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Figure 1 only refers to cohort studies, in practice we could have other data structures. Although not required, our setup is most interesting in the case where all datasets to be combined contain at least one variable that is also measured in at least one of the other datasets. 3. Background knowledge The temporal stru... | https://arxiv.org/abs/2503.21526v2 |
LetG= (V,E)be a graph and let τbe a tiered ordering of the nodes in V. Then, if τ(A)< τ(B)the edge {A∗−∗B} ∈Eis referred to as a cross-tier edge. Importantly, tiered orderings not only inform us of possible edge directions in the form of a set of forbidden edges, they also imply certain restrictions on the separations.... | https://arxiv.org/abs/2503.21526v2 |
tFCI outputs a PMG that may or may not be maximally informative (i.e. the algorithm is not complete). It does represent a superset of the MAGs in the restricted equivalence class, i.e. the algorithm is sound (cf. Proposition 8 below). We provide the tFCI as a pseudo-algorithm (Algorithm 2) in Appendix B, where the simp... | https://arxiv.org/abs/2503.21526v2 |
IOD algorithm returns a setof PAGs for which the marginalised independence models onto the nodes in V1, . . . ,Vnare equal to the independence models learned from the individual datasets. Furthermore, this set is guaranteed (in the oracle case) to contain the PAG of M. The IOD algorithm consists of two parts: The first... | https://arxiv.org/abs/2503.21526v2 |
all edges are candidates for this, and the algorithm considers each combination of edge removals (32 candidate skeletons in total). Consider the case of the removal of edge A◦−◦D: The algorithm proceeds with two possible graphs, Figure 2 (b) and (e). Now, the IOD algorithm considers all possible v-structures. A v-struc... | https://arxiv.org/abs/2503.21526v2 |
maximally informative graphs that are compatible with the marginal independence models and en- code the tiered background knowledge. However, for the same reasons as the tFCI, the algorithm is only sound, and not necessarily complete: The tIOD outputs a set of PMGs that might not represent restricted equivalence classe... | https://arxiv.org/abs/2503.21526v2 |
statistical errors than the original FCI/IOD algorithms. 9 BANG DIDELEZ However, the simple tIOD algorithm has an interesting additional benefit. Even the simple modifications alone, without orienting cross-tier edges, outputs a set of PAGs that should often be more informative than with the original IOD algorithm in t... | https://arxiv.org/abs/2503.21526v2 |
that Pτis a subset of Ponly if at least one of the conditions holds. 10 CAUSAL DISCOVERY WITH TIERED BACKGROUND KNOWLEDGE AND OVERLAPPING DATASETS dataset 1dataset 2 tier 1 tier 2B CD A Figure 3: Left: A MAG Mwith tiered ordering τ, where the variables are measured in two datasets, dataset 1 and dataset 2. Right: All g... | https://arxiv.org/abs/2503.21526v2 |
example graph output by the IOD algorithm, with (a) as input), that would not have been output by the simple tIOD. learned (marginal) independence models. The only independence found was XB⊥ ⊥XC|XA. This implicitly requires the graph to have a path between BandC, that can be separated by Awithout also conditioning on D... | https://arxiv.org/abs/2503.21526v2 |
datasets, are not consistent on the overlapping variables even under this assumption. This can easily be fixed by combining the statistical tests and / or the datasets on the overlapping variables before inputting the estimated independence models into the algorithm; one such solution is given in (Tillman and Spirtes, ... | https://arxiv.org/abs/2503.21526v2 |
https://github.com/ QixShawnChen/tfci . Tom Claassen, Joris M Mooij, and Tom Heskes. Learning sparse causal models is not np-hard. InProceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence , pages 172–181, 2013. Diego Colombo and Marloes H Maathuis. Order-independent constraint-based causal... | https://arxiv.org/abs/2503.21526v2 |
variables and selection bias, pages 211–252. MIT press, 1999. Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, prediction, and search . MIT press, 2000. Robert Tillman and Peter Spirtes. Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapp... | https://arxiv.org/abs/2503.21526v2 |
all edges in Ethat contain an arrowhead, such that EU⊆E. The induced subgraph ofG overV′⊆Vis defined as GV′= (V′,EV′)where EV′⊆Econtains all edges between nodes in V′. ByGE′= (V,E′)we denote the subgraph of Gobtained by removing all edges not in E′⊆. The skeleton of Gis obtained by replacing every {A∗−∗B} ∈Ewith{A◦−◦B}... | https://arxiv.org/abs/2503.21526v2 |
