  Bayesian statistical inference for Generalized Linear Models (GLMs) with
parameters lying on a constrained space is of general interest (e.g., in
monotonic or convex regression), but often constructing valid prior
distributions supported on a subspace spanned by a set of linear inequality
constraints can be challenging, especially when some of the constraints might
be binding leading to a lower dimensional subspace. For the general case with
canonical link, it is shown that a generalized truncated multivariate normal
supported on a desired subspace can be used. Moreover, it is shown that such
prior distribution facilitates the construction of a general purpose product
slice sampling method to obtain (approximate) samples from corresponding
posterior distribution, making the inferential method computationally efficient
for a wide class of GLMs with an arbitrary set of linear inequality
constraints. The proposed product slice sampler is shown to be uniformly
ergodic, having a geometric convergence rate under a set of mild regularity
conditions satisfied by many popular GLMs (e.g., logistic and Poisson
regressions with constrained coefficients). One of the primary advantages of
the proposed Bayesian estimation method over classical methods is that
uncertainty of parameter estimates is easily quantified by using the samples
simulated from the path of the Markov Chain of the slice sampler. Numerical
illustrations using simulated data sets are presented to illustrate the
superiority of the proposed methods compared to some existing methods in terms
of sampling bias and variances. In addition, real case studies are presented
using data sets for fertilizer-crop production and estimating the SCRAM rate in
nuclear power plants.

Methodology;Applications