Seite - 299 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 442 If1/2≤α<1, then θ¯=αθi+(1−α)θ′jbelongs to thenaturalparameterspaceMθ. ThereforeAαi,j(I) isboundedandcanbecomputedfromtheCDFof p(x; θ¯)as Aαi,j(I)= c α i (c ′ j) 1−αexp(F ( θ¯ )−αF(θi)−(1−α)F(θ′j))∫ I p ( x; θ¯ ) dx. (41) Theothercaseα> 1 ismoredifficult: if θ¯= αθi+(1−α)θ′j still lies inMθ, thenAαi,j(I) canbe computedbyEquation(41).Otherwisewetry tosolve itbyanumerical integrator. This isnot idealas the integralmaydiverge,orourapproximationmaybe too loose toconclude.Wepoint the reader to [42] andEquations (61)–(69) in [35] for relatedanalysiswithmoredetails. As computingAαi,j(I) only requiresO(1) time, theoverall computational complexity (without considering the envelope computation) isO( ). 4.2.AdaptiveBounds Thissectionderives theshape-dependentboundswhich improve thebasicbounds inSection4.1. Wecanrewriteamixturemodelm(x) inaslab Ir as m(x)=wζ(r)pζ(r)(x) ⎛⎝1+ ∑ i =ζ(r) wipi(x) wζ(r)pζ(r)(x) ⎞⎠ , (42) wherewζ(r)pζ(r)(x) isaweightedcomponent inm(x) servingasa reference.Weonlydiscuss thecase that the reference is chosenas thedominatingcomponent, i.e., ζ(r)= δ(r). However it isworth to note that theproposedboundsdonotdependonthisparticularchoice. Therefore theratio wipi(x) wζ(r)pζ(r)(x) = wi wζ(r) exp (( θi−θζ(r) ) t(x)−F(θi)+F(θζ(r)) ) (43) canbebounded ina sub-rangeof [0,1]byanalyzing theextremevaluesof t(x) in the slab Ir. This canbedonebecause t(x)usually consistsofpolynomial functionswithfinite criticalpointswhich canbesolvedeasily.Correspondingly the function ( 1+∑i =ζ(r) wipi(x) wζ(r)pζ(r)(x) ) in Ir canbeboundedina subrangeof [1,k],denotedas [ωζ(r)(Ir),Ωζ(r)(Ir)].Hence ωζ(r)(Ir)wζ(r)pζ(r)(x)≤m(x)≤Ωζ(r)(Ir)wζ(r)pζ(r)(x). (44) This formsbetterboundsofm(x) thanEquation (32)becauseeachcomponent in the slab Ir is analyzedmore accurately. Therefore,we refine the fundamental boundsofm(x)by replacing the Equations (34)and(35)with cν(r)pν(r)(x) :=ωζ(r)(Ir)wζ(r)pζ(r)(x), cδ(r)pδ(r)(x) :=Ωζ(r)(Ir)wζ(r)pζ(r)(x). (45) (46) Then, the improvedboundsofHα aregivenbyEquations (38)and(39)accordingto theabove replaceddefinitionof cν(r)pν(r)(x)and cδ(r)pδ(r)(x). To evaluate ωζ(r)(Ir) andΩζ(r)(Ir) requires iterating through all components in each slab. Therefore thecomputationalcomplexity is increasedtoO( (k+k′)). 4.3.Variance-ReducedBounds This section further improves the proposed bounds based on variance reduction [43]. Byassumption, α≥ 1/2, thenm(x)αm′(x)1−α ismore similar tom(x) rather thanm′(x). The ratio 299