bsl {BSL}

Performing BSL, uBSL, semiBSL and BSLmisspec

Description

This is the main function for performing MCMC BSL (with a standard or non-standard likelihood estimator) to sample from the approximate posterior distribution. A couple of extentions to the standard approach are available by changing the following arguments, method, shrinkage, whitening, misspecType. Parallel computing is supported with the R package foreach.

Usage

bsl(
  y,
  n,
  M,
  model,
  covRandWalk,
  theta0,
  fnSim,
  fnSum,
  method = c("BSL", "uBSL", "semiBSL", "BSLmisspec"),
  shrinkage = NULL,
  penalty = NULL,
  fnPrior = NULL,
  simArgs = NULL,
  sumArgs = NULL,
  logitTransformBound = NULL,
  standardise = FALSE,
  GRC = FALSE,
  whitening = NULL,
  misspecType = NULL,
  tau = 1,
  parallel = FALSE,
  parallelArgs = NULL,
  thetaNames = NULL,
  plotOnTheFly = FALSE,
  verbose = 1L
)

Arguments

y

The observed data. Note this should be the raw dataset NOT the set of summary statistics.

n

The number of simulations from the model per MCMC iteration for estimating the synthetic likelihood.

M

The number of MCMC iterations.

model

A “MODEL” object generated with function newModel. See newModel.

covRandWalk

The covariance matrix of a multivariate normal random walk proposal distribution used in the MCMC.

theta0

Deprecated, will be removed in the future, use model instead. Initial guess of the parameter value, which is used as the starting value for MCMC.

fnSim

Deprecated, will be removed in the future, use model instead. A function that simulates data for a given parameter value. The first argument should be the parameters. Other necessary arguments (optional) can be specified with simArgs.

fnSum

Deprecated, will be removed in the future, use model instead. A function for computing summary statistics of data. The first argument should be the observed or simulated dataset. Other necessary arguments (optional) can be specified with sumArgs.

method

A string argument indicating the method to be used. The default, “BSL”, runs standard BSL. “uBSL” uses the unbiased estimator of a normal density of Ghurye and Olkin (1969). “semiBSL” runs the semi-parametric BSL algorithm and is more robust to non-normal summary statistics. “BSLmisspec” estimates the Gaussian synthetic likelihood whilst acknowledging that there may be incompatibility between the model and the observed summary statistic (Frazier and Drovandi 2021).

shrinkage

A string argument indicating which shrinkage method to be used. The default is NULL, which means no shrinkage is used. Shrinkage estimation is only available for methods “BSL” and “semiBSL”. Current options are “glasso” for the graphical lasso method of Friedman et al. (2008) and “Warton” for the ridge regularisation method of Warton (2008).

penalty

The penalty value to be used for the specified shrinkage method. Must be between zero and one if the shrinkage method is “Warton”.

fnPrior

Deprecated, will be removed in the future, use model instead. A function that computes the log prior density for a parameter. The default is NULL, which uses an improper flat prior over the real line for each parameter. The function must have a single input: a vector of parameter values.

simArgs

Deprecated, will be removed in the future, use model instead. A list of additional arguments to pass into the simulation function. Only use when the input fnSim requires additional arguments. The default is NULL.

sumArgs

Deprecated, will be removed in the future, use model instead. A list of additional arguments to pass into the summary statistics function. Only use when the input fnSum requires additional arguments. The default is NULL.

logitTransformBound

A p by 2 numeric matrix indicating the upper and lower bounds of parameters if a logit transformation is used on the parameter space, where p is the number of parameters. The default is NULL, which means no logit transformation is used. It is also possible to define other transformations within the simulation and prior function from model. The first column contains the lower bound of each parameter and the second column contains the upper bound. Infinite lower or upper bounds are also supported, eg. matrix(c(1,Inf,0,10,-Inf,0.5),3,2,byrow=TRUE).

standardise

A logical argument that determines whether to standardise the summary statistics before applying the graphical lasso. This is only valid if method is “BSL”, shrinkage is “glasso” and penalty is not NULL. The diagonal elements will not be penalised if the shrinkage method is “glasso”. The default is FALSE.

