create_sampler: Create a sampler object

In mcmcsae: Markov Chain Monte Carlo Small Area Estimation

Create a sampler object

Description

This function sets up a sampler object, based on the specification of a model. The object contains functions to draw a set of model parameters from their prior and conditional posterior distributions, and to generate starting values for the MCMC simulation. The functions share a common environment containing precomputed quantities such as design matrices based on the model and the data. The sampler object is the main input for the MCMC simulation function MCMCsim.

Usage

create_sampler(
  formula,
  data = NULL,
  family = "gaussian",
  ny = NULL,
  ry = NULL,
  r.mod,
  sigma.fixed = NULL,
  sigma.mod = NULL,
  Q0 = NULL,
  formula.V = NULL,
  logJacobian = NULL,
  linpred = NULL,
  compute.weights = FALSE,
  block = compute.weights,
  prior.only = FALSE,
  control = sampler_control()
)

Arguments

formula	formula to specify the response variable and additive model components. The model components form the linear predictor part of the model. A model component on the right hand side can be either a regression term specified by reg(...), a covariates subject to error term specified by mec(...), or a generic random effect term specified by gen(...). See for details the help pages for these model component creation functions. An offset can be specified as offset(...). Other terms in the formula are collectively interpreted as ordinary regression effects, treated in the same way as a reg(...) term, but without the option to change the prior.
data	data frame with n rows in which the variables specified in model components (if any) can be found.
family	character string describing the data distribution. The default is 'gaussian'. Other options are 'binomial' for the binomial distribution, 'negbinomial' for the negative binomial distribution, and "poisson" for the Poisson distribution. See mcmcsae-family for the functional way of specifying family and associated settings. For the binomial distribution logistic and probit link functions are supported, the latter only for binary data. For the negative binomial and Poisson distributions a log link function is assumed. Note that posterior inferences for family = 'poisson' are only approximate, as they are implemented using the negative binomial distribution with its (reciprocal) overdispersion parameter set to a very large value. For categorical or multinomial data, family = "multinomial" can be used. The implementation is based on a stick-breaking representation of the multinomial distribution, and the logistic link function relates each category except the last to a linear predictor. The categories can be referenced in the model specification formula by 'cat_'.
ny	in case family="binomial" the (vector of) numbers of trials. It can be either a numeric vector or the name of a variable in data. Defaults to a vector of 1s.
ry	in case family="negbinomial" the known, i.e. fixed part of the (reciprocal) dispersion parameter. It can be specified either as a numeric vector or the name of a numeric variable in data. The overall dispersion parameter is the product of ry with a positive scalar factor modelled as specified by argument r.mod. By default ry is taken to be 1. For family = "poisson" a single value can be specified, determining how well the Poisson distribution is approximated by the negative binomial distribution. The value should be large enough such that the negative binomial's overdispersion becomes negligible, but not too large as this might result in slow MCMC mixing. The default is ry=100 in this case.
r.mod	prior specification for a scalar (reciprocal) dispersion parameter of the negative binomial distribution. The prior can be specified by a call to a prior specification function. Currently pr_invchisq, pr_gig and pr_fixed are supported. The default is a chi-squared prior with 1 degree of freedom. To set the overall dispersion parameter to the value(s) specified by ry, use r.mod = pr_fixed(value=1).
sigma.fixed	for Gaussian models, if TRUE the residual standard deviation parameter 'sigma_' is fixed at 1. In that case argument sigma.mod is ignored. This is convenient for Fay-Herriot type models with (sampling) variances assumed to be known. Default is FALSE.
sigma.mod	prior for the variance parameter of a gaussian sampling distribution. This can be specified by a call to one of the prior specification functions pr_invchisq, pr_exp, pr_gig or pr_fixed for inverse chi-squared, exponential, generalized inverse gaussian or degenerate prior distribution, respectively. The default is an improper prior pr_invchisq(df=0, scale=1). A half-t prior on the standard deviation can be specified using pr_invchisq with a chi-squared distributed scale parameter.
Q0	n x n data-level precision matrix for a Gaussian model. It defaults to the unit matrix. If an n-vector is provided it will be expanded to a (sparse) diagonal matrix with Q0 on its diagonal. If a name is supplied it will be looked up in data and subsequently expanded to a diagonal matrix.
formula.V	a formula specifying the terms of a variance model in the case of a Gaussian likelihood. Currently two types of terms are supported: a regression term for the log-variance specified with vreg(...), and a term vfac(...) for multiplicative modeled factors at a certain level specified by a factor variable. By using unit-level inverse-chi-squared factors the marginal sampling distribution becomes a Student-t distribution, and by using unit-level exponential factors it becomes a Laplace or double exponential distribution.
logJacobian	if the data are transformed the logarithm of the Jacobian can be supplied so that it is incorporated in all log-likelihood computations. This can be useful for comparing information criteria for different transformations. It should be supplied as a vector of the same size as the response variable, and is currently only supported if family="gaussian". For example, when a log-transformation is used on response vector y, the vector -log(y) should be supplied.
linpred	a list of matrices defining (possibly out-of-sample) linear predictors to be simulated. This allows inference on e.g. (sub)population totals or means. The list must be of the form list(name_1=X_1, ...) where the names refer to the model component names and predictions are computed by summing X_i %*% p[[name_i]]. Alternatively, linpred="fitted" can be used as a short-cut for simulations of the full in-sample linear predictor.
compute.weights	if TRUE weights are computed for each element of linpred. Note that for a large dataset in combination with vector-valued linear predictors the weights can take up a lot of memory. By default only means are stored in the simulation carried out using MCMCsim.
block	if TRUE all coefficients are sampled in a single block. Alternatively, a list of character vectors indicating which coefficients should be sampled together in blocks.
prior.only	whether a sampler is set up only for sampling from the prior or for sampling from both prior and posterior distributions. Default FALSE. If TRUE there is no need to specify a response in formula. This is used by generate_data, which samples from the prior predictive distribution.
control	a list with further computational options, see details section.

