Prior Distributions - Revisited

Prior distribution does not significantly affect the parametric identification (both identified values and associated uncertainty) results if it is sufficiently fiat in the range with significant likelihood values. Therefore, it is common to absorb the prior distribution into the normalizing constant and the results are equivalent to the maximum likelihood solution. However, it is not appropriate to absorb the prior distribution into the normalizing constant for model class 

It is clear that the prior distribution affects the modal class selection results. Therefore, the choice of prior distribution is important for model class selection because it offers a reference for comparison in quantifying the information gained from the data. The prior distribution expresses how much previous experience or information a user has about a model class. A more informative prior distribution is used if the user has more experience of the model class. The evidence of such a model class is surplus due to the lifting of the prior PDF. However, inappropriate previous information on the parameters will be penalized by the small value of the inner product of the prior distribution and the likelihood function. In general, it is more difficult to give the prior distribution for empirical models since the physical meaning of the parameters are not as obvious as physical models. More investigations are needed to explore further in this direction. 

Note that inf p(D 9, Cj) = 0 if the range of any of the parameters is infinite. Equalities hold 0 

Bayesian Methods for Structural Dynamics and Civil Engineering 

Note that it is appropriate to select the prior distribution and its boundaries by previous experience or physical laws but not by observing the measurements. In other words, it is inappropriate to select the prior distribution by observing the bounds of the measurements. 

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