  Skewed probit regression is but one example of a statistical model that
generalizes a simpler model, like probit regression. All skew-symmetric
distributions and link functions arise from symmetric distributions by
incorporating a skewness parameter through some skewing mechanism. In this work
we address some fundamental issues in skewed probit regression, and more
genreally skew-symmetric distributions or skew-symmetric link functions. We
address the issue of identifiability of the skewed probit model parameters by
reformulating the intercept from first principles. A new standardization of the
skew link function is given to provide and anchored interpretation of the
inference. Possible skewness parameters are investigated and the penalizing
complexity priors of these are derived. This prior is invariant under
reparameterization of the skewness parameter and quantifies the contraction of
the skewed probit model to the probit model. The proposed results are available
in the R-INLA package and we illustrate the use and effects of this work using
simulated data, and well-known datasets using the link as well as the
likelihood.
