A statistical method for the elicitation of priors in Bayesian generalised
linear models (GLMs) and extensions is proposed. Probabilistic predictions are
elicited from the expert to parametrise a multivariate t prior distribution for
the unknown linear coefficients of the GLM and an inverse gamma prior for the
dispersion parameter, if unknown. The elicited predictions condition on defined
elicitation scenarios. Dependencies among scenarios are then elicited from the
expert by additionally conditioning on hypothetical experiments. Elicited
conditional medians efficiently parametrise a canonical vine copula model of
dependence that may be truncated for efficiency. The statistical elicitation
method permits prior parametrisation of GLMs with alternative choices of design
matrices or observation models from the same elicitation session. Extensions of
the method apply to multivariate data, data with bounded support,
semi-continuous data with point mass at zero, and count data with
overdispersion or zero-inflation. A case study elicits a prior for an extended
GLM embedded in a statistical model of overdispersed counts described by a
binomial-simplex mixture distribution. The elicited canonical vine model of
dependence is found to incorporate substantial information into the prior. The
procedures of the statistical elicitation method are implemented in the R
package eglm