Can you test likelihood ratio between different models?

Question

Supposing I have a Null hypothesis $H_{0}$ that my data comes from model $M_{1}$, with parameter $\theta = \theta_{1} $ (so that the model is parametrized as $M(\theta_{1})$ ), and an alternative hypothesis $H_{1}$ for model $M_{1}(\theta)$ where $\theta = \theta_{2}$.

Can I use the likelihood ratio test in case like this, where I am comparing the likelihood between two models?

If not, can I use Bayes factor to do the hypothesis test as alternative ?

Answer 1

That is what a likelihood ratio test is used for: to compare the fit of two models. Suppose we have some parameter space $\Theta$. Let $\Theta_0 = \{\theta_1 \}$. Then the hypothesis test would be $$H_{0}: \theta \in \Theta_0 \\ H_{1}: \theta \notin \Theta_0$$

Then consider $L(\theta|x)$ and the likelihood ratio statistic $$\Lambda(x) = \frac{\sup \{L(\theta|x): \theta \in \Theta_0 \}}{\sup \{L(\theta|x): \theta \in \Theta \}}$$