of Vis inS, and (ii) no non-collider on πis inS thenπis m-connecting given S. For a MAG G= (V,E), if there exists a path from a A∈VtoB∈Vwhere B̸=Athat is m-connecting given S, we say that AandBare m-connected given S. If no such path exists, 17 BANG DIDELEZ we say that AandBare m-separated given SinGand denote this by ... | https://arxiv.org/abs/2503.21526v2 |
the following set: 18 CAUSAL DISCOVERY WITH TIERED BACKGROUND KNOWLEDGE AND OVERLAPPING DATASETS Definition 17 ( poss.dsepG(A, B)(Spirtes et al., 1999)) LetM= (V,E)be a PMG and Aand Bdistinct nodes in V. Then V∈Vis inposs.dsepG(A, B)if and only if there is a path π=⟨A= V1, . . . , V n=V⟩between VandAthat in Gsuch that ... | https://arxiv.org/abs/2503.21526v2 |
lines 30-34 in the simple tFCI algorithm 30foreach ordered pair of adjacent nodes Vi, Vjdo 31 ifτ(Vi)< τ(Vj)then 32 orient Vi∗ −◦VjasVi∗ →Vj 33 end 34end 35apply orientation rules R1-R4 (Figure 5) and R8-R10 (Figure 6) repeatedly to Guntil none applies 21 BANG DIDELEZ Algorithm 3: The tiered IOD (tIOD) algorithm input ... | https://arxiv.org/abs/2503.21526v2 |
and (ii) each set in sepset corresponds to an independence in I(M), (iii) for every lfor each ⟨{Vi, Vj},Vl⟩ ∈InducingPaths there is an inducing path between Viand Vjwith respect to V\VlinM, and (iv) if for every non-adjacent pair AandBinMthere exists a set S′⊆pastτ V(A, B)such that AandBare m-separated by S′inMthen 66 ... | https://arxiv.org/abs/2503.21526v2 |
only if they can be m-separated by dsepG(A, B)or dsepG(B, A)by Lemma 16. Recall that dsepG(A, B)⊆anG({A, B})(Definition 15). Note that anG(A)⊆pastτ V(A)andanG(B)⊆pastτ V(B)for any τconsistent with G, and it follows that dsepG(A, B)⊆pastτ V(A)∪pastτ V(B). Assume without loss of generality that τ(A)≤τ(B), thenpastτ V(A)⊆... | https://arxiv.org/abs/2503.21526v2 |
the same skeleton and v-structures as M. We only need to argue that the additional arrowheads and tails obtained in the full tFCI algorithm are correct. The arrowhead orientation in line 32 is sound due the background knowledge being correct. The orien- tation rules R1-R4 and R8-R10 are sound in the sense that they pre... | https://arxiv.org/abs/2503.21526v2 |
the IOD (algorithm). Consider a fixed i∈ {1, . . . , n }, letA, B∈Viand let pdsGi(A, B)τandpdsGi(B, A)τbe sets of nodes considered at line 20 of the tIOD algorithm. Let Mibe the true MAG over Vi. Then, AandBare m-separated in Miif and only if they are m-separated by dsepMi(A, B)or dsepMi(B, A)(Definition 15) by Lemma 1... | https://arxiv.org/abs/2503.21526v2 |
Tillman and Spirtes (2011) that Gcontains a subset of the v-structures in M. Let⟨A, B, C ⟩be an unshielded triple in GandM, and assume that this is a v-structure in Mbut not in G. Ifτ(B)> max( τ(A), τ(C)), then every graph considered at line 49 must have ⟨A, B, C ⟩oriented as a v- structure. If τ(B)≤max( τ(A), τ(C))and... | https://arxiv.org/abs/2503.21526v2 |
skeleton and v- structures as M. Assume that G′also contains arrowheads or tails that are not in M: Then this has been oriented at line 27. Following the proof of Theorem 5.2 in Tillman and Spirtes (2011), this must be an arrowhead that is into the collider in a v-structure in some MAG Mi= (Vi,Ei), for some i∈ {1, . . ... | https://arxiv.org/abs/2503.21526v2 |
Spirtes (2011)), and let RemoveEdgeτandOrientVstructureτbe obtained in the tIOD algorithm. 29 BANG DIDELEZ Then the tIOD algorithm visits fewer graphs than the IOD algorithm if and only if either |RemoveEdge |>|RemoveEdgeτ|,|OrientVstructure |>|OrientVstructureτ|, or both|RemoveEdge |>|RemoveEdgeτ|and|OrientVstructure ... | https://arxiv.org/abs/2503.21526v2 |
arXiv:2503.21576v1 [math.PR] 27 Mar 2025Empirical Measures and Strong Laws of Large Numbers in Categorical Probability Tobias Fritz1, Tom´ aˇ s Gonda1, Antonio Lorenzin2, Paolo Perrone3, and Areeb Shah Mohammed1 1Department of Mathematics, University of Innsbruck, Austr ia 2Department of Computer Science, University Co... | https://arxiv.org/abs/2503.21576v1 |
. . 18 3 Empirical Sampling Morphisms 22 3.1 The Idea and the Categorical Definition . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Empirical Sampling Morphisms for Standard Borel Spaces . . . . . . . . . . . . . . . 24 3.3 Empirical Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2503.21576v1 |
