defaultpost {rjmcmc}

Perform Post-Processing Using Default Bijections

Description

Performs Bayesian multimodel inference, estimating Bayes factors and posterior model probabilities for N candidate models. Unlike rjmcmcpost, this function uses a default bijection scheme based on approximating each posterior by a multivariate normal distribution. The result is reminiscent of the algorithm of Carlin & Chib (1995) with a multivariate normal pseudo-prior. Transformation Jacobians are computed using automatic differentiation so do not need to be specified.

Usage

defaultpost(posterior, likelihood, param.prior, model.prior,
  chainlength = 10000, TM.thin = chainlength/10, progress = TRUE,
  save.all = TRUE)

Arguments

posterior

A list of N matrices containing the posterior distributions under each model. Generally this will be obtained from MCMC output. Note that each parameter should be real-valued so some parameters may need to be transformed, using logarithms for example.

likelihood

A list of N functions specifying the log-likelihood functions for the data under each model.

param.prior

A list of N functions specifying the prior distributions for each model-specific parameter vector.

model.prior

A numeric vector of the prior model probabilities. Note that this argument is not required to sum to one as it is automatically normalised.

chainlength

How many iterations to run the Markov chain for.

TM.thin

How regularly to calculate transition matrices as the chain progresses.

progress

A logical determining whether a progress bar is drawn.

save.all

A logical determining whether to save the value of the universal parameter at each iteration, as well as the corresponding likelihoods, priors and posteriors. If TRUE, the output object occupies significantly more memory.

Value

Returns an object of class rj (see rjmethods). If save.all=TRUE, the output has named elements result, densities, psidraws, progress and meta. If save.all=FALSE, the densities and psidraws elements are omitted.

result contains useful point estimates, progress contains snapshots of these estimates over time, and meta contains information about the function call.

References

Carlin, B. P. and Chib, S. (1995) Bayesian Model Choice via Markov Chain Monte Carlo Methods. Journal of the Royal Statistical Society, Series B, 473-484.

Barker, R. J. and Link, W. A. (2013) Bayesian multimodel inference by RJMCMC: A Gibbs sampling approach. The American Statistician, 67(3), 150-156.

See Also

adiff rjmcmcpost

Examples

## Comparing two binomial models -- see Barker & Link (2013) for further details.

y=c(8,16); sumy=sum(y)
n=c(20,30); sumn=sum(n)

L1=function(p){if((all(p>=0))&&(all(p<=1))) sum(dbinom(y,n,p,log=TRUE)) else -Inf}
L2=function(p){if((p[1]>=0)&&(p[1]<=1)) sum(dbinom(y,n,p[1],log=TRUE)) else -Inf}

p.prior1=function(p){sum(dbeta(p,1,1,log=TRUE))}
p.prior2=function(p){dbeta(p[1],1,1,log=TRUE)+dbeta(p[2],17,15,log=TRUE)}

draw1=matrix(rbeta(2000,y+1,n-y+1), 1000, 2, byrow=TRUE)  ## full conditional posterior
draw2=matrix(c(rbeta(1000,sumy+1,sumn-sumy+1),rbeta(1000,17,15)), 1000, 2)

out=defaultpost(posterior=list(draw1,draw2), likelihood=list(L1,L2), 
                param.prior=list(p.prior1,p.prior2), model.prior=c(1,1), chainlength=1000)