This book points the attention on a very crucial topic in Statistics - Model Selection - from a Bayesian point of view. In particular we are interested in analyzing the way in which we have to think and rationalize, when dealing with a problem of model choice. In the Classical background, this problem is strictly related to the field of hypotheses testing, since most of the tools used by Classical statisticians, to support such type of decision, are tests over parameters in the model or likelihood ratios. In spite of this, Bayesian theory allows us to tackle this problem in a more general setting that does not necessarily coincide with the hypotheses testing approach, leading us to the point that a threshold between these two settings is needed. However it is not clear yet where the hypotheses testing ends, and the model selection begins. A possible key to the solution of this matter lies on the definition of a statistical model, and more specifically of a nested model. A model selection problem with nested models identified by inequality constraints will be considered to illustrate this idea, with the support of an application implemented with Matlab.