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Consider a model with a parameter <math>\theta</math> and Gaussian prior <math>p(\theta)=N(0,0.5^2)</math>, which is the Normal distribution with mean zero and standard deviation 0.5 (illustrated in the Figure, left). This prior says that without any data, this parameter is expected to have value zero, but we are willing to entertain positive or negative values (with a 99% confidence interval [-1.16 1.16]). This model is fitted to the data to provide an estimate of the parameter <math>q(\theta)</math> and the model evidence <math>p(y)</math>.
To assess whether the parameter contributed to the model evidence, i.e. whether we have learnt anything about this parameter from the data, an alternative 'reduced' model is specified in which the parameter has a prior with a smaller variance: e.g. <math>\tilde{p}_0=N(0,1000^2)</math>. This is illustrated in the Figure (right). This prior effectively 'switches off' the parameter, saying that we are almost certain that it has value zero. The parameter <math>\tilde{q}(\theta)</math> and evidence <math>\tilde{p}(y)</math> for this reduced model are rapidly computed from the full model using Bayesian model reduction.
The hypothesis that the parameter contributed to the model is then tested by comparing the full and reduced models via the [[Bayes factor]], which is the ratio of model evidences:
<math>BF=\frac{p(y)}{\tilde{p}(y)}</math>
The larger this ratio, the greater the evidence for the full model, which included the parameter as a free parameter. Conversely, the stronger the evidence for the reduced model, the more confidently we can conclude that the parameter did not contribute and can eliminated from the model.
== Applications ==
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