DPbetabinom: Bayesian Semiparametric Beta-Binomial Model using a DP prior
In DPpackage: Bayesian Nonparametric Modeling in R

Description Usage Arguments Details Value Author(s) References Examples

This function generates a posterior density sample for a semiparametric version of the Beta-Binomial model using a Dirichlet process prior for the mixing distribution.

1 2	DPbetabinom(y,ngrid,prior,mcmc,state,status, data=sys.frame(sys.parent()),work.dir=NULL)

`y`	a matrix giving the binomial data. The first column must include the number of sucess and the second column the number of trials.
`ngrid`	number of grid points where the predictive density estimate is evaluated.
`prior`	a list giving the prior information. The list includes the following parameter: `a0` and `b0` giving the hyperparameters for prior distribution of the precision parameter of the Dirichlet process prior, `alpha` giving the value of the precision parameter (it must be specified if `a0` is missing, see details below), and `a1` and `b1` giving the parameters of the beta centering distribution.
`mcmc`	a list giving the MCMC parameters. The list must include the following integers: `nburn` giving the number of burn-in scans, `nskip` giving the thinning interval, `nsave` giving the total number of scans to be saved, and `ndisplay` giving the number of saved scans to be displayed on screen (the function reports on the screen when every `ndisplay` iterations have been carried out).
`state`	a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis.
`status`	a logical variable indicating whether this run is new (`TRUE`) or the continuation of a previous analysis (`FALSE`). In the latter case the current value of the parameters must be specified in the object `state`.
`data`	data frame.
`work.dir`	working directory.

This generic function fits a semiparametric version of the Beta-Binomial model (Liu, 1996):

yi | ni, pi ~ Binom(ni,pi), i=1,…,n

pi | G ~ G

G | alpha, G0 ~ DP(alpha G0)

where, the baseline distribution is the beta distribution,

G0 = Beta(a1,b1)

To complete the model specification, the following hyperprior can be assumed for the total mass parameter:

alpha | a0, b0 ~ Gamma(a0,b0)

Notice that the baseline distribution, G0, is a conjugate prior in this model specification. Therefore, standard algorihtms for conjugate DP models are used (see, e.g., Escobar and West, 1995; MacEachern, 1998).

An object of class DPbetabinom representing the DP Beta-Binomial model fit. Generic functions such as print, summary, and plot have methods to show the results of the fit. The results include the baseline parameters, alpha, and the number of clusters.

The MCMC samples of the parameters in the model are stored in the object thetasave. The object is included in the list save.state and are matrices which can be analyzed directly by functions provided by the coda package. The subject-specific binomial probabilities are stored in the object randsave.

The list state in the output object contains the current value of the parameters necessary to restart the analysis. If you want to specify different starting values to run multiple chains set status=TRUE and create the list state based on this starting values. In this case the list state must include the following objects:

`ncluster`	an integer giving the number of clusters.
`p`	a vector of dimension (no. observations+1) giving the current value of the binomial probabilities.
`ss`	an interger vector defining to which of the `ncluster` clusters each observation belongs.
`alpha`	giving the value of the precision parameter.

Alejandro Jara <atjara@uc.cl>

Fernando Quintana <quintana@mat.puc.cl>

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.

Liu, J.S. (1996). Nonparametric Hierarchical Bayes via Sequential Imputations. The Annals of Statistics, 24: 911-930.

MacEachern, S.N. (1998) Computational Methods for Mixture of Dirichlet Process Models, in Practical Nonparametric and Semiparametric Bayesian Statistics, eds: D. Dey, P. Muller, D. Sinha, New York: Springer-Verlag, pp. 1-22.

## Not run: 
    # Data
      data(rolling)
      y <- cbind(rolling$y1,rolling$y2)


    # Prior information

      prior<-list(alpha=1,
                  a1=1,
                  b1=1)

    # Initial state
      state <- NULL

    # MCMC parameters

      mcmc <- list(nburn=5000,
                   nsave=10000,
                   nskip=3,
                   ndisplay=100)

    # Fitting the model

      fit <- DPbetabinom(y=y,ngrid=100, 
                         prior=prior, 
                         mcmc=mcmc, 
                         state=state, 
                         status=TRUE)

      fit
      summary(fit)

    # density estimate
      plot(fit,output="density")

    # parameters
      plot(fit,output="param")

## End(Not run)