Biomedical Engineering Reference
In-Depth Information
contrast, using a set of appropriately selected fixed knots requires much less
computational effort while maintaining enough modeling flexibility. While it
is true in general that using more knots can provide more modeling flexibility,
using too many knots requires additional computational cost and may cause
over-fitting problems. Based on our experience, using 10 to 30 knots (equally
spaced or based on quantiles) provides adequate modeling flexibility for data
sets containing up to thousands of observations.
6.3
Proposed Methods
Bayesian methods typically require sampling all unknown parameters from
their posterior distributions that are formed by combining the likelihood func-
tions and the priors distributions. However, using the likelihood function in
Equation (6.1) causes dicult sampling because the posterior distributions
of the unknowns are nonstandard. Metropolis{Hastings algorithms can be
applied but are complicated and require tuning parameters. Our proposed
Bayesian methods do not require any complicated Metropolis{Hastings steps.
All the parameters are sampled either from standard distributions or by using
the automatic adaptive rejection method (ARS) (Gilks and Wild, 1992) under
each model. This is achieved by expanding the observed likelihoods utilizing
the data structure and the model properties.
The priors for and are taken to be essentially the same for the three
models considered. Specifically, we assign a multivariate normal N(
0
;
0
)
prior for the regression coecients . We adopt independent exponential priors
E() for all f
l
g
l=1
. To allow for more flexibility, we assign a G(a
;b
) prior, a
Gamma distribution with mean a
=b
, for . Such prior specication allows for
penalizing large basis coecients and shrinking the coecients of unnecessary
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