GRC

A logical argument indicating whether the Gaussian rank correlation matrix (Boudt et al. 2012) should be used to estimate the covariance matrix in “BSL” method. The default is FALSE, which uses the sample covariance by default.

whitening

This argument determines whether Whitening transformation should be used in “BSL” method with Warton's shrinkage. Whitening transformation helps decorrelate the summary statistics, thus encouraging sparsity of the synthetic likelihood covariance matrix. This might allow heavier shrinkage to be applied without losing much accuracy, hence allowing the number of simulations to be reduced. By default, NULL represents no Whitening transformation. Otherwise this is enabled if a Whitening matrix is provided. See estimateWhiteningMatrix for the function to estimate the Whitening matrix.

misspecType

A string argument indicating which type of model misspecification to be used. The two options are "mean" and "variance". Only used when method is “BSLmisspec”. The default, NULL, means no model misspecification is considered.

tau

A numeric argument, parameter of the prior distribution for "BSLmisspec" method. For mean adjustment, tau is the scale of the Laplace distribution. For variance inflation, tau is the mean of the exponential distribution. Only used when method is “BSLmisspec”.

parallel

A logical value indicating whether parallel computing should be used for simulation and summary statistic evaluation. The default is FALSE. When model simulation is fast, it may be preferable to perform serial or vectorised computations to avoid significant communication overhead between workers. Parallel computation can only be used if not using a vectorised simulation function, see MODEL for options of vectorised simulation function.

parallelArgs

A list of additional arguments to pass into the foreach function. Only used when parallel computing is enabled, default is NULL.

thetaNames

Deprecated, will be removed in the future, use model instead. A string vector of parameter names, which must have the same length as the parameter vector. The default is NULL.

plotOnTheFly

A logical or numeric argument defining whether or by how many iterations a posterior figure will be plotted during running. If TRUE, a plot of approximate univariate posteriors based on the current accepted samples will be shown every one thousand iterations. The default is FALSE.

verbose

An integer indicating the verbose style. 0L means no verbose messages will be printed. 1L uses a custom progress bar to track the progress. 2L prints the iteration numbers (1:M) to track the progress. The default is 1L.

Value

An object of class bsl is returned, see BSL for more information of the S4 class.

Author(s)

Ziwen An, Leah F. South and Christopher Drovandi

References

Boudt K, Cornelissen J, Croux C (2012). “The Gaussian Rank Correlation Estimator: Robustness Properties.” Statistics and Computing, 22(2), 471–483. doi: 10.1007/s11222-011-9237-0.

Frazier DT, Drovandi C (2021). “Robust Approximate Bayesian Inference with Synthetic Likelihood.” Journal of Computational and Graphical Statistics (In Press). https://arxiv.org/abs/1904.04551.

Friedman J, Hastie T, Tibshirani R (2008). “Sparse Inverse Covariance Estimation with the Graphical Lasso.” Biostatistics, 9(3), 432–441.

Ghurye SG, Olkin I (1969). “Unbiased Estimation of Some Multivariate Probability Densities and Related Functions.” Ann. Math. Statist., 40(4), 1261–1271.

Warton DI (2008). “Penalized Normal Likelihood and Ridge Regularization of Correlation and Covariance Matrices.” Journal of the American Statistical Association, 103(481), 340–349. doi: 10.1198/016214508000000021.

Price LF, Drovandi CC, Lee A, Nott DJ (2018). “Bayesian Synthetic Likelihood.” Journal of Computational and Graphical Statistics, 27, 1–11. doi: 10.1080/10618600.2017.1302882.

An Z, South LF, Nott DJ, Drovandi CC (2019). “Accelerating Bayesian Synthetic Likelihood With the Graphical Lasso.” Journal of Computational and Graphical Statistics, 28(2), 471–475. doi: 10.1080/10618600.2018.1537928.

An Z, Nott DJ, Drovandi C (2019). “Robust Bayesian Synthetic Likelihood via a Semi-Parametric Approach.” Statistics and Computing (In Press).

See Also

ma2, cell, mgnk and toad for examples. selectPenalty for a function to tune the BSLasso tuning parameter and plot for functions related to visualisation.

Examples

## Not run: 
# This is just a minimal test run, please see package built-in examples for more
# comprehensive usages of the function
toy_sim <- function(n, theta) matrix(rnorm(n, theta), nrow = n)
toy_sum <- function(x) x
model <- newModel(fnSimVec = toy_sim, fnSum = toy_sum, theta0 = 0)

result_toy <- bsl(y = 1, n = 100, M = 1e4, model = model, covRandWalk = matrix(1),
    method = "BSL", plotOnTheFly = TRUE)
summary(result_toy)
plot(result_toy)

## End(Not run)