Definition: Log-likelihood ratio
Index:
The Book of Statistical Proofs ▷
General Theorems ▷
Frequentist statistics ▷
Likelihood theory ▷
Log-likelihood ratio
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Metadata: ID: D199 | shortcut: llr | author: JoramSoch | date: 2024-06-14, 14:45.
Definition: Let $m_0$ and $m_1$ be two generative models describing the same measured data $y$ using different model parameters $\theta_0 \in \Theta_0$ and $\theta_1 \in \Theta_1$. Then, the logarithmized quotient of the maximized likelihood functions of these two models is denoted as $\log \Lambda_{01}$ and is called the log-likelihood ratio of $m_0$ relative to $m_1$:
\[\label{eq:llr} \log \Lambda_{01} = \log \frac{\operatorname*{max}_{\theta_0 \in \Theta_0} \mathcal{L}_{m_0}(\theta_0)}{\operatorname*{max}_{\theta_1 \in \Theta_1} \mathcal{L}_{m_1}(\theta_1)} = \log p(y|\hat{\theta}_0,m_0) - \log p(y|\hat{\theta}_1,m_1) \; .\]- Wikipedia (2024): "Likelihood-ratio test"; in: Wikipedia, the free encyclopedia, retrieved on 2024-06-14; URL: https://en.wikipedia.org/wiki/Likelihood-ratio_test#Definition.
Metadata: ID: D199 | shortcut: llr | author: JoramSoch | date: 2024-06-14, 14:45.