Weights pi_k proportional to exp(-c k log n) make
sum_k pi_k exp(n eps_k^2) converge whenever n eps_k^2 is at most of
order k log n. That summability is the universal-weight condition
behind every adaptation theorem in the chapter: it lets one prior
serve every smoothness at once. The series converges exactly when
the penalty c beats the rate scale.
Usage
Ghosalunivweights(n = 100, c = 2, K_max = 200, eps_scale = 1)
Arguments
- n
Sample size entering both the weights and the rate.
- c
Penalty scale of the weights.
- K_max
Number of terms summed.
- eps_scale
Scale of n eps_k^2 in units of k log n.
Value
List with estimate (the full sum),
partial_sums, converges, method.
Details
Formula: S = sum_{k=1}^{K} pi_k exp(n eps_k^2),
log pi_k = -c k log n - log Z, n eps_k^2 = eps_scale k log n.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, section 10.2.1.
Examples
Ghosalunivweights()
#> $estimate
#> [1] 101
#>
#> $partial_sums
#> [1] 101 101 101
#>
#> $converges
#> [1] TRUE
#>
#> $method
#> [1] "universal weights (GvdV 2017 sec. 10.2.1)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
