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by Malliavin and Mancino [ 14,15]. Furthermore, Ogihara and Yoshida [ 18] and Ogihara [ 17] constructed a quasi-log-likelihood function that accounts for nonsynchronous observations in parametric diffusion process models and derived maximum likelihood-type estimators for the param eters. Innonsynchronousobservationmodel... | https://arxiv.org/abs/2503.18400v1 |
the main results. Section 2.1outlines the model settings and assumptions, and introduces the q uasi-likelihood functions necessary for constructing the test statistics. In Se ction2.2, we establish the consistency of the test statistics, while Section 2.3discusses the AUMPI property of the proposed test. Section 2.4det... | https://arxiv.org/abs/2503.18400v1 |
aim to introduce test statistics that achieve cons istency for nonsynchronous observations of diffusion processes. For 1≤i≤Ml, we define Il i= (Sn,l i−1,Sn,l i]. The increment of Xl tover the interval Il iis given by ∆l iX= Xl Sn,l i−Xl Sn,l i−1. We then define ∆lX= (∆l iX)1≤i≤Mland ∆X= ((∆1X)⊤,(∆2X)⊤)⊤. In other words, ∆... | https://arxiv.org/abs/2503.18400v1 |
[0,∞) satisfying sup k≥1|sk−sk−1| ≤1,inf k≥1|sk−sk−1|>0. For (sk)∞ k=0∈S, we define Ml,k= #{i; supIl i∈(sk−1,sk]}, qn= max{k;sk≤nhn}. We also define the matrix El (k)by [El (k)]ij=/braceleftigg 1,ifi=jand supIl i∈(sk−1,sk], 0,otherwise. Furthermore, we define G=/braceleftigg |I1 i∩I2 j| |I1 i|1/2|I2 j|1/2/bracerightigg... | https://arxiv.org/abs/2503.18400v1 |
of AUMPI as defined by Ch oi et al. [ 2], and we present the results demonstrating that the tests constructed from T1 nandT2 nare AUMPI. Consider a general parameter space Θ ⊂Rdand, for 1 ≤r≤d, let the parameter γ= (ϑ,η)∈Θ be such thatϑ∈Rrandη∈Rd−r. Here,ηrepresents a nuisance parameter; in the case where r=d, we ignore... | https://arxiv.org/abs/2503.18400v1 |
the quasi-log- likelihood functions H1 n(σ) andH2 n(θ) overσandθ, respectively. This necessitates repeated evaluations of these functions. Since the size of Snincreases with n, the computation of its inverse can be highly burdensome. 8 Toreducethe computationalcost ofthe matrixinversion,let Lbe apositive integerand con... | https://arxiv.org/abs/2503.18400v1 |
compare the test statistic T1 nwith a test statistic based on the Hayashi–Yoshida estimator presented in the Appendix. 3.1 Test for σwith time-invariant diffusion coefficients Consider the two-dimensional diffusion process ( X1 t,X2 t)⊤governed by the stochastic differential equation /parenleftbiggdX1 t dX2 t/parenrightbigg... | https://arxiv.org/abs/2503.18400v1 |
a√n-neighborhood of 0; that is, we consider σ(3)= 0+u√n, u∈ {1,2,3,4}. The results are presented in Tables 4and5in the same format as Tables 2and3. Note that, as specified in Table 1, we setn= 103so that 1/√n≈0.032; in other words, the tests are conducted for true values muc h closer to the null than, for example, σ(3)=... | https://arxiv.org/abs/2503.18400v1 |
Proportion of samples with T2 n≥χ2 1(α) under local alternatives True ValueThresholdχ2 1(0.10)χ2 1(0.05)χ2 1(0.01) θ(1)= 0 0.090 0.054 0.018 θ(1)= 0+1/√nhn 0.192 0.108 0.034 θ(1)= 0+2/√nhn 0.406 0.310 0.122 θ(1)= 0+3/√nhn 0.680 0.576 0.340 θ(1)= 0+4/√nhn 0.876 0.812 0.614 13 Then, define Y1(σ) = lim T→∞1 T/integraldispl... | https://arxiv.org/abs/2503.18400v1 |
−/integraldisplay1 01 n∂2 P0σH1 n/parenleftig σ0+u/parenleftbig ˜σn−σ0/parenrightbig/parenrightig du=˜Γ0 1,3,n(˜σn,σ0). Hence, combining with ( 4.4), we obtain 1√n∂P0σH1 n(˜σn)−1√n∂P0σH1 n(σ0) =−˜Γ0 1,3,n(˜σn,σ0)√nP0/parenleftbig ˜σn−σ0/parenrightbig . Since ˜σn∈Θ1implies∂P0σH1 n(˜σn) = 0, the claim follows. Lemma 4.... | https://arxiv.org/abs/2503.18400v1 |
