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# BAYESIAN GOODNESS OF FIT TESTING USING
INFINITE DIMENSIONAL EXPONENTIAL FAMILIES

**Isabella Verdinelli and Larry Wasserman**

### Abstract:

*We develop a nonparametric Bayes factor for
testing the fit of a parametric model.
We begin with a nominal parametric family
which we then embed
into an infinite
dimensional exponential family.
The new model then has a parametric and nonparametric component.
We give the log density
of the nonparametric component a Gaussian process prior.
An asymptotic consistency requirement puts a restriction on the
form of the prior leaving us with a single
hyperparameter for which we suggest a default value
based on simulation experience.
Then we construct a Bayes factor to test
the nominal model versus the semiparametric alternative.
Finally, we show that the Bayes factor is consistent.
The proof of the consistency is based on approximating
the model by a sequence of exponential families.
**
** Keywords:* Bayes factor, Consistency, Gaussian process prior,
Markov chain Monte Carlo, Nonparametric Bayesian Inference,
Sieve.

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