Le Cam's bound on the posterior mass of an alternative set

The last term divides by Pi(U), so a prior that starves the neighbourhood of the truth destroys the bound however good the test is.

Usage

Lecam(dtv, p0_phi, prior_mass, integral)

Arguments

dtv

d_TV(P_0, P_U), in [0, 1].

p0_phi

P_0 phi, the type I error, in [0, 1].

prior_mass

Pi(U), strictly positive.

integral

int_V P(1 - phi) dPi(P), non-negative.

Value

List with bound, term_tv, term_test, term_prior, informative.

Details

Formula: P_0 Pi(V | X) <= d_TV(P_0, P_U) + P_0 phi

• (1/Pi(U)) int_V P(1 - phi) dPi(P)

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, Lemma 6.46 (Le Cam). Read from the copy of the book held in the corpus. NOTE: the worklist filed this under "Appendix D"; in the book it is Lemma 6.46 in Section 6.8.2.

Examples

Lecam(dtv = 0.3, p0_phi = 0.4, prior_mass = 0.5, integral = 0.2)
#> $bound
#> [1] 1.1
#> 
#> $term_tv
#> [1] 0.3
#> 
#> $term_test
#> [1] 0.4
#> 
#> $term_prior
#> [1] 0.4
#> 
#> $informative
#> [1] 0
#> 
#> $method
#> [1] "Le Cam posterior inequality, Ghosal Lemma 6.46"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"