Details

The right hand side of the formula argument to create_sampler can be used to specify additive model components. Currently four model components are supported: reg(...) for regression or 'fixed' effects, gen(...) for generic random effects, mec(...) for measurement in covariates effects, and bart(...) for a Bayesian additive regression trees component. Note that an offset can be added separately, in the usual way using offset(...).

For gaussian models, formula.V can be used to specify the variance structure of the model. Currently two specialized variance model components are supported, vreg(...) for regression effects predicting the log-variance and vfac(...) for modeled variance factors.

Value

A sampler object, which is the main input for the MCMC simulation function MCMCsim. The sampler object is an environment with precomputed quantities and functions. The main functions are rprior, which returns a sample from the prior distributions, draw, which returns a sample from the full conditional posterior distributions, and start, which returns a list with starting values for the Gibbs sampler. If prior.only is TRUE, functions draw and start are not created.

References

J.H. Albert and S. Chib (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association 88(422), 669-679.

D. Bates, M. Maechler, B. Bolker and S.C. Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software 67(1), 1-48.

S.W. Linderman, M.J. Johnson and R.P. Adams (2015). Dependent multinomial models made easy: Stick-breaking with the Polya-Gamma augmentation. Advances in Neural Information Processing Systems, 3456-3464.

P.A. Parker, S.H. Holan and R. Janicki (2023). Conjugate Modeling Approaches for Small Area Estimation with Heteroscedastic Structure. Journal of Survey Statistics and Methodology, smad002.

N. Polson, J.G. Scott and J. Windle (2013). Bayesian Inference for Logistic Models Using Polya-Gamma Latent Variables. Journal of the American Statistical Association 108(504), 1339-1349.

H. Rue and L. Held (2005). Gaussian Markov Random Fields. Chapman & Hall/CRC.

M. Zhou and L. Carin (2015). Negative Binomial Process Count and Mixture Modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence 37(2), 307-320.

Examples

# first generate some data
n <- 200
x <- rnorm(n)
y <- 0.5 + 2*x + 0.3*rnorm(n)
# create a sampler for a simple linear regression model
sampler <- create_sampler(y ~ x)
sim <- MCMCsim(sampler)
(summary(sim))

y <- rbinom(n, 1, 1 / (1 + exp(-(0.5 + 2*x))))
# create a sampler for a binary logistic regression model
sampler <- create_sampler(y ~ x, family="binomial")
sim <- MCMCsim(sampler)
(summary(sim))

mcmcsae documentation built on Oct. 11, 2023, 1:06 a.m.