When f0 sits in a d-dimensional submodel the hierarchical posterior
attains the parametric sqrt(d/n) rate: the selected dimension stays
near d and the posterior risk falls like d/n, i.e. with log-log slope
one against n. Recovering a slope of one is the whole content of the
section – a nonparametric prior paying no price on a parametric truth.
Usage
Ghosalparamrate(d_true = 2, ns = c(100, 1000, 10000), lam = 1, seed = 42)
Arguments
- d_true
Dimension of the true submodel.
- ns
Increasing vector of sample sizes.
- lam
Complexity-penalty scale.
- seed
Seed for the deterministic draws.
Value
List with estimate (log-log slope),
risk_by_n, parametric, method.
Details
Formula: K-hat = argmax_K [log m_K(y) - lam K log n];
risk = K-hat/(n + 1) + sum_{k=K-hat}^{d-1} theta_k^2.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, section 10.2.2.
Examples
Ghosalparamrate()
#> $estimate
#> [1] 0.997861
#>
#> $risk_by_n
#> [1] 0.019801980 0.001998002 0.000199980
#>
#> $parametric
#> [1] TRUE
#>
#> $method
#> [1] "parametric adaptation (GvdV 2017 sec. 10.2.2)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