H1 n(ˆσn)−H1 n(σ0)/parenrightig −Y1(ˆσn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|Y1(ˆσn)|. By Ogihara [ 17] Proposition 3.8, sup σ∈Θ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 n/parenleftig H1 n(σ)−H1 n(σ0)/parenrightig −Y1(σ)/vextendsingle/vextendsingle/vextendsingle/vextendsingleP→0... | https://arxiv.org/abs/2503.18400v1 |
n(ˆθn) = 0. Then, by a Taylor expansion, 1√nhn∂θH2 n(˜θn)−1√nhn∂θH2 n(ˆθn) =−˜Γ0 2,n(˜θn,ˆθn)/radicalbig nhn/parenleftbig˜θn−ˆθn/parenrightbig . Hence,/radicalbig nhn/parenleftbig˜θn−ˆθn/parenrightbig =−(˜Γ0 2,n)−1(˜θn,ˆθn)1√nhn∂θH2 n(˜θn) on{det˜Γ0 2,n(˜θn,ˆθn)>0}. (4.13) Together with Lemma 4.7, (4.12) and (4.13), we... | https://arxiv.org/abs/2503.18400v1 |
Assume conditions (A1)–(A6). For each k∈ {0,1,2,3}, sup σ∈Θ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√n/parenleftig ∂k σHn(σ,θ0)−∂k σH1 n(σ)/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingleP→0, sup θ∈Θ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√nhn/parenleftig ∂k θ(... | https://arxiv.org/abs/2503.18400v1 |
(A6). Then, ˆ σ∗ nP→σ0, and under H(σ) 0, ˜σ∗ nP→σ0as n→ ∞. Proof.Following the discussion in Section 3.2 of Ogihara [ 17], for anyδ>0 there exists η>0 such that inf |σ−σ0|≥δ/parenleftig −Y1(σ)/parenrightig ≥η. Since, by definition, Hn(ˆσ∗ n,θ0)−Hn(σ0,θ0)≥0, it follows from Lemma 4.12that for any ǫ>0, for sufficiently l... | https://arxiv.org/abs/2503.18400v1 |
2003 . [2] S. Choi, W. J. Hall, and A. Schick. Asymptotically unifor mly most powerful tests in parametric and semiparametric models. Ann. Statist. , 24(2):841–861, 1996. [3] A. De Gregorio and S. M. Iacus. Divergences test statisti cs for discretely observed diffusion processes. Journal of Statistical Planning and Infe... | https://arxiv.org/abs/2503.18400v1 |
and Yoshida [ 10], the authors showed that, for a fixed observation interval [0 ,T] (withT >0) and as max i,l|Il i| →0, one has√n/parenleftig HYn−/a\}bracketle{tX1,X2/a\}bracketri}htT/parenrightigd→ N(0,c) (asn→ ∞), (A.1) wherecis a positive constant. Here, for any interval I, we define v(I) =/integraldisplay I[btb⊤ t(... | https://arxiv.org/abs/2503.18400v1 |
arXiv:2503.18515v1 [math.AP] 24 Mar 2025Recovering a (1+1)-dimensional wave equation from a single white noise boundary measurement Emilia L.K. Bl˚ asten∗,1, Tapio Helin†,1, Antti Kujanp¨ a¨ a‡,1, Lauri Oksanen§,2, and Jesse Railo¶,1,3 1Computational Engineering, School of Engineering Sciences, Lapp eenranta-Lahti Univ... | https://arxiv.org/abs/2503.18515v1 |
passive imaging, in two and higher dimensions, is explored in recent work on the inverse problem of corner scattering, where boundary measurements performed with a single arbitrary w ave recover, for example, information about the geometry and location of the scatterer [BPS14; HSV16; BPS20]. In this paper, we adopt a d... | https://arxiv.org/abs/2503.18515v1 |
measurements (a δ-function as an incident wave). The problem ( P) correspondstosimplylisteningtosomeonespeakavowel,andtodet erminetheirvocaltractconfiguration based on the audio alone. It is something that humans are quite adep t at doing automatically (especially people good at imitating!) but machines and algorithms s... | https://arxiv.org/abs/2503.18515v1 |
Theorem 1.3is derivedin Section 4. The keyelement is Proposition 5.3which givesa relationbetween random white noise measurements and the standard Neumann-to- Dirichlet map. This result relies on energy estimates studied separately in Section 5. The actual proof of Theorem 1.3is a combination of Proposition 5.3and the t... | https://arxiv.org/abs/2503.18515v1 |