that is of primary interest to us is the Glivenko–Cantelli theorem . It applies to xivalued in Rand establishes the uniform convergence of the relative frequencies of Equation ( 1.2) asTranges over all intervals in R. We regard such theorems on the convergence of empirical meas ures as the conceptually most funda- ment... | https://arxiv.org/abs/2503.21576v1 |
proofs that they satisfy the two axioms mentioned above. Together, our abstract theorems and the concrete construct ions recover the de Finetti theorem for standard Borel spaces [ FGP21 ], the Glivenko–Cantelli theorem (Corollary 4.8) and the strong law 1However, making this precise for unbounded frequires dealing with... | https://arxiv.org/abs/2503.21576v1 |
For example, the sequence (1,2,3,...)overX=Nhas limiting relative frequency zerofor each singleton subset TofN. To address this second issue, we require the limits in Equati on (1.4) to be uniform on sets of the form T={1,...,t},3and take esNto be undefined on all sequences that do not satisfy this requi rement (see Pro... | https://arxiv.org/abs/2503.21576v1 |
kernels to more general Markov categories. Many of these not ions can be captured in terms of uni- versal properties. To illustrate this, consider the Markov category BorelStoch , which has standard Borel spaces as objects and Markov kernels between them as mo rphisms and is the Markov category of primary interest for ... | https://arxiv.org/abs/2503.21576v1 |
process as nvaries, proving various results on when this process converges to th e underlying distribution. 5This fresh perspective has also gained popularity in the com puter science community, as it allows for a more systematic treatment of the logic underlying probability t heory [ JZ20,Ste21] as well as application... | https://arxiv.org/abs/2503.21576v1 |
to [ FR20]. Section 2.5then introduces distribution objects in quasi-Markov cate gories. Finally, following the arguments laid out in [ FL23], we show how a specific equalizer of permutations of a counta ble sequence of objects (a de Finetti object) is in fact also a dis tribution object (Theorem 2.23). 2.1 Basic Definit... | https://arxiv.org/abs/2503.21576v1 |
s atisfy the following weaker condition, which will play an important role in our proofs. Definition 2.5 ([DLR23 , Definition 3.1]) .A morphism f:A→Xin a CD category is quasi-total if it satisfies fX A=fX Af (2.9) A CD category is a quasi-Markov category if every morphism is quasi-total. Lemma 2.6. Every monomorphism m:A→... | https://arxiv.org/abs/2503.21576v1 |
partializ ation ofBorelStoch in the sense of [ SM]. 2.2 Determinism, Almost Sure Equality and Positivity To get a suitable language for discussing probability, we ne ed some natural generalizations of existing notions for Markov categories. Namely, we define what it mean s for a morphism to be deterministic, almost sure... | https://arxiv.org/abs/2503.21576v1 |
that coarse-graining a state m:I→Yto a copyable state g◦m:I→ZmakesYandZnecessarily 12 independent in the sense that g = mZ Y Z Y mmg (2.18) holds. Thus the positivity axiom can be also seen as a require ment that a deterministic variable cannot display correlation with other variables [ FGHL+23, Section 2.2]. Our main ... | https://arxiv.org/abs/2503.21576v1 |
g◦h, positivity gives hhg = hg (2.23) and thus we find hhgf = hgf =g h(2.24) as desired. To show compatibility with post-composition by a morphism k, we need to prove k◦f⊒k◦g. Using the quasi-totality of gin combination with f⊒g, we get7 gfk = ggfk = ggk (2.25) and therefore ggk = =gfk gkk k (2.26) holds as desired. InP... | https://arxiv.org/abs/2503.21576v1 |
tensor factor in the source [ FR20, Section 5]. We will mainly consider Xσin the case where σis afinite permutation, meaning that it leaves almost all elements of Nfixed. In this way, we obtain an action of the finite permutation group S∞onXN. Givenf:A→X, a common type of morphism A→XNappearing in categorical probability ... | https://arxiv.org/abs/2503.21576v1 |
category [ Gir82]; • The unit δ:X→PXis the measurable function assigning to each x∈Xthe Dirac delta δx∈PX; • The counit samp:PX→Xis the Markov kernel that takes a measure µ∈PXas input and returns a sample from µas output. Formally, for all measurable T⊆X, we have samp(T|µ) =µ(T) (2.37) for all measurable subsets A⊆X. 2... | https://arxiv.org/abs/2503.21576v1 |
for a de Fi netti object. In other words, this means that every exchangeable morphism p:A→XNfactors across samp(N). If the representability assumption is strengthened to observational representabi lity, then every Xhas a de Finetti object given by PX[MP22 , Section 8.2].10,11The universal property is illustrated by the... | https://arxiv.org/abs/2503.21576v1 |
