Simultaneous Inference Under the Vacuous Orientation Assumption Ruobin Gong Department of Statistics, Rutgers University ISIPTA 2019, Ghent, Belgium July 3, 2019 1 / 7
S IMULTANEOUS I NFERENCE U NDER THE V ACUOUS O RIENTATION A SSUMPTION Ruobin Gong Department of Statistics, Rutgers University, USA · ruobin.gong@rutgers.edu I. M OTIVATION II. N OTATION & M ODEL E ′ E ∼ χ 2 E | E ′ E ∼ isotropic Y is a k -vector of observable measurements, E ∼ Normal ( 0 , I k ) = + k and corresponding M its unknown true values. E is a vector of measurement errors and S 2 an (configuration) (orientation) associated variance parameter. Posit E , the fol- lowing body of marginal model evidence: 1. Y − M = E : additive measurement error 2. Y = y : precisely observed measurement 3. Error configuration: E ′ E = S 2 U, where U ∼ χ 2 k 4. Fixed error variance: S 2 = s 2 (4’. Random error variance: S 2 ∼ U s ) Auxiliary variables U and U s are means to ex- Precise simultaneous inference for k unknown quantities must rely on a known correlational struc- press evidence in stochastic form. E is judged ture such as error independence, i.e. E ∼ Normal ( 0 , I k ) . We relax this assumption by keeping the to be independent suitable for DS-ECP (see IV). χ 2 k configuration component while ridding the isotropic orientation component. No assumption on error orientation is made. III. E VIDENCE P ROJECTION AND C OMBINATION IV. DS-ECP Combination of evidence E results in a class of Projection of R E onto the margin of interest M , Central to Dempster-Shafer Extended Calculus of Probability ( DS-ECP ) is the processing of subsets of the full model state space def = { M ∈ Ω M : ( M − y ) ′ ( M − y ) = s 2 U } R M | E = bodies of independent marginal evidence. def Y , M , E , S 2 � R E = = { � ∈ Ω : Y = y , Y − M = E , E ′ E = S 2 U, S 2 = s 2 } , where U ∼ µ E , the χ 2 k distribution. R M | E is D EFINITION 1. A body of marginal evidence E again a random subset of Ω M whose distribu- consisting of J pieces is said to be independent , which is a multi-valued map from U to subsets tion is dictated by U . For every realization U = if the marginal auxiliary variables (a.v.s) asso- of Ω . Since U ∼ χ 2 ciated with each piece are all statistically inde- k , R E is a random subset of Ω u , R M | E ( u ) is a k -sphere centered at y with ra- dius s √ u . We say that R M | E embodies posterior pendent. That is, for U j ∼ µ j , j = 1 , · · · , J , with distribution inherited from U . The density function of U dictates the mass function of R E . inference for M given evidence E . ( U 1 , · · · , U J ) ∼ µ 1 × · · · × µ J . Notably, deterministic pieces of evidence are V. P OSTERIOR I NFERENCE associated with degenerate a.v.s, thus always independent of other pieces of evidence. Linear forms of hypotheses are expressed by a Rectangular regions of the form consistent system of equations CM = a , where Dempster’s Rule of Combination amounts to M ∈ Ω M : M ∈ ⊗ k C α = � i =1 ( y i ± c α · s ) � 1) taking the product of marginal a.v.s, and 2) C is a real-valued p by k matrix with arbitrary p . Define summary statistic parallels Bonferroni simultaneous confidence applying domain revision to the joint a.v. to ex- t y = ( a − Cy ) ′ ( CC ′ ) − 1 ( a − Cy ) , regions. Probabilities associated with C α are clude values that result in algebraic incompati- functions of the standardized half width c α . bility, i.e. empty intersections of marginal focal where in case p > rank ( C ) , the inverse is taken sets. Denote µ the prior probability of U , the E XAMPLE 3 (test for all pairwise contrasts). The si- to be the Moore-Penrose pseudoinverse. joint a.v. for E measurable w.r.t. σ (Ξ) . A poste- multaneous test for all pairwise means are iden- riori E , revise µ to µ E measurable w.r.t. σ (Ξ E ) ⊂ T HEOREM 3. Posterior probabilities concerning tical has null hypothesis σ (Ξ) where Ξ E = { u ∈ Ξ : R E ( u ) � = ∅} , and one-sided linear hypothesis H : CM ≤ a are H = ∩ 1 ≤ i<j ≤ k H i,j , H i,j : M i = M j . µ E = ( µ × 1 Ξ E ) /µ (Ξ E ) , { p ( H ) , q ( H ) , r ( H ) } = { F ( t y ) , 0 , 1 − F ( t y ) } The number of pairwise contrasts tested is on where 1 A ( S ) = 1 if S ⊆ A and 0 otherwise. For if Cy ≤ a , and quadratic order of k , but the compound hypoth- the current model, domain revision of the a.v. { p ( H ) , q ( H ) , r ( H ) } = { 0 , F ( t y ) , 1 − F ( t y ) } esis H always spans a 1 -dimensional subspace is trivial, namely µ E = µ . of Ω M . As k increases, the distribution of r ( H ) otherwise. F is the CDF of scaled χ 2 k with scal- Stochastic three-valued logic . Posterior in- ing factor s 2 (fixed error variance case). (Figure 2 left) approaches uniform, which is that ference about assertions concerning the state of a correctly calibrated p -value under the null Posterior (1 − α ) credible regions of the form space is expressed through a probability triple model, whereas the Bonferroni procedure (Fig- M ∈ Ω M : ( M − y ) ′ ( M − y ) ≤ F − 1 ( p , q , r ) , representing weights of evidence “for”, A α = � � , ure 2 right) becomes increasingly conservative 1 − α “against”, and “don’t know” about that asser- for larger k . The vacuous orientation model where F − 1 is the α th -quantile of µ E . tion. Set functions p , q , r : Ω M → [0 , 1] are such α captures the logical connection among the large T HEOREM 6. A α is a sharp posterior credible re- that for all H ∈ σ (Ω M ) , number of hypotheses (collinearity), and deliv- gion in the sense that r ( A α ) = 0 for all α . � ers posterior inference reflective of the geometry p ( H ) = dµ E , T HEOREM 7. A α is calibrated to the i.i.d. error of the hypothesis space. { u ∈ Ξ E : R M | E ( u ) ⊆ H } model, P ∗ , in the sense that for all M ∗ and all The ( p , q , r ) representation is an alternative to a α , p ( A ) = P ∗ ( M ∗ ∈ A ) = 1 − α and q ( A ) = pair of belief and plausibility functions on Ω M , P ∗ ( M ∗ ∈ A c ) = α . where p is the belief function and 1 − q (equiva- lently p + r ) is its conjugate plausibility function. VI. F UTURE D IRECTIONS Figure 2: Distribution of r ( H ) (left) and Bonferroni p - The vacuous orientation model may extend to value (right) for all pairwise contrasts under the null • Elliptical distributions; sampling model. For larger k , r ( H ) resembles a cor- • Multivariate and multiple regression; Figure 1: Focal sets that constitute p ( H ) for one-sided rectly calibrated p -value, whereas the Bonferroni p - • Partially vacuous orientation models linear (left) and rectangular (right) hypotheses. value becomes more conservative. based on finer variance decomposition. 2 / 7
Motivation: simultaneous inference/meta analysis M = ( M 1 , . . . , M k ) : vector of unknown parameters Y = ( Y 1 , . . . , Y k ) a vector of observable data aimed at measuring M Each Y i is a statistic from an experiment which we understand well, but we do not how they relate to one another. 3 / 7
Let E = Y − M denote the vector of measurement errors. I. M OTIVATION E | E ′ E ∼ isotropic E ′ E ∼ χ 2 E ∼ Normal ( 0 , I k ) = + k (configuration) (orientation) Precise simultaneous inference for k unknown quantities must rely on a known correlational struc- ture such as error independence, i.e. E ∼ Normal ( 0 , I k ) . We relax this assumption by keeping the χ 2 k configuration component while ridding the isotropic orientation component. 4 / 7
Posterior Inference = { M ∈ Ω M : ( M − y ) ′ ( M − y ) = s 2 U } , def = R M | E is a random subset of Ω M (concentric hyperspheres), whose distribution is dictated by the auxiliary variable U ∼ χ 2 k . R M | E embodies posterior inference for M given E . 5 / 7
Testing many collinear hypotheses Example 3. The simultaneous test for all pairwise means being identical: H = ∩ 1 ≤ i < j ≤ k H i , j , H i , j : M i = M j . For larger k , P ( H | E ) approaches uniformity as if a well-calibrated p -value. 6 / 7
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