Numerical Checks for the WS Posterior-Mean Rule

Runs a standard collection of numerical checks for endpoint handling, symmetry, support bounds, finiteness, and Gaussian monotonicity of the posterior-mean shrinkage rule.

ws_numerical_checks(quadrature_n = 128L)

Arguments

quadrature_n

The number of Gauss-Legendre nodes used in the numerical integrations. Larger values can improve accuracy but increase computation time.

Details

The routine evaluates Gaussian and Laplace posterior means over a fixed coefficient grid for all combinations of spike probability \(\pi\in\{0,0.5,1\}\) and Wendland weight \(\omega\in\{0,0.5,1\}\). It checks the numerical versions of $$\delta(-d)=-\delta(d)$$ and $$|\delta(d)|\leq\beta$$. For the Gaussian likelihood, min_first_difference checks the expected nondecreasing behavior on the grid. The endpoint columns verify the exact no-spike and all-spike rules.

Value

A data frame with one row for each likelihood, spike probability, and slab weight combination. Important columns are max_oddness_error, min_first_difference, max_support_excess, all_finite, pi0_exact_slab, and pi1_exact_zero.

See also

level_posterior, wswavelet

References

Sanyal, N. (2026). Resolution-Adaptive Compact-Support Priors for Bayesian Wavelet Denoising: A Wendland-Semicircle Slab Mixture for Low-SNR Signal Recovery. Axioms, 15(9), 678. <DOI:10.3390/axioms15090678>

Author

Nilotpal Sanyal

Examples

# \donttest{
checks <- ws_numerical_checks(quadrature_n = 48L)
checks
#>    likelihood  pi omega max_oddness_error min_first_difference
#> 1    gaussian 0.0   0.0      1.021405e-14         9.141978e-04
#> 2     laplace 0.0   0.0      3.108624e-15        -2.220446e-15
#> 3    gaussian 0.5   0.0      1.021405e-14         9.142390e-04
#> 4     laplace 0.5   0.0      1.554312e-15        -1.776357e-15
#> 5    gaussian 1.0   0.0      0.000000e+00         0.000000e+00
#> 6     laplace 1.0   0.0      0.000000e+00         0.000000e+00
#> 7    gaussian 0.0   0.5      1.021405e-14         9.743291e-04
#> 8     laplace 0.0   0.5      2.746692e-13        -1.776357e-15
#> 9    gaussian 0.5   0.5      1.021405e-14         9.744105e-04
#> 10    laplace 0.5   0.5      1.736389e-13        -1.221245e-15
#> 11   gaussian 1.0   0.5      0.000000e+00         0.000000e+00
#> 12    laplace 1.0   0.5      0.000000e+00         0.000000e+00
#> 13   gaussian 0.0   1.0      6.661338e-15         2.177574e-03
#> 14    laplace 0.0   1.0      7.304157e-13        -3.330669e-16
#> 15   gaussian 0.5   1.0      6.661338e-15         2.180499e-03
#> 16    laplace 0.5   1.0      4.086731e-13        -2.220446e-16
#> 17   gaussian 1.0   1.0      0.000000e+00         0.000000e+00
#> 18    laplace 1.0   1.0      0.000000e+00         0.000000e+00
#>    max_support_excess all_finite pi0_exact_slab pi1_exact_zero
#> 1                   0       TRUE           TRUE             NA
#> 2                   0       TRUE           TRUE             NA
#> 3                   0       TRUE             NA             NA
#> 4                   0       TRUE             NA             NA
#> 5                   0       TRUE             NA           TRUE
#> 6                   0       TRUE             NA           TRUE
#> 7                   0       TRUE           TRUE             NA
#> 8                   0       TRUE           TRUE             NA
#> 9                   0       TRUE             NA             NA
#> 10                  0       TRUE             NA             NA
#> 11                  0       TRUE             NA           TRUE
#> 12                  0       TRUE             NA           TRUE
#> 13                  0       TRUE           TRUE             NA
#> 14                  0       TRUE           TRUE             NA
#> 15                  0       TRUE             NA             NA
#> 16                  0       TRUE             NA             NA
#> 17                  0       TRUE             NA           TRUE
#> 18                  0       TRUE             NA           TRUE
# }