ℓ(N)⊗idis monic by virtue of the limitQXbeing preserved by tensor products, the morphism ℓ(N)⊗ℓ(N)is also monic. Therefore, Equation ( 2.49) implies that µis copyable, which proves the surjectivity of the map ℓ◦. 3 Empirical Sampling Morphisms The key idea of this paper is that the limiting behaviour of in finite sequen... | https://arxiv.org/abs/2503.21576v1 |
esX ιι···:= esYY YN(3.5) is an empirical sampling morphism for Y. Proof. The permutation invariance of esYis a direct consequence of the naturality of braiding and the permutation invariance of esX. Concerning empirical adequacy, let f:A→YN⊗Zbe exchangeable in the first factor. Then we have es(N) X f AZYN = f AYNZ ιN ιN... | https://arxiv.org/abs/2503.21576v1 |
for the sequence (1,2,3,...). 14For more details about the concept of tight sequences, see [ Bil95, Section 25]. 25 Proof. The tightness condition (Property (i)) implies that the limits lim n→∞|{i≤n|xi≥t}| n(3.13) are uniform in t: For a given ε >0and large t, the right-hand side is less than εfor every n, and therefor... | https://arxiv.org/abs/2503.21576v1 |
empirical sampling morphism for R. Definition 3.9. LetRbe equipped with its Borel σ-algebra. We define esR:RN→Ras the partial Markov kernel where: •A sequence (xi)belongs to dom(esR)if and only if the limits lim n→∞|{i≤n|xi≤t}| n(3.17) exist uniformly in t∈Q. •In this case, we define esR(|(xi))to be the unique probability... | https://arxiv.org/abs/2503.21576v1 |
the question of whether empirical sampling mor phisms exist for other kinds of measurable spaces, and what kind of structure is needed in or der to construct them. Before we conclude this section, let us also discuss why the n aturality of empirical sampling mor- phisms cannot hold in Partial(BorelStoch ). We already s... | https://arxiv.org/abs/2503.21576v1 |
finite a nalogue trivially holds: The expec- tation value of fwith respect to the finite empirical measure from ( 1.3) is the empirical average 1 n/summationtextn i=1f(xi). It is not hard to show that also full version of Equation ( 3.23) holds for any finite X. However, it is too much to hope for in general: For instance... | https://arxiv.org/abs/2503.21576v1 |
empirical average) .Consider the sequence (xn)overNgiven by xn:=/braceleftBigg n+1 ifnis a power of 2, 1 otherwise .(3.25) Then the limit lim n→∞|{i≤n|xi≤t}| n(3.26) of relative frequencies is equal to 1for every t∈N, since almost all elements of the sequence are one. The convergence is uniform by monotonicity in t. Th... | https://arxiv.org/abs/2503.21576v1 |
balanced idempotents has been proven synth etically for any Markov category in [ FGL+, Theorem 4.4.3] under the assumptions of positivity, obser vational representabil- ity, and the equalizer principle defined therein. If one gene ralized this result to quasi-Markov cat- egories, one would obtain alternative synthetic r... | https://arxiv.org/abs/2503.21576v1 |
f:A→XN⊗Ythat is invariant under all finite permutations of XN, i.e. one that is exchangeable. By the empirical adequacy of esand the above splitting of es(N), we have Y Aes(N)XN =Y AπXN ι f fY AXN =f(4.5) i.e.ffactors through ι⊗idYas required. Observational representability now follows from Theorem 2.23. We record as th... | https://arxiv.org/abs/2503.21576v1 |
Glivenko–Cantelli theorem) .Consider an arbitrary exchangeable mor- phismf:A→XNand an arbitrary morphism p:A→Xin a quasi-Markov category. Given As- 16That result is stated for Markov categories, but the proof wo rks in quasi-Markov categories just as well. 35 sumption 4.1, they satisfy f Asamp(N)PXXN es♯PXXN f A= es♯an... | https://arxiv.org/abs/2503.21576v1 |
but we have not yet worked out the details. We turn to a few more consequences that can be formulated and p roven synthetically. Comparing the empirical adequacy of empirical sampling with the state ment of one version of the synthetic de Finetti theorem [ CFG+24, Theorem 2.11] suggests that escould play the role of th... | https://arxiv.org/abs/2503.21576v1 |
es♯ R((xi))has finite first moment. (ii) In this case, we have /parenleftBig E◦es♯ R/parenrightBig/parenleftbig (xi)/parenrightbig = lim n→∞1 nn/summationdisplay i=1xi. (4.23) Proof. Property (i)holds by the definition of Eand Property (ii)by the definition of esR, and in particular by Equation ( 3.30). Based on this, we c... | https://arxiv.org/abs/2503.21576v1 |
