Check the three conditions that deliver a posterior contraction rate

Note the asymmetry: the prior-mass and sieve-mass conditions use epsilon-bar, the entropy condition uses epsilon, and the theorem also needs n epsilon-bar^2 -> infinity.

Usage

Testcond(prior_ball, log_entropy, sieve_mass, eps_bar, eps, n, Cconst)

Arguments

prior_ball

Pi_n(B_2(p_0, ebar_n)), in (0, 1].

log_entropy

log N(xi eps_n, P_{n,1}, d), non-negative.

sieve_mass

Pi_n(P_{n,2}), in [0, 1].

eps_bar

ebar_n, positive.

eps

eps_n, at least eps_bar.

n

Sample size.

Cconst

The constant C > 0.

Value

List with holds, cond_prior, cond_entropy, cond_sieve, slack_prior, slack_entropy, slack_sieve, n_eps_bar_sq.

Details

Formula: (i) Pi_n(B_2(p_0, ebar_n)) >= e^{-C n ebar_n^2}; (ii) log N(xi eps_n, P_{n,1}, d) <= n eps_n^2; (iii) Pi_n(P_{n,2}) <= e^{-(C + 4) n ebar_n^2}

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, Theorem 8.9, Section 8.2, conditions (8.4), (8.5) and (8.6). Read from the copy of the book held in the corpus.

Examples

Testcond(prior_ball = 0.5, log_entropy = 10, sieve_mass = 0.1,
         eps_bar = 0.1, eps = 0.2, n = 100, Cconst = 1)
#> $holds
#> [1] 0
#> 
#> $cond_prior
#> [1] 1
#> 
#> $cond_entropy
#> [1] 0
#> 
#> $cond_sieve
#> [1] 0
#> 
#> $slack_prior
#> [1] 0.3068528
#> 
#> $slack_entropy
#> [1] -6
#> 
#> $slack_sieve
#> [1] -2.697415
#> 
#> $n_eps_bar_sq
#> [1] 1
#> 
#> $method
#> [1] "Contraction-rate conditions, Ghosal Theorem 8.9"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"