The sqrt(n)-rescaled posterior approaches N(theta-hat, I^(-1)/n) in
TOTAL VARIATION – not merely weakly. Total variation is the strong
mode that makes the theorem useful, because it transfers to every
derived quantity including credible sets. The Beta-Bernoulli case
admits the exact posterior, so the distance to the BvM normal can be
integrated rather than argued.
Usage
Ghosalinfdimbvm(theta0 = 0.4, n = 2000, seed = 42)
Arguments
- theta0
True success probability, in (0, 1).
- n
Number of Bernoulli trials.
- seed
Seed for the deterministic draws.
Value
List with estimate (total variation distance),
bvm_holds, method.
Details
Formula: TV = (1/2) int |Beta(t; 1+S, 1+n-S) - phi(t; S/n, I^(-1)/n)| dt
over theta-hat +/- 6 sd, by midpoint rule.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, section 12.1.
Examples
Ghosalinfdimbvm()
#> $estimate
#> [1] 0.0026185
#>
#> $bvm_holds
#> [1] TRUE
#>
#> $method
#> [1] "parametric BvM (GvdV 2017 sec. 12.1)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