M arkov categories. arXiv:2502.03477 .↑6,13 [EP23] N. Ensarguet and P. Perrone. Categorical probabilit y spaces, ergodic decompositions, and transitions to equilibrium. 2023. arXiv:2310.04267 .↑6 [FGHL+23] T. Fritz, T. Gonda, N. G. Houghton-Larsen, A. Lorenzin, P . Perrone, and D. Stein. Dilations and Information Flow ... | https://arxiv.org/abs/2503.21576v1 |
doi: 10.1017/etds.2023.6 . ↑6 [Per24] P. Perrone. Markov Categories and Entropy. Transactions on Information Theory , 70(3), 2024. doi: 10.1109/tit.2023.3328825 .↑3 [SM] A. Shah Mohammed. Partializations of Markov categorie s. (In preparation). ↑6,7, 10,11,13,15,16,32,39 [Ste21] D. Stein. Structural Foundations for Pro... | https://arxiv.org/abs/2503.21576v1 |
out the sixth power and taking expectations, each term ends up being an expecta- tion of a product of six of the Zi’s. When all six factors are distinct, then we can use exchang eability to rewrite this expectation as E[Z1···Z6]. Similarly by exchangeability, all other terms are multi- ples of one of E/bracketleftbig Z... | https://arxiv.org/abs/2503.21576v1 |
just before (if Fnis below Fmat the supremum). Hence there are 2npossibilities for where the supremum can be attained, and we can write the event on the lef t-hand side of Equation ( A.10) as ∃j≤n:/parenleftbigg|{i≤n|Wi≤Wj}| n−|{i≤m|Wi≤Wj}| m> ε ∨|{i≤n|Wi< Wj}| n−|{i≤m|Wi< Wj}| m<−ε/parenrightbigg By Lemma A.1, we have... | https://arxiv.org/abs/2503.21576v1 |
of Inequality ( A.15), then the two maxima coincide by exchangeability. Therefore we obtain E[1EnZ1]≥0. (A.16) Since the sequence of events Enis increasing in n, we obtain the same inequality for E∞. Plugging in the definition of Z1intoE[1E∞Z1]≥0then gives the second step in E[Y1]≥E[1E∞Y1]≥E[1E∞r] =rP/bracketleftBigg su... | https://arxiv.org/abs/2503.21576v1 |
n = lim n→∞1 nmn/summationdisplay i1,...,im=1T1(xi1)···Tm(xim). Thinking of k/mapsto→ikas a map {1,...,m} → {1,...,n}shows that this amounts to averaging over all such maps, of which there are nmmany. Since the fraction of injections among these maps goes to 1asn→ ∞, we obtain the same limit by averaging over all injec... | https://arxiv.org/abs/2503.21576v1 |
particular, for fixed mthe limit in Equation ( B.3) must exist uniformly for sets of the form Ti={1,...,ti}. The analogue of Equation ( B.4) holds because the relevant limits are all uniform inS, as is immediate from the bound Equation ( A.2) in Lemma A.1. Proof of Theorem 3.10.We denote the limiting function of ( 3.17)... | https://arxiv.org/abs/2503.21576v1 |
kernel esR+:RN +→R+. Lemma B.3. The partial Markov kernel esR+, defined as above, is an empirical sampling morphism forR+inPartial(BorelStoch ), Proof. The proof that esR+is a partial Markov kernel is analogous to the previous cases . While permutation invariance is clear, empirical adequacy is whe re the main work lies... | https://arxiv.org/abs/2503.21576v1 |
arguments as in the proof of Lemma B.3. However, we still need to show p/parenleftbig dom(esR)×Y/parenrightbig = 1 (B.24) for any exchangeable p:I→RN⊗Y, i.e. that the conditions in Definition 3.18that are additionally imposed, on top of those in Definition 3.9, hold almost surely for a random sequence (xi)with an exchang... | https://arxiv.org/abs/2503.21576v1 |
Locally minimax optimal and dimension-agnostic discrete argmin inference Ilmun Kim1and Aaditya Ramdas2,3 1Department of Statistics and Data Science, Yonsei University, 2Department of Statistics and Data Science, Carnegie Mellon University, 3Machine Learning Department, Carnegie Mellon University ilmun@yonsei.ac.kr aram... | https://arxiv.org/abs/2503.21639v2 |
this dependence explicitly, we denote it by dn, though we omit the subscript when the distinction is not essential. Noting the duality between confidence sets and hypothesis tests, a large part of the paper will focus on solving the following dual testing problem: given some fixed r∈[d], we test the null and alternativ... | https://arxiv.org/abs/2503.21639v2 |
the lim inf n) but pointwise in Θ. To elaborate the latter point, it is helpful to consider the following three types of coverage discussed in the literature: 1.Weak coverage : At least one element of Θis covered, i.e., P(Θ∩bΘ̸=∅)≥1−α; 2.Pointwise coverage : Every element of Θis covered, i.e., infr∈ΘP(r∈bΘ)≥1−α; 3.Unif... | https://arxiv.org/abs/2503.21639v2 |
independent-component bootstrap to address issues of inconsistency and dependence. While their approach provides valuable insights into ranking variability, it does not directly target argmin inference or provide formal confidence sets for the best-performing index. Xieetal.(2009)addressedinferenceinthepresenceoftiesan... | https://arxiv.org/abs/2503.21639v2 |
