Pi = sum_k pi_k Pi_k with pi_k proportional to exp(-lam k log n): the posterior over the model index k concentrates, so the contraction rate adapts to the truth without the truth being known. The evidence of each sub-model is available in closed form here (conjugate normal means), so the model posterior is exact rather than sampled.
Arguments
- y
Numeric vector of observed coordinates; simulated when NULL.
- n
Precision (sample size) of each coordinate.
- K_true
Number of nonzero coordinates in the simulated truth.
- lam
Complexity-penalty scale of the model prior.
- K_max
Largest model dimension considered.
- seed
Seed for the deterministic draw of
y.
Details
Formula: log m_K(y) = sum_k [-log(2 pi s_k^2)/2 - y_k^2/(2 s_k^2)], s_k^2 = 1/n + tau2 1{k < K}; pi_K propto m_K(y) exp(-lam K log n).
References
Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 10.1.
Examples
Ghosaladaptthm()
#> $estimate
#> [1] 3
#>
#> $model_posterior
#> [1] 1.615165e-128 3.472216e-86 1.053545e-51 9.995909e-01 4.088065e-04
#> [6] 2.576932e-07 4.387080e-10 2.972506e-13 1.084305e-16 3.644088e-19
#> [11] 1.788374e-22 6.530961e-26 3.019763e-29
#>
#> $K_true
#> [1] 3
#>
#> $method
#> [1] "adaptive model prior (GvdV 2017 sec. 10.1)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"