treated in the appendix of the book. 2.2 The wave front set of a distribution Denote˙Rn:=Rn\{0}.By Paley-Wiener theorem, a distribution u∈ D′(Rn) is smooth in a neighbour- hood ofx∈Rnif and only if there is φ∈C∞ c(Rn) such that φ(x)/\e}atio\slash= 0 and /hatwiderφu(ξ) =O(|ξ|−N), (2.2) for everyξ∈˙RnandN∈Nas|ξ| → ∞. Her... | https://arxiv.org/abs/2503.18515v1 |
noise. See e.g. [HS08; Kuo96] for more details on the topic. Let us write Wφ:=/a\}b∇acketle{t·,φ/a\}b∇acket∇i}htforφ∈ S(Rd). It follows from the definition above that white noise admits the isometry EW2 φ=/ba∇dblφ/ba∇dbl2 L2 (2.4) (cf.ˆIto isometry). Consequently, E(WφWψ) =/a\}b∇acketle{tφ,ψ/a\}b∇acket∇i}htL2. Combining... | https://arxiv.org/abs/2503.18515v1 |
0in(0,∞)×R, (3.6) u= 0,in(0,∞)×(−∞,0), (3.7) ∂µ xu/vextendsingle/vextendsingle x=0=f,in(−∞,2δ+s). (3.8) that vanishes in the wedge W:={(x,t)∈R2:|t|<x+s}. Moreover, the solution depends continuously onfwith respect to the topology (see Section 2.1) ofE′(R). Before proving the lemma, let us show how to derive Proposition... | https://arxiv.org/abs/2503.18515v1 |
limit limε→0vεexists in D′(R2), we may choose FεandGεsuch that they converge individually to some distributions F0andG0. Hence, uε(x,t) =Fε(t−x)+Gε(t+x) (3.24) in (−∞,δ]×Rfor someFε,Gεthat converge in D′(R) asǫ→0. Formally, we have u0(x,t) =F0(t−x)+G0(t+x) (3.25) for (x,t)∈(−∞,δ)×R. For the rigorous expression, define ρ... | https://arxiv.org/abs/2503.18515v1 |
further extends the identity u0= (−1)µρ∗ −f(−µ)to V∪Xt>0∪W. Considering u0as a distribution in D′((0,∞)×R), we may cut off everything below W∩(0,∞)×Rwithout conflicting /squareAu0= 0. Indeed, /squareAu= 0 in (0,∞)×Rforu:=χu0, where χ∈C∞satisfiesχ|t>−s/4= 1 andχ|t<−s/2= 0. Moreover, uvanishes in (0 ,∞)×(−∞,0) sinceu0= 0 in... | https://arxiv.org/abs/2503.18515v1 |
We have 0 =∂µ xv|x=0=∂µ xv→ /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =F(t−x)|x=0+v← /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ∈C∞|x=0≡(−1)µF(µ)modC∞. Hencev→∈C∞((0,δ)×R) and therefore v→∈C∞((0,∞)×R) by the propagation of singularities. In conclusion, v=v→+v←∈C∞... | https://arxiv.org/abs/2503.18515v1 |
forwards propagating solution u. One checks that the relation between them is ˜uφ(x,t) =uψ(x,t0−t), (4.8) whereψ(t) :=φ(t0−t). We define the time-reversedNeumann-to-Dirichlet map ˜ΛND:C∞ c(R)→C∞(R) by˜ΛND(φ) = ˜uφ|x=0. The following proposition shall be applied. 13 Proposition 4.1. LetA∈C∞(R)be admissible. Let f∈ D′(R)b... | https://arxiv.org/abs/2503.18515v1 |
form 1 T2/integraldisplayT 0/integraldisplaytψ 0/a\}b∇acketle{t˜ΛND(φ),(τ∗ ˜sχ)2τ∗ ˜s+r˜ΛND(φ)/a\}b∇acket∇i}ht/a\}b∇acketle{tψ,τ∗ ˜sψ/a\}b∇acket∇i}htd˜sdr, (4.32) wheretψ:= diam(supp( ψ))<∞and ˜s=s−r. We would like to show that this converges to zero as T→ ∞. Since the Cauchy-Schwarz inequality implies |/a\}b∇acketle{t... | https://arxiv.org/abs/2503.18515v1 |
everyω∈U. Moreover, the left hand side is lim k→∞lim j→∞/a\}b∇acketle{tCTj1(φk)−CTj2(φk),ψ/a\}b∇acket∇i}ht by (4.34). Thus, lim k→∞lim j→∞/a\}b∇acketle{tCTj1(φk)−CTj2(φk),ψ/a\}b∇acket∇i}ht=/a\}b∇acketle{t˜ΛND1(ϕ)−˜ΛND2(ϕ),ψ/a\}b∇acket∇i}ht The test functions ϕ,ψ∈C∞ c(R) above can be chosen freely. Since A1/\e}atio\slas... | https://arxiv.org/abs/2503.18515v1 |
2x /radicalBig λ2+(M/2)2/parenleftBigg/radicalBig λ2+(M/2)2cosh/parenleftbigg/radicalBig λ2+(M/2)2x/parenrightbigg (5.11) −(λ+M/2)sinh/parenleftbigg/radicalBig λ2+(M/2)2x/parenrightbigg/parenrightBigg (5.12) Theng1andg2are strictly positive on [0,ℓ], and they satisfy ( 5.6) and (5.7). As a consequence, ( 5.8) holds for... | https://arxiv.org/abs/2503.18515v1 |
Let us show, that not only h1=h2but alsoA1=A2holds provided that Λ 1= Λ2. Leta>0. The equation ( A.5) yields A(x)p(x,t0+a) =/integraldisplayt0+a t0A(x)∂tp(x,t)dt=−/integraldisplayt0+a t0∂xu(x,t)dt, where we used the initial value p(x,t0) = 0. The integral is well defined provided enough regularity in f. Integrating agai... | https://arxiv.org/abs/2503.18515v1 |