in Figure 3 suggest that its validity may be sensitive to the problem context, particularly in maintaining type I error control. Moreover, as shown in Figure 4, their method exhibits significant power loss in certain regimes, indicating that the test may not achieve a minimax separation rate and highlighting the need f... | https://arxiv.org/abs/2503.21639v2 |
moment conditions. Section 5 proposes and analyzes DA model confidence sets with uniform coverage. Section 6 presents empirical results demonstrating the competitive performance of the proposed method compared to existing approaches. We conclude in Section 7 by summarizing the paper and discussing potential directions ... | https://arxiv.org/abs/2503.21639v2 |
We propose two different approaches for this purpose. The first estimator is the plug-in estimator, defined as bsplug:= sargmin k∈[d]\{r}X(2) k, which directly selects the index corresponding to the smallest sample mean in the second half of the data. Alternatively, we propose a noise-adjusted estimator that accounts f... | https://arxiv.org/abs/2503.21639v2 |
k EP[W2 k]min( 1,|Wk| n1/2(EP[W2 k])1/2)# =o(1)asn→ ∞. (5) This condition serves a similar role to the remainder term in a Berry–Esseen bound, but with a lighter tail requirement that allows for a broader class of distributions. For example, the t- distribution with 3degrees of freedom lacks a finite third moment, yet ... | https://arxiv.org/abs/2503.21639v2 |
sha1_base64="RWOOr1K8HtVTRRzSSEBgVi3Rr8M=">AAAB7nicbVDJSgNBEK2JW4xb1KOXxiB4CjMSl2NADx4jmAWSIfR0epIm3T1DL0IY8hFePCji1e/x5t/YSeagiQ8KHu9VUVUvSjnTxve/vcLa+sbmVnG7tLO7t39QPjxq6cQqQpsk4YnqRFhTziRtGmY47aSKYhFx2o7GtzO//USVZol8NJOUhgIPJYsZwcZJ7Z6w/exy2i9X/Ko/B1olQU4qkKPRL3/1BgmxgkpDONa6G/ipCTOsDCOcTks9q2mKyRgPaddRiQXVYTY/d4rOn... | https://arxiv.org/abs/2503.21639v2 |
particular, the variance of ⟨v,X⟩is at most σ2for every unit norm v∈Rd. Now define a class of local alternatives that share the same cardinality of the confusion set as P1,r(ε;τ):= P∈ P:µr−µ⋆≥εand|Cr|=τ , where ε >0is a positive constant and τ∈ {0,1, . . . , d −2}. We aim to characterize the condition onεunder which t... | https://arxiv.org/abs/2503.21639v2 |
argmin test. Then the index ris excluded from bΘDA with probability tending to one: lim n→∞inf P∈P1,r(ε;τ)P r /∈bΘDA = 1. As formally established later in Corollary 3.4, the above result is minimax optimal in the sense that no asymptotically valid confidence set can reliably exclude the index r /∈Θwhen the mean gapµr... | https://arxiv.org/abs/2503.21639v2 |
idea is to replace bswith a robust alternative that is less sensitive to outliers. To this end, we employ the median-of-means (MoM) estimator for estimating the argmin s. The MoM estimator, which traces back to Nemirovsky and Yudin (1983); Jerrum et al. (1986), has been extensively studied in the literature (e.g., Alon... | https://arxiv.org/abs/2503.21639v2 |
be established for any robust estimator that exhibits sub-Gaussian tails under finite second moment conditions—suchasCatoni’sM-estimator(Catoni,2012)andthetruncatedempiricalmean(Bubeck et al., 2013), with only minor changes to the proof in order to incorporate minor differences be- tween the formal guarantees of these ... | https://arxiv.org/abs/2503.21639v2 |
confusion set, so we propose the following two-step construction. Two-step construction of a DA-MCS with uniform coverage. To provide a better bench- mark with uniform coverage, we now introduce a modified version of our own confidence set con- struction that attains a uniform coverage guarantee. Let ψk(S, c)denote the... | https://arxiv.org/abs/2503.21639v2 |
only for the indices in this data-adaptive subset bΘuni DAof size bd:=|bΘuni DA|, to obtain the interval C2=" min k∈bΘuni DA X(1) k−z1−α 2bdbσ(1) k√n ,min k∈bΘuni DA X(1) k+z1−α 2bdbσ(1) k√n# , where X(1) kandbσ(1) kare the sample mean and sample standard deviation for the k-th popu- lation based on D1. This data-a... | https://arxiv.org/abs/2503.21639v2 |
on real-world data are presented in Section 6.4 and in Section 6.5, respectively. Experiments on uni- form coverage are in Section 6.6 and Appendix B.2 and on heavy-tailed settings in Appendix B.1. We also refer to Table 1 for a summary of execution times of all methods. 18 Computational Efficiency. Table 1 summarizes ... | https://arxiv.org/abs/2503.21639v2 |
