Finite-difference score from a log-density

Builds a score function – the gradient of the log density – by central finite differences, for use as the score argument of ksd_test(). The target need only be expressible as a function returning a (possibly unnormalised) log density; any additive normalising constant cancels in the gradient, so the constant may be omitted.

Usage

numeric_score(log_density, h = 1e-04)

Arguments

log_density

A function accepting an n x d numeric matrix and returning a numeric vector of length n: the log density (up to an additive constant) evaluated row-wise.

h

Positive numeric. Finite-difference step. Default 1e-4. Too large biases the gradient; too small amplifies floating-point noise.

Value

A function of one argument (an n x d numeric matrix) returning the n x d matrix of finite-difference scores, suitable as the score argument of ksd_test().

Details

This makes ksd_test() usable against any target a user can write down as a log density, not only targets with a hand-derived score. It is also the kernR-side adapter for a log-posterior contract: wrapping an exported log-posterior evaluator in numeric_score() yields the score ksd_test() needs to check whether posterior draws are calibrated against that posterior, with no dependency added on the producer – the evaluator is passed in as an ordinary function.

Central differences cost 2 * d evaluations of log_density per call, where d is the dimension. For a cheap closed-form target prefer gaussian_score() or a hand-written score; numeric_score() is the general fallback.

See also

ksd_test(), gaussian_score()

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

Author

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

Examples

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

# Standard-normal log density (unnormalised): -||x||^2 / 2
ld <- function(z) -0.5 * rowSums(z^2)
s <- numeric_score(ld)

# Matches the closed-form standard-normal score -x to finite-difference order
max(abs(s(x) - (-x)))
#> [1] 5.821565e-12

# Use directly in a goodness-of-fit test
ksd_test(x, score = numeric_score(ld), n_boot = 199L, seed = 1L)
#> 
#>    KSD Test
#> 
#> Statistic: 7.48743e-05 
#> P-value:   0.4000 
#> N:         100 
#> Perms:     199 
#> Kernel X:  imq
#> 
#> Goodness-of-fit verdict
#>   Stein kernel: imq (beta = -0.5)
#>   Bandwidth:     1.57 (median heuristic)
#>   Bootstrap:    wild, B = 199
#>   Surprise:     1.322 bits
#>   Verdict:      consistent with target
#>