The contraction rate of a Gaussian process prior is the smallest eps
solving phi_{w0}(eps) <= n eps^2. When the concentration function
behaves like eps^(-a) – a = 2 for Brownian motion, Lemma 11.27 –
the solution is eps_n = n^(-1/(2+a)), i.e. n^(-1/4) for BM. Nothing
about the process enters beyond that single exponent, which is why
the rate equation is stated once and reused for every prior in the
chapter.
Usage
Ghosalgpcrtthm(phi_exponent = 2, n = 10000)
Arguments
- phi_exponent
The exponent a in phi(eps) ~ eps^(-a).
- n
Sample size.
Value
List with estimate (eps_n), rate_exponent,
balance_gap, method.
Details
Formula: eps_n = n^(-1/(2 + a)), the point where eps^(-a) = n eps^2.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, Theorem 11.20, eq. (11.12).
Examples
Ghosalgpcrtthm()
#> $estimate
#> [1] 0.1
#>
#> $rate_exponent
#> [1] 0.25
#>
#> $balance_gap
#> [1] 2.842171e-16
#>
#> $method
#> [1] "Gaussian rate equation (GvdV 2017 Thm 11.20)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
