Borel-Kolmogorov paradox and the Savage-Dickey ratio

<div style="clear:both;"></div>A simple illustration of the Borel-Kolmogorov paradox, which concerns the fact that probabilities for the same events can be very different depending on the parameterization, if one conditions on an event of probability zero. A particular context in which this plays a role is in the computation of Bayes Factors where one wishes to compare a null hypothesis that a certain function of a set of parameters is equal to a particular value versus that it has any value likely under a given prior. In the note it is determined in which cases this Bayes Factor (computed via the Savage Dickey ratio) depends on the chosen parameterization in the model. A particular situation in which this problem arises, involves the so called 'encompassing prior' approach, in which the null hypothesis to be tested is specified as the limit of a nested truncation process of the encompassing prior distribution towards the null hypothesis. (Part of a paper submitted by <a href="http://www.ruudwetzels.com/index.php?src=publications">Ruud Wetzels and Eric-Jan Wagenmakers and me.</a>) <br/> <!-- a hrefold="http://users.fmg.uva.nl/rgrasman/worknotes/uploaded_files/Borel%20Paradox%20and%20Savage%20Dickey%20ratio.pdf">Borel Paradox and Savage Dickey ratio.pdf</a --> <a href="http://dl.getdropbox.com/u/609029/Borel/stukje%20Emcompasing%20prior%20v3.pdf">Borel Paradox and Savage Dickey ratio.pdf</a><div style="clear:both; padding-bottom:0.25em"></div>