gaussianSynLike {BSL}

Estimate the Gaussian synthetic (log) likelihood

Description

This function estimates the Gaussian synthetic log-likelihood (see Wood 2010 and Price et al. 2018). Several extensions are provided in this function: shrinkage enables shrinkage estimation of the covariance matrix and is helpful to bring down the number of model simulations (see An et al. (2019) for an example of BSL with glasso (Friedman et al. 2008) shrinkage estimation); GRC uses Gaussian rank correlation (Boudt et al. 2012) to find a more robust correlation matrix; whitening (Kessy et al. 2018) could further reduce the number of model simulations upon Warton's shrinkage (Warton 2008) by decorrelating the summary statistics.

Usage

gaussianSynLike(
  ssy,
  ssx,
  shrinkage = NULL,
  penalty = NULL,
  standardise = FALSE,
  GRC = FALSE,
  whitening = NULL,
  ssyTilde = NULL,
  log = TRUE,
  verbose = FALSE
)

Arguments

ssy

The observed summary statisic.

ssx

A matrix of the simulated summary statistics. The number of rows is the same as the number of simulations per iteration.

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”.

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.

ssyTilde

The whitened observed summary statisic. If this is not NULL, it will be used to save computation effort. Only used if Whitening is enabled.

log

A logical argument indicating if the log of likelihood is given as the result. The default is TRUE.

verbose

A logical argument indicating whether an error message should be printed if the function fails to compute a likelihood. The default is FALSE.

Value

The estimated synthetic (log) likelihood value.

References

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.

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.

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

Kessy A, Lewin A, Strimmer K (2018). “Optimal Whitening and Decorrelation.” The American Statistician, 72(4), 309–314. doi: 10.1080/00031305.2016.1277159.

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.

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.

Wood SN (2010). “Statistical Inference for Noisy Nonlinear Ecological Dynamic Systems.” Nature, 466, 1102–1107. doi: 10.1038/nature09319.

See Also

Other available synthetic likelihood estimators: gaussianSynLikeGhuryeOlkin for the unbiased synthetic likelihood estimator, semiparaKernelEstimate for the semi-parametric likelihood estimator, synLikeMisspec for the Gaussian synthetic likelihood estimator for model misspecification.

Examples

data(ma2)
ssy <- ma2_sum(ma2$data)
m <- newModel(fnSim = ma2_sim, fnSum = ma2_sum, simArgs = ma2$sim_args,
              theta0 = ma2$start)
ssx <- simulation(m, n = 300, theta = c(0.6, 0.2), seed = 10)$ssx

# the standard Gaussian synthetic likelihood (the likelihood estimator used in BSL)
gaussianSynLike(ssy, ssx)
# the Gaussian synthetic likelihood with glasso shrinkage estimation
# (the likelihood estimator used in BSLasso)
gaussianSynLike(ssy, ssx, shrinkage = 'glasso', penalty = 0.1)
# the Gaussian synthetic likelihood with Warton's shrinkage estimation
gaussianSynLike(ssy, ssx, shrinkage = 'Warton', penalty = 0.9)
# the Gaussian synthetic likelihood with Warton's shrinkage estimation and Whitening transformation
W <- estimateWhiteningMatrix(20000, m)
gaussianSynLike(ssy, ssx, shrinkage = 'Warton', penalty = 0.9, whitening = W)