sqrt(n)(L(theta_post) - L(theta0)) tends to N(0, ||L||^2) for a
BOUNDED linear functional. Boundedness is the whole condition: the
full infinite-dimensional BvM can fail while every bounded linear
functional of the same posterior still satisfies it, which is the
distinction section 12.4 is drawing.
Usage
Ghosalwnlinbvm(
L_coefs = c(0.6, 0.8),
n = 500,
prior_var = 50,
n_sim = 500,
seed = 42
)
Arguments
- L_coefs
Coefficients of the linear functional.
- n
Precision (sample size).
- prior_var
Prior variance per coordinate.
- n_sim
Number of replicates.
- seed
Seed for the deterministic draws.
Value
List with estimate (empirical variance),
norm2_L, gap, method.
Details
Formula: theta_post = Y prior_var / (prior_var + 1/n);
Var(sqrt(n) L(theta_post - theta0)) -> sum_k L_k^2.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, section 12.4.2.
Examples
Ghosalwnlinbvm()
#> $estimate
#> [1] 1.071105
#>
#> $norm2_L
#> [1] 1
#>
#> $gap
#> [1] 0.07110458
#>
#> $method
#> [1] "linear-functional BvM (GvdV 2017 sec. 12.4.2)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
