Abstract:
Setting prior distributions on model parameters is the act of
characterising the nature of our uncertainty and has proven a
critical issue in applied Bayesian statistics. Although the prior
distribution should ideally encode the users' uncertainty about the parameters, this level of knowledge transfer seems to be
unattainable in practice and applied statisticians are forced to
search for a ``default'' prior. Despite the development of
objective priors, which are only available explicitly for a small
number of highly restricted model classes, the applied
statistician has few practical guidelines to follow when choosing
the priors. An easy way out of this dilemma is to re-use prior
choices of others, with an appropriate reference.
In this talk, I will introduce a new concept for constructing
prior distributions. We exploit the natural nested structure
inherent to many model components, which defines the model
component to be a flexible extension of a base model. Proper
priors are defined to penalise the complexity induced by deviating
from the simpler base model and are formulated after the input of
a user-defined scaling parameter for that model component,
both in the univariate and the multivariate case. These priors are
invariant to reparameterisations, have a natural connection to
Jeffreys' priors, are designed to support Occam's razor and seem
to have excellent robustness properties, all which are highly
desirable and allow us to use this approach to define default
prior distributions. Through examples and theoretical results, we
demonstrate the appropriateness of this approach and how it can be
applied in various situations, like random effect models, spline
smoothing, disease mapping, Cox proportional hazard models with
time-varying frailty, spatial Gaussian fields and multivariate
probit models. Further, we show how to control the overall
variance arising from many model components in hierarchical
models.