ination of the Shape of a Scattering Screen From a Passive Measurement”. In: Mathematics 8.7 (July 2020), p. 1156. issn: 2227– 7390.doi:10.3390/math8071156 . [Che+09] J Cheng, J Nakagawa, M Yamamoto, and T Yamazaki. “Uniqu eness in an inverse prob- lem for a one-dimensional fractional diffusion equation”. In: Inverse pr... | https://arxiv.org/abs/2503.18515v1 |
equation with one measurement and the pseudorandom source”. In: Analysis & PDE 5.5 (2012), pp. 887–912. [HLO14] T Helin, M Lassas, and L Oksanen. “Inverse problem for the wave equation with a white noise source”. In: Communications in Mathematical Physics 332 (2014), pp. 933–953. [H¨ or71] L H¨ ormander. “FourierIntegr... | https://arxiv.org/abs/2503.18515v1 |
Minimax Rate-Optimal Inference for Individualized Quantile Treatment Effects in High-dimensional Models Jiachen Sun and Yin Xia Department of Statistics and Data Science, Fudan University Abstract The quantification of treatment effects plays an important role in a wide range of applications, including policy making an... | https://arxiv.org/abs/2503.18523v1 |
many scientific fields (e.g., Abadie et al., 2002). As an example, a recent study examining environmental factors and body mass in- dex (BMI) finds that while the middle distributions of BMI are similar between breastfed 2 and formula-fed children aged 5-6 years, significant differences are present at the upper and low... | https://arxiv.org/abs/2503.18523v1 |
further expands these results to a distributed setting. Additionally, Yan et al. (2023) integrates debiasing estimation with convolution-type smoothed quantile regression to enhance computational efficiency. Leveraging these debiasing approaches, significant advancements have been achieved in high-dimensional treatment... | https://arxiv.org/abs/2503.18523v1 |
framework for confidence intervals of the IQTE ∆ τ,new, which, to the best of our knowledge, is not available in the current literature. Finally, a minimax optimal rate of the detection boundary is developed for the hypothesis testing of ∆ τ,new. This result generalizes the minimax framework for the IATE (Cai et al., 2... | https://arxiv.org/abs/2503.18523v1 |
is interested in the following quantity: ∆τ,new=xT new βτ,1−βτ,2 . From the definition of ∆ τ,new, we can clearly observe the aforementioned two aspects of heterogeneities: the personalized heterogeneity captured by the inclusion of xnew, and the quantile heterogeneity addressed by selecting different quantile levels... | https://arxiv.org/abs/2503.18523v1 |
More precisely, the estimation error of \xT newβτ,kin (7) can be decomposed into two main components: \xT newβτ,k−xT newβτ,k=Uτ,k+ ∆ τ,k, (9) where Uτ,kis a dominating variance term defined by Uτ,k=1 nkcMT τ,knkX i=11 fk,i(F−1 k,i(τ))Xk,iφτ(Yk,i−XT k,iβτ,k), (10) and ∆ τ,kis a bias term defined by ∆τ,k=xT new(bβτ,k−βτ,... | https://arxiv.org/abs/2503.18523v1 |
specific quantile: a smaller value of the sparsity function indicates a higher concentration of the data at this quantile. Therefore, our target linear functionals at the quantile of in- terest are easier to be estimated if the corresponding sparsity function is small. This coincides with the fact that a small sparsity... | https://arxiv.org/abs/2503.18523v1 |
sufficient conditions. Define the covariance matrix asΣk=E Xk,iXT k,i and the precision matrix as Ωk=Σ−1 kfork= 1,2. Denote by fa general density function with support on R, and let Fbe the corresponding CDF. Denote by Ua compact set satisfying U⊊U ⊂(0,1). Condition 1. Suppose that Xk,i∈Rpis sub-Gaussian, i.e. ∥Xk,i