GEM stick-breaking weights

Formula: GEM(alpha) = stick-breaking with V_k ~ Beta(1, alpha)

Usage

Dpgem(alpha = 1, K = 10, seed = 42)

Arguments

alpha

Concentration, strictly positive.

K

Truncation level.

seed

Seed of the deterministic stream.

Value

List with estimate (largest weight), weights, V, remaining, expected_remaining, K, method.

Details

w_1 = V_1 and w_k = V_k prod_{j<k} (1 - V_j), so the weights and the unbroken remainder sum to exactly one at every truncation. The marginal mean of the first weight is 1/(1 + alpha), and the expected remaining mass after K breaks is (alpha/(1 + alpha))^K.

References

Sethuraman (1994), Statistica Sinica 4(2):639-650; Pitman (2002), Poisson-Dirichlet and GEM invariant distributions, Technical Report 621, U.C. Berkeley.

Examples

Dpgem()
#> $estimate
#> [1] 0.4060138
#> 
#> $weights
#>  [1] 2.886946e-01 4.060138e-01 1.125119e-02 2.803644e-01 3.093107e-03
#>  [6] 7.816353e-03 2.759363e-03 1.949039e-06 2.736728e-06 1.468755e-06
#> 
#> $V
#>  [1] 0.2886946 0.5708010 0.0368539 0.9534894 0.2261706 0.7385843 0.9974084
#>  [8] 0.2718402 0.5242012 0.5912790
#> 
#> $remaining
#> [1] 1.015275e-06
#> 
#> $expected_remaining
#> [1] 0.0009765625
#> 
#> $K
#> [1] 10
#> 
#> $method
#> [1] "GEM stick-breaking weights"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"