19 0.0 0.1 0.2 0.3 0.4 0.5 αRejection Rate 0.01 0.10 0.20 0.30 0.40 0.50n=250, d=4 LOO Bonferroni csranks DA-plug DA-adj y=α 0.0 0.1 0.2 0.3 0.4 0.5 αRejection Rate 0.01 0.10 0.20 0.30 0.40 0.50n=1000, d=4 0.0 0.1 0.2 0.3 0.4 0.5 αRejection Rate 0.01 0.10 0.20 0.30 0.40 0.50n=2500, d=40.0 0.1 0.2 0.3 0.4 0.5 αRejection... | https://arxiv.org/abs/2503.21639v2 |
MCS 0.000 0.004 0.008 0.048 0.054 0.140 0.352 0.354 0.646 DA-plug 0.219 0.305 0.501 0.371 0.424 0.679 0.205 0.238 0.426 DA-plug×100.307 0.401 0.727 0.593 0.674 0.957 0.310 0.359 0.655 DA-adj 0.232 0.448 0.931 0.365 0.506 0.932 0.207 0.250 0.477 DA-adj×100.307 0.589 0.988 0.585 0.728 0.994 0.300 0.370 0.697 Table 3: Emp... | https://arxiv.org/abs/2503.21639v2 |
0.4ρ= 0.8 LOO 0.084 0.115 0.380 0.000 0.001 0.181 0.258 0.351 0.703 Bonferroni 0.171 0.130 0.055 0.166 0.103 0.030 0.017 0.006 0.003 csranks 0.184 0.381 0.962 0.162 0.363 0.961 0.019 0.041 0.223 MCS 0.004 0.002 0.004 0.000 0.000 0.000 0.140 0.156 0.166 DA-plug 0.049 0.052 0.042 0.062 0.067 0.059 0.098 0.128 0.202 DA-pl... | https://arxiv.org/abs/2503.21639v2 |
methods across varying dimensions under the settings described in Section 6.4. The dashed line represents the nominal level of 0.05. The results demonstrate the superior performance of the DA argmin tests in the considered high-dimensional settings, consistently maintaining strong power across all dimensions while cont... | https://arxiv.org/abs/2503.21639v2 |
indicate that the DA-adjmethod consistently produces smaller inclusion sets across both competition years compared to the alternatives. This advantage is particularly pronounced in the 2023 competition, where the average size of the inclusion set produced by DA-adjis17.35±1.17 with the number after ±indicating the stan... | https://arxiv.org/abs/2503.21639v2 |
The first row reports the coverage rates of methods designed for pointwise coverage, while the bottom two rows report those of methods tar- geting uniform coverage. The results demonstrate that the DA-MCS method with the two-step construction consistently achieves superior uniform coverage compared to its one-step coun... | https://arxiv.org/abs/2503.21639v2 |
general rank- kinference problems, where the objective is to identify the index corresponding to the k-th smallest mean. Such an extension would broaden the applicability of our methodology and introduce new theoretical challenges. Second, it may be worthwhile to ex- plore thresholding-based approaches for constructing... | https://arxiv.org/abs/2503.21639v2 |
Learning Research , 17(18):1–40. Hung, K. and Fithian, W. (2019). Rank verification for exponential families. The Annals of Statis- tics, 47(2). Ingster, Y. I. (1987). Minimax testing of nonparametric hypotheses on a distribution density in the Lpmetrics. Theory of Probability & Its Applications , 31(2):333–337. Jerrum... | https://arxiv.org/abs/2503.21639v2 |
A Novel Approach of High Dimensional Linear Hypothesis Testing Problem. Journal of the American Statistical Association , (to appear). 30 A Proofs and Technical Lemmas In this section, we collect the proofs of the main results and some technical lemmas. A.1 Proof of Theorem 2.1 This result is almost a direct consequenc... | https://arxiv.org/abs/2503.21639v2 |
+P√n(µ1−µbs)≤(z1−α+ 1)√ 4σ2δ−1∩bs∈Cc b ≤P√n 2(µ1−µ2)≤(z1−α+ 1)√ 4σ2δ−1 +P(bs∈Cc b). To deal with P(bs∈Cc b), define the event E4,δ:=d\ k=2 X(2) k−µk <s 2σ2 nlog2d δ . Another application of the sub-Gaussian tail bound together with the union bound yields P(Ec 4,δ) =P d[ k=2 X(2) k−µk ≥s 2σ2 nlog2d δ! ≤δ. Fr... | https://arxiv.org/abs/2503.21639v2 |
(z1−α+ 1)√ 4σ2δ−1∩eE3,δ∩ {bs∈C}! +δ, where the last inequlity uses (p∨r)/(q∨r)≤1 +r−1pfor any p, q≥0andr >0. Moreover, we define another event eE1,δ:= γ⊤ 2bΣ(2)γ2≤4σ2 δ , which holds with probability at least 1−δ, similarly to E1,δ. By incorportaing this event into the above inequality for (I)using the union bound, w... | https://arxiv.org/abs/2503.21639v2 |
over the i.i.d. samples X1, . . . ,Xnby adding the superscript ⊗ntoP. As in the proof of Theorem 3.1, we set r= 1 without loss of generality. Form∈Z>0, the mean vector µ(0)consists of the first m+ 1components set to zero, followed by the remaining d−m−1components set to bn>0, that is µ(0)= (0,0, . . . , 0|{z} mentries,... | https://arxiv.org/abs/2503.21639v2 |
of Theorem 3.3. A.5 Proof of Corollary 3.4 Given r∈[d], define Aα,r:=n bΘ : lim inf n→∞inf P∈P0,rP(r∈bΘ)≥1−αo , which satisfies Aα⊆ A α,r. Now consider the test ψthat rejects the null hypothesis H0:r∈Θ if and only if r /∈bΘ. This establishes a one-to-one correspondence between Aα,rand the set of asymptotic level- αtest... | https://arxiv.org/abs/2503.21639v2 |
