In many applications the model is not a finite set of hypotheses, but rather a continuum labelled by a real-valued parameter. This leads to some subtle changes in the definition of the distribution over hypotheses and the likelihoods. The prior and posterior must be written down as a so-called probability density function, \(P(h_{\theta}) d\theta\). The likelihoods need to be defined by a limit process: the probability \(P(h_{\theta})\) is infinitely small so that we cannot define \(P(s \mid h_{\theta})\) in the normal manner. But other than that the Bayesian machinery works exactly the same:  \[ P(h_{\theta} \mid s) d\theta \;\; = \;\; \frac{P(s \mid h_{\theta})}{P(s)} P(h_{\theta}) d\theta. \]  Finally, summations need to be replaced by integrations:  \[ P(s) \; = \; \int_{\theta \in \Theta} P(h_{\theta}) P(s \mid h_{\theta}) d\theta . \]  This expression is often called the marginal likelihood of the model: it expresses how probable the data is in the light of the model as a whole.