Does the simpler "small" model fit the data, or is the more general "big" model is needed?
Since it has more parameters to adjust, the big model, \(\mathcal{M}_B\), always fits better than the small model, \(\mathcal{M}_S\), but does it fit significantly better?
A model's fit can be described by its maximum possible likelihood,
\[ L(\mathcal{M}) \;\;=\;\; \underset{\text{all models in }\mathcal{M}}{\operatorname{max}} P(data \mid model) \]This is the likelihood with all unknown parameters replaced by maximum likelihood estimates.
Likelihood ratio
Because \(\mathcal{M}_S\) is a special case of \(\mathcal{M}_B\),
\[ L(\mathcal{M}_B) \;\;\ge\;\; L(\mathcal{M}_S) \]Equivalently, the likelihood ratio is always at least one,
\[ R \;\;=\;\; \frac{L(\mathcal{M}_B)}{L(\mathcal{M}_S)} \;\;\ge\;\; 1 \]Big values of \(R\) suggest that \(\mathcal{M}_S\) does not fit as well as \(\mathcal{M}_B\). Equivalently,
\[ \log(R) \;\;=\;\; \ell(\mathcal{M}_B) - \ell(\mathcal{M}_S) \;\;\ge\;\; 0 \]and again, big values suggest that \(\mathcal{M}_S\) does not fit as well as \(\mathcal{M}_B\).