Dirichlet-process Bernstein-von Mises (Brownian bridge)

sqrt(n)(F_post - F0) converges to B(F0) with B a Brownian bridge, so at a fixed t the limit is N(0, F0(t)(1 - F0(t))). The bridge, not a plain Brownian motion, is what appears because the posterior CDF is pinned at both 0 and 1 – and the concentration parameter alpha washes out entirely in the limit, which is the substantive content.

Usage

Ghosaldpbvm(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), bridge_variance, gap, method.

Details

Formula: F_post(t) = (alpha t + N_t) / (alpha + n); Var(sqrt(n)(F_post(t) - t)) -> t(1 - t).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 12.2.

Examples

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