k−z1−α 2bdbσ(1) k√n ≤µ⋆≤min k∈bΘuni DA X(1) k+z1−α 2bdbσ(1) k√n ,Θ⊆bΘuni DA =P min k∈bΘuni DA X(1) k−z1−α 2bdbσ(1) k√n ≤min k∈bΘuni DAµk≤min k∈bΘuni DA X(1) k+z1−α 2bdbσ(1) k√n ,Θ⊆bΘuni DA ≥P ∀k∈bΘuni DA:X(1) k−z1−α 2bdbσ(1) k√n≤µk≤X(1) k+z1−α 2bdbσ(1) k√n,Θ⊆bΘuni DA ≥1−P [ k∈bΘuni DA(√n|X(1) k−µk| bσ(1)... | https://arxiv.org/abs/2503.21639v2 |
the data from a heavy- tailed distribution—specifically a multivariate t-distribution with 3degrees of freedom, which has a finite second moment but an infinite third moment. For each sample, a standard normal vector Z∼N(0,Id)is drawn and a chi-squared random variable U∼χ2 3is generated independently. The observed data... | https://arxiv.org/abs/2503.21639v2 |
appropriate rejection rates (correct coverage). Methodµ(a,0)+ equal variance µ(b,0)+ equal variance µ(c,0)+ equal variance ρ= 0 ρ= 0.4ρ= 0.8ρ= 0 ρ= 0.4ρ= 0.8ρ= 0 ρ= 0.4ρ= 0.8 DA-plug 0.0210.023 0.0280.0300.029 0.026 0.052 0.049 0.053 DA-plug-mom 0.0150.014 0.0190.0280.029 0.025 0.054 0.052 0.053 DA-plug-catoni 0.0220.0... | https://arxiv.org/abs/2503.21639v2 |
rates close to the nominal level remain unshaded. DA-plug (pointwise) ρ= 0 ρ= 0.4 ρ= 0.8 |Θ|= 20.971 (82.31)0.971 (83.11)0.959 (81.72) |Θ|= 50.822 (45.50)0.829 (46.42)0.841 (48.33) |Θ|= 100.718 (33.31)0.720 (33.57)0.748 (33.36) |Θ|= 150.662 (31.69)0.669 (31.63)0.688 (31.53) |Θ|= 200.623 (32.75)0.616 (32.84)0.651 (32.56... | https://arxiv.org/abs/2503.21639v2 |
the DA-MCS-adj2procedure to construct the screening set bΘuni DA. Table 13 reports the average widths of C1andC2across varying |Θ|and correlation levels ρ. When |Θ|is small, the data-adaptive interval C2is narrower than the non-adaptive C1. However, as |Θ|grows, this advantage diminishes and C2becomes wider due to the ... | https://arxiv.org/abs/2503.21639v2 |
1 Wasserstein bounds for non-linear Gaussian filters Toni Karvonen and Simo Särkkä Abstract —Most Kalman filters for non-linear systems, such as the unscented Kalman filter, are based on Gaussian approxi- mations. We use Poincaré inequalities to bound the Wasserstein distance between the true joint distribution of the ... | https://arxiv.org/abs/2503.21643v1 |
(X,eY)and the true joint distribution Z= (X,Y)is1 DKL(Z∥eZ) =1 2log det I+ ΣΣΣ−1 V Cov[Y]−ΣΣΣV −Cov(X,Y)TP−1Cov(X,Y) .(1) However, since the Kullback–Leibler divergence is not a metric, it is difficult to incorporate other errors, such as those propagated from previous time-steps or due to numerical integration i... | https://arxiv.org/abs/2503.21643v1 |
are affine. We use prto denote Gaussian projection that transforms a given random variable into a Gaussian one while preserving the mean and covariance. That is, for an arbitrary random variable Zwith finite mean E(Z)and covariance Cov(Z)we have prZ∼ N(E(Z),Cov(Z)), (4) which is to say that the moments of the Gaussian ... | https://arxiv.org/abs/2503.21643v1 |
two most important members of this class are the Wasserstein distance dW(X,Z) =W1(X,Z) = sup Lip(h)≤1 E[h(X)]−E[h(Z)] , where Lip(h)is the maximal Lipschitz constant defined as Lip(h) = sup x̸=y|h(x)−h(y)| ∥x−y∥, and the total variation distance dTV(X,Z) = sup |h|≤1/2 E[h(X)]−E[h(Z)] . Note that the total variation dis... | https://arxiv.org/abs/2503.21643v1 |
to say thatg(x) =Ax+bfor some A∈Rd2×d1andb∈Rd2. For our purposes the recent second order Poincaré inequality , proven first by Chatterjee [31] for the total variation distance, is essential. Multivariate extensions for the Wasserstein distance appear in [32] and [33]. The version in Theorem 5 below is a corollary of [3... | https://arxiv.org/abs/2503.21643v1 |
E[Jf(Xp)]TE Jh[f(Xp) +U]Jf(Xp)T , B1= In×n E Jh[f(Xp) +U]! ΣΣΣU × In×nE Jh[f(Xp) +U]T , A2=E"In×n Jh[f(Xp) +U] Jf(Xp)PpJf(Xp)T+ ΣΣΣU × In×nJh[f(Xp) +U]T , CV= 0n×n0n×m 0m×nΣΣΣV . (13) Proof. The Jacobian of gin Equation (9) is Jg(x) = Jf(x1) In×n 0n×m Jh[f(x1) +x2]Jf(x1)Jh[f(x1) +x2]Im×m by the chai... | https://arxiv.org/abs/2503.21643v1 |
coincide because Jacobians are constant matrices. Therefore Cov(Z) = Cov( eZ) =A1+B1+CV=A2+CV andtr Cov( Z) = tr PX+ trPY. Also recall from (13) that trCV= trΣΣΣV. From these equations it follows that dW(Z,eZ) ≤ ∥E(Y)−E(eY)∥2+ trA2+ trΣΣΣV+ trPX+ trPeY −2 trp L A1+B1+CV L =∥E(Y)−E(Y)∥2+ tr Cov( Z) + tr Cov( Z) −2 t... | https://arxiv.org/abs/2503.21643v1 |
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