beta_mode: Posterior Mode
In conting: Bayesian Analysis of Contingency Tables

Description Usage Arguments Details Value Note Author(s) References Examples

This function finds the posterior mode of the log-linear parameters of a log-linear model with a given design matrix and prior distribution.

1	beta_mode(X, prior = "SBH", y, IP , a = 0.001 , b = 0.001)

`X`	The n by p design matrix where n is the number of cells and p is the number of log-linear parameters.
`prior`	The prior distribution. It can be one of `c("UIP","SBH")`, where `"UIP"` = unit information prior; and `"SBH"` = Sabanes-Bove & Held prior. The default value is `"SBH"`.
`y`	The n by 1 vector of cell counts.
`IP`	A p by p matrix giving the inverse of the prior scale matrix.
`a`	The shape hyperparameter of the Sabanes-Bove & Held prior, see Overstall & King (2014).
`b`	The scale hyperparameter of the Sabanes-Bove & Held prior, see Overstall & King (2014).

The posterior mode is found by maximising the log unnormalised posterior pdf given by the sum of the log-likelihood and the log of the prior pdf. This optimisation is achieved using a quasi Newton-Raphson method.

For details on the unit information and Sabanes-Bove & Held priors for generalised linear models see Ntzoufras et al (2003) and Sabanes-Bove & Held (2011), respectively. See Overstall & King (2014), and the references therein, for their application to log-linear models and contingency tables.

The posterior mode is required for the reversible jump algorithm implemented from Forster et al (2012).

beta_mode will return a p by 1 vector containing the posterior mode of the log linear parameters.

This function will not typically be called by the user.

Antony M. Overstall A.M.Overstall@soton.ac.uk.

Sabanes-Bove, D. & Held, L. (2011) Hyper-g priors for generalized linear models. Bayesian Analysis, 6 (3), 387–410.

Forster, J.J., Gill, R.C. & Overstall, A.M. (2012) Reversible jump methods for generalised linear models and generalised linear mixed models. Statistics and Computing, 22 (1), 107–120.

Nztoufras, I., Dellaportas, P. & Forster, J.J. (2003) Bayesian variable and link determination for generalised linear models. Journal of Statistical Planning and Inference, 111 (1), 165–180.

Overstall, A.M. & King, R. (2014) conting: An R package for Bayesian analysis of complete and incomplete contingency tables. Journal of Statistical Software, 58 (7), 1–27. http://www.jstatsoft.org/v58/i07/

data(AOH) ## loads the AOH data

X<-model.matrix(~alc+hyp+obe,data=AOH,
contrasts=list(alc="contr.sum",hyp="contr.sum",obe="contr.sum"))
## Sets up the design matrix for the independence model

IP<-(t(X)%*%X)/dim(X)[1]
## Set up inverse of prior scale matrix

beta_mode(X=X,prior="UIP",y=AOH$y,IP=IP)
## Finds the posterior mode of the log-linear parameters under the
## independence model with the unit information prior. Will get:
#X(Intercept)        Xalc1        Xalc2        Xalc3        Xhyp1        Xobe1 
# 2.894270420 -0.045859743 -0.071775824  0.089541068 -0.504141954  0.008163604 
#       Xobe2 
#-0.016327209

beta_mode(X=X,prior="SBH",y=AOH$y,IP=IP)
## Finds the posterior mode of the log-linear parameters under the
## independence model with the Sabanes-Bove & Held prior. Will get:
#X(Intercept)        Xalc1        Xalc2        Xalc3        Xhyp1        Xobe1 
# 2.908298763 -0.043704371 -0.068212247  0.085338704 -0.473628107  0.007762839 
#       Xobe2 
#-0.015525678