Various other prominent model selection tools are based on methods from Bayesian statistics. They all start from the idea that the quality of a model is expressed in the performance of the model on the sample data: the model that, on the whole, makes the sampled data most probable is to be preferred.  Because of this, there is a close connection with the hierarchical Bayesian modelling referred to earlier (Gelman 2013). The central notion in the Bayesian model selection tools is thus the marginal likelihood of the model, i.e., the weighted average of the likelihoods over the model, using the prior distribution as a weighing function:  \[ P(s \mid M_{i}) \; = \; \int_{\theta \in \Theta_{i}} P(h_{\theta}) P(s \mid h_{\theta}) d\theta . \]  Here \(\Theta_{i}\) is the parameter space belonging to model \(M_{i}\).  The marginal likelihoods can be combined with a prior probability over models, \(P(M_{i})\), to derive the so-called posterior model probability, using Bayes' theorem. One way of evaluating models, known as Bayesian model selection, is by comparing the models on their marginal likelihood, or else on their posteriors (cf. Kass and Raftery 1995).