Semiparametric Bernstein-von Mises for a smooth functional

sqrt(n)(psi(G_post) - psi(F0)) tends to N(0, sigma_eff^2) for a smooth functional psi. The variance in the limit is the SEMIPARAMETRIC EFFICIENCY BOUND, not merely some variance: the Bayes procedure is asymptotically efficient without being told the functional in advance. Here psi is the mean and the truth uniform, so the bound is var(X) = 1/12.

Usage

Ghosalsemiparabvm(n = 2000, alpha = 2, n_sim = 400, seed = 42)

Arguments

n

Sample size per replicate.

alpha

DP concentration parameter, positive.

n_sim

Number of replicates.

seed

Seed for the deterministic draws.

Value

List with estimate (empirical variance), efficient_variance, gap, method.

Details

Formula: psi(G_post) = (alpha/2 + sum_i X_i) / (alpha + n); Var(sqrt(n)(psi(G_post) - 1/2)) -> 1/12.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 12.3.

Examples

# \donttest{
set.seed(1)
r <- Ghosalsemiparabvm(); TRUE
#> [1] TRUE
# }