\theta \in \Theta_1 | x) = P(H_1|x) = 1\).

In the Bayesian approach, these probabilities depend on the sample \(x\) and can be obtained using the posterior distribution \(\pi(\theta | x)\), which can be calculated using Bayes theorem.

Types of hypothesis test

Simple

\(H_0: \theta = \theta_0 \text{ and } H_1: \theta = \theta_1\)

\[\pi(H_0 |x) = \frac{f(x|H_0) \pi(H_0 )}{f(x|H_0 ) \pi(H_0 ) + f(x|H_1) \pi(H_1 )}\]

\[\pi(H_1 |x) = \frac{f(x|H_1) \pi(H_1 )}{f(x|H_0 ) \pi(H_0 ) + f(x|H_1) \pi(H_1 )}\]

\[\text{with } \pi(H_0) + \pi(H_1) = 1\]

Then

\[\text{Posterior odds }=\frac{\pi(H_0 |x)}{\pi(H_1 |x)} = \frac{f(x|H_0)}{f(x|H_1 )} \frac{\pi(H_0)}{\pi(H_1)} = \text{Bayes factor x Priod odds} = B \text{ x Priod odds}\]

Composite

\[B = \frac{\int_{\Theta_0} f(x|\theta)g_0(\theta) \, d\theta}{\int_{\Theta_1} f(x|\theta)g_1(\theta) \, d\theta}\]

where \(g_i(\theta)\) is the prior distribution of \(\theta\) under \(H_i\).

One-side

Two-sides

\(H_0: \theta = \theta_0 \text{ and } H_1: \theta \neq \theta_0\)

\[B = \frac{\int_{\Theta_0} f(x|\theta) \delta(\theta-\theta_0) \, d\theta}{\int_{\Theta_1} f(x|\theta)g_1(\theta) \, d\theta} = \frac{f(x|\theta_0)}{\int_{\Theta_1} f(x|\theta)g_1(\theta) \, d\theta}\]

Decision rule

In the frequentist approach, the rejection region is often specified by establishing a criterium on a test statistic \(W(x)\) (which often is \(\Lambda(x)\)):

\[R=\{x: W(x) < c\}\]

In the Bayesian approach, given that \(\pi(H_0 |x)\) and \(\pi(H_1 |x)\) are probabilities, a natural test statistic is the posterior odds and a natural criterium is 1

\[R=\left\{ x: \frac{\pi(H_0 |x)}{\pi(H_1 |x)} < 1 \right\} =\left\{ x: \pi(H_1 |x) > 1/2 \right\}\]