1.3. Bayesian Estimation in Binary Regression

Considering the distribution Bernoulli for the variable response, the likelihood function is given by

Where is the cdf of an asymmetric distribution indexed by the shape parameter associated with the asymmetric link. For Power Logit and Scobit . For skew probit and skew logit

The logit, probit, cloglog, scobit and power logit links consider this likelihood function; however skew probit and skew logit links consider other versions of the likelihood function considering augmented versions that are discussed in the specific references of these models.

In the Bayesian Inference, the parameters of interest are assumed like random variables and so is need establishes a priori probability distributions that reflects our previous knowledge of its behavior. Combining the likelihood function and the priori distributions we can obtain the posteriori distributions of the parameters of interest. In the present work, we consider priors that they are vague proper priors with known distributions but variance big as well as independence between priors (see Nzoufras, 2009).

In Binary Regression models there is consensus about the prior specification for regression coefficients, thus is assumed for with to be large, and this case is considered . In relation with the shape parameter associated with the link: for Scobit and Power logit models is assumed (, ) and for skew probit models , where and . In addition is assumed independent priors as

With as indicated above.

Bayesian Inference is facilitated with the use of different MCMC methods implemented in WinBUGS or OpenBUGS software using a minimum programming. For more details about the use of these software for Bayesian Inference and usual Binary regression models we suggest the book of Congdon (2005), Congdon (2010) and Ntzoufras (2009). An introduction to MCMC methods is given in Gilks, Richardson, and Spiegelhalter (1996).

Also, Bayesian Inference for some traditional Binary models considering R packages as arm, bayesm, DPpackage, LaplacesDemon, MCMCpack are available.

However, for most of the models presented here, there is no program that generates codes for WinBUGS or OpenBUGS with excecption of BRMUW (Bazán 2010) a previous version of this program in Spanish. In contrast, BUGS codes for all Binary regression models presented here are facilitated using BayesianModeling. The Binary regression models implemented in BayesianModeling classified according to its links are:

• Symmetric: probit, logit.

• Asymmetric: cloglog, scobit, power logit, skew logit, skew probit (CDS, BBB and standard).

All codes for Binary regression models are established considering the likelihood function presented here and considering the priors suggested with the exception of the skew logit and skew probit models which use an augmented likelihood function version. In BayesianModeling, when a specific code is generated for a Binary regression model, also References to justify the model and the choices of priors are showed.