Score function for a multivariate normal target

Builds a score function – the gradient of the log density, \(\nabla_x \log p(x) = -\Sigma^{-1}(x - \mu)\) – for a multivariate normal target, suitable for the score argument of ksd_test().

Usage

gaussian_score(mean = NULL, sigma = NULL)

Arguments

mean

Numeric vector of length d, the target mean. NULL (default) uses the zero vector of the sample dimension.

sigma

Numeric d x d covariance matrix. NULL (default) uses the identity matrix of the sample dimension. Must be symmetric and invertible.

Value

A function of one argument (an n x d numeric matrix) returning the n x d matrix of scores. The mean and covariance are captured by the closure.

Details

The returned closure accepts the sample matrix and returns the score evaluated row-wise. Leaving mean or sigma at NULL defaults them to the zero vector and the identity matrix of the dimension seen at call time, so gaussian_score() with no arguments is the standard-normal score in any dimension.

References

Liu, Q., Lee, J. D., & Jordan, M. I. (2016). A kernelized Stein discrepancy for goodness-of-fit tests. Proceedings of the 33rd International Conference on Machine Learning, PMLR 48, 276-284.

See also

ksd_test()

Other goodness-of-fit tests: calibration_suite(), concordance_test(), concordance_test_nystrom(), coverage_test(), joint_coverage_test(), ksd_test(), ksd_test_nystrom(), numeric_score()

Author

Max Moldovan, max.moldovan@adelaide.edu.au

Examples

set.seed(1)
x <- matrix(stats::rnorm(200L), ncol = 2L)

# Standard-normal target in two dimensions
s0 <- gaussian_score()
str(s0(x))
#>  num [1:100, 1:2] 0.626 -0.184 0.836 -1.595 -0.33 ...

# Correlated-normal target
sig <- matrix(c(1, 0.5, 0.5, 1), nrow = 2L)
s1 <- gaussian_score(mean = c(0, 0), sigma = sig)