  In the variational relevance vector machine, the gamma distribution is
representative as a hyperprior over the noise precision of automatic relevance
determination prior. Instead of the gamma hyperprior, we propose to use the
inverse gamma hyperprior with a shape parameter close to zero and a scale
parameter not necessary close to zero. This hyperprior is associated with the
concept of a weakly informative prior. The effect of this hyperprior is
investigated through regression to non-homogeneous data. Because it is
difficult to capture the structure of such data with a single kernel function,
we apply the multiple kernel method, in which multiple kernel functions with
different widths are arranged for input data. We confirm that the degrees of
freedom in a model is controlled by adjusting the scale parameter and keeping
the shape parameter close to zero. A candidate for selecting the scale
parameter is the predictive information criterion. However the estimated model
using this criterion seems to cause over-fitting. This is because the multiple
kernel method makes the model a situation where the dimension of the model is
larger than the data size. To select an appropriate scale parameter even in
such a situation, we also propose an extended prediction information criterion.
It is confirmed that a multiple kernel relevance vector regression model with
good predictive accuracy can be obtained by selecting the scale parameter
minimizing extended prediction information criterion.
