Bayesian Estimation of Univariate Probability Distributions

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Description

For a given dataset, this function serves to approximate (using a Metropolis algorithm) the posterior distribution of the parameters for some specified parametric probability distribution.

Usage

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metropolis(model, iter = 1000, tun = 2, trans.list = NULL,
	start = NULL, variance = NULL, prior = NULL, burn = 0,
	uniroot.interval = c(-100, 100),pass.down.to.C=FALSE)

Arguments

model

mle object corresponding to the fitted (by maximum likelihood) model.

A list(x=dataset, dist=distribution) object may also be provided, but the user will then have to make sure to specify the arguments start and variance. Moreover, the latter two arguments will have to be specified on their transformed scales (see trans.list).

iter

The requested number of iterations - the Markov Chain's length.

tun

A tuning constant; value by which the covariance matrix of the multivariate normal proposal will be multiplied - see References.

trans.list

A list object containing a function for each parameter that is to be estimated. For each parameter, the function must correspond to the inverse transformation that will determine the parametrization for which the simulation will be carried out (see Example and Details).

start

A vector of starting values for the algorithm. If NULL, the maximum likelihood parameter estimates will be used as starting values for the Markov Chain. If model is not an object from the class mle, this argument will have to be specified, along with the argument variance. Moreover, as already stated above, the user will have to make sure that both start and variance are those for the transformed parameters (see trans.list).

variance

Covariance matrix of the multivariate normal proposal distribution. If NULL, the observed Fisher's information will be used and multiplied by the specified tun. As for start, this argument needs to be specified if model is not from the class mle.

prior

A function that corresponds to the joint prior distribution (see Example). Note that the prior distribution will be evaluated on the transformed parameter space(s).

burn

Burn-in period (see References).

uniroot.interval

Default is c(-100,100). This interval is used by R's function uniroot to search for the inverse of each element in trans.list.

pass.down.to.C

If TRUE, the iterative task is passed down to a C program for faster implementation of the MCMC algorithm.

Details

This function uses a single block Metropolis algorithm with multivariate normal proposal. For this function to work properly, all parameters should be defined on the real line - parameter transformation(s) might be required. If trans.list is not specified, the function will assume that the parameter distributions are all defined on the real line (i.e., function(x) x will be used for each parameter). If no prior distribution is provided, an improper prior distribution - uniform on the interval )-Inf,+Inf( - will be used for all parameters (i.e., prior distribution proportional to 1 - function(x) 1).

In order to minimize the number of arguments for metropolis, the function automatically computes the inverse of trans.list: this suppresses the need for the user to provide both the "inverse transformation" and the "transformation". However, problems may occur, and it is why the user is allowed to alter uniroot.interval. Depending on the number of errors reported, future versions of this package may end up requesting that a list for both the "inverse transformation" and the "transformation" be provided by the user.

A nice list of references is provided below for more information on topics such as: MCMC algorithms, tuning of Metropolis-Hastings algorithms, MCMC convergence diagnostics, the Bayesian paradigm ...

Value

rate

MCMC acceptance rate. This value is computed before applying the burn-in; i.e., it is computed for sims.all.

total.time

Total computation time.

sims.all

Array containing all iterations.

sims

Array containing iterations after burn-in.

input

Inputted mle object.

iter

Number of iterations.

prior

Prior distribution.

burn

Integer corresponding to the number of iterations to be discarded - burn-in period.

M

Parameter vector whose elements correspond to the parameter values (on the scales specified by trans.list) obtained at the last iteration of the Metropolis sampler; i.e. sims[iter,].

V

Covariance matrix computed using, after removing the burn-in period, the joint posterior distribution of the parameters (on the scales specified by trans.list). This matrix might be used to tune the MCMC algorithm.

References

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004). Bayesian data analysis, 2nd edition, Chapman & Hall/CRC.

Carlin, B.P, and Louis, T.A. (2009). Bayesian methods for data analysis. Chapman & Hall/CRC.

Gamerman, D., and Lopes H.F. (2006). Markov Chain Monte Carlo: Stochastic simulation for Bayesian inference. 2nd edition, Chapman & Hall/CRC.

Gilks, W.R., Richardson, S., and Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall.

See Also

plot.metropolis, mle

Examples

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### These examples should be re-run with, e.g., iter > 2000.
data(yarns)
x <- yarns$x
fit.x <- mle(x,'gamma',c(.1,.1))
bayes.x.no.prior <- metropolis(model=fit.x,iter=150,
	trans.list=list(function(x) x,function(x) exp(x)))
plot(bayes.x.no.prior)

# examples of prior distributions (note that these prior distribution
#	are specified for the transformated parameters;
#	i.e., in this case, 'meanlog' -> 'meanlog' and 'sdlog' -> 'ln.sdlog')
# for the scale parameter only
prior.1 <- function(x) dnorm(x[2],.8,.1)
# for both parameters (joint but independent in this case)
prior.2 <- function(x) dunif(x[1],3.4,3.6)*dnorm(x[2],1,1)

bayes.x.prior.2 <- metropolis(model=fit.x,iter=150,
	trans.list=list(function(x) x,function(x) exp(x)),prior=prior.2)
plot(bayes.x.prior.2)

# Example where 'model' is not from the class 'mle'; i.e.
# both 'start' and 'variance' need to be specified!
#x <- rweibull(5,2,1)
x <- c(0.9303492,1.0894917,0.9628029,0.6145032,0.4756699)
# Here 'fit.x <- mle(x,'weibull',c(.1,.1))' is not used,
model.x <- list(x=x,dist='weibull')
# and an informative prior distribution is considered to ensure a proper posterior distribution
prior.x <- function(x) dnorm(x[1],log(2),.1)*dnorm(x[2],log(1),.1)
trans.list.x <- list(function(x) exp(x), function(x) exp(x))
bayes.x <- metropolis(model=model.x,iter=150,prior=prior.x,trans.list=trans.list.x,
            pass.down.to.C=TRUE,start=c(0,0),variance=diag(.1,2,2))