Random-series prior with random dimension

K ~ pi_n and beta_k | K iid N(0, sigma^2). Putting a prior on the number of retained basis coefficients makes the posterior on K concentrate near the effective dimension of the truth, and that is what lets the contraction rate adapt to an unknown smoothness. The evidence for each K is conjugate, so the whole posterior on K is exact.

Usage

Ghosalrndseriespr(K_true = 4, n = 1000, lam = 0.5, K_max = 15, seed = 42)

Arguments

K_true

Number of nonzero coefficients in the truth.

n

Precision (sample size) of each coefficient.

lam

Complexity-penalty scale.

K_max

Largest dimension considered.

seed

Seed for the deterministic draw.

Value

List with estimate (posterior mean of K), K_posterior, mode_K, method.

Details

Formula: pi(K | y) propto exp(log m_K(y) - lam K log n); the reported estimate is the posterior mean sum_K K pi(K | y).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 10.4.

Examples

Ghosalrndseriespr()
#> $estimate
#> [1] 4.001805
#> 
#> $K_posterior
#>  [1]  0.000000e+00  0.000000e+00  0.000000e+00 3.048198e-180  9.982039e-01
#>  [6]  1.787406e-03  8.678385e-06  1.670822e-08  1.727542e-11  1.660310e-13
#> [11]  2.312295e-16  2.393500e-19  3.139860e-22  3.390151e-25  3.438252e-28
#> [16]  3.460868e-31
#> 
#> $mode_K
#> [1] 4
#> 
#> $method
#> [1] "random series prior (GvdV 2017 sec. 10.4)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"