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**A Reference Bayesian Test for Nested Hypotheses**

And its Relationship to the Schwarz Criterion

**Robert E. Kass and Larry Wasserman**

### Abstract:

*To compute a Bayes factor for testing
in the presence of
a nuisance parameter , priors under the null and alternative
hypotheses must be chosen.
As in Bayesian estimation, an important problem has been to define
automatic or ``reference'' methods for determining priors based only
on the structure of the model. In this paper we apply the heuristic device
of taking
the amount of
information in the prior on
equal to the amount of information in a single observation. Then, after
transforming to be ``null orthogonal'' to , we
take the marginal priors on to be equal under the null and
alternative hypotheses.
Doing so, and taking the prior on to be Normal, we
find that the
log of the Bayes factor may be approximated by the Schwarz criterion with an
error of order , rather than the usual error
of order . This result
suggests the
Schwarz criterion should provide sensible approximate solutions to
Bayesian testing problems, at least when the
hypotheses are nested. When, instead, the prior on is
elliptically Cauchy a constant correction term must be added to the
Schwarz criterion; the result then becomes a multidimensional
generalization of Jeffreys's method.
**
**Keywords:* Bayes Information Criterion, Laplace's method,
Model selection, Null-orthogonal parameters, Orthogonal
parameters.

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