Inductive inference with parametric models is a very important tool in the natural sciences.

• Consider \( N \) different models \( M_i \) (\( i = 1, \ldots, N \)), each with parameters \( \boldsymbol{\alpha}_i \). Each of them implies a sampling distribution (conditional predictive distribution for possible data)

$$ $$ p(D|\boldsymbol{\alpha}_i, M_i) $$ $$

• The \( \boldsymbol{\alpha}_i \) dependence when we fix attention on the actual, observed data (\( D_\mathrm{obs} \)) is the likelihood function:

$$ $$ \mathcal{L}_i (\boldsymbol{\alpha}_i) \equiv p(D_\mathrm{obs}|\boldsymbol{\alpha}_i, M_i) $$ $$

• We may be uncertain about \( i \) (model uncertainty),
• or uncertain about \( \boldsymbol{\alpha}_i \) (parameter uncertainty).