BP(c, H0) has Levy measure nu(du, dt) = c u^(-1) (1 - u)^(c-1) du
dH0(t). The u^(-1) factor makes nu infinite near zero – infinitely
many infinitesimal jumps – yet the expected total jump mass is
finite and equals H0 exactly, because the u in "u dnu" cancels it.
That cancellation is what the quadrature here confirms.
Usage
Ghosalbpcont(c = 2, t_max = 1, n_grid = 2000)
Arguments
- c
Concentration, positive.
- t_max
Upper time limit, with H0(t) = t.
- n_grid
Number of midpoint cells.
Value
List with estimate (expected total jump mass),
H0_t_max, gap, method.
Details
Formula: int_0^1 u nu(du) = int_0^1 c (1 - u)^(c-1) du = 1,
so the expected mass on [0, t_max] is H0(t_max) = t_max.
References
Ghosal & van der Vaart (2017), Fundamentals of
Nonparametric Bayesian Inference, CUP, section 13.3.2.
Examples
Ghosalbpcont()
#> $estimate
#> [1] 1
#>
#> $H0_t_max
#> [1] 1
#>
#> $gap
#> [1] 0
#>
#> $method
#> [1] "BP Levy measure (GvdV 2017 sec. 13.3.2)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
