  Bayesian model selection provides a formal method of determining the level of
support for new parameters in a model. However, if there is not a specific
enough underlying physical motivation for the new parameters it can be hard to
assign them meaningful priors, an essential ingredient of Bayesian model
selection. Here we look at methods maximizing the prior so as to work out what
is the maximum support the data could give for the new parameters. If the
maximum support is not high enough then one can confidently conclude that the
new parameters are unnecessary without needing to worry that some other prior
may make them significant. We discuss a computationally efficient means of
doing this which involves mapping p-values onto upper bounds of the Bayes
factor (or odds) for the new parameters. A p-value of 0.05 ($1.96\sigma$)
corresponds to odds less than or equal to 5:2 which is below the `weak' support
at best threshold. A p-value of 0.0003 ($3.6\sigma$) corresponds to odds of
less than or equal to 150:1 which is the `strong' support at best threshold.
Applying this method we find that the odds on the scalar spectral index being
different from one are 49:1 at best. We also find that the odds that there is
primordial hemispherical asymmetry in the cosmic microwave background are 9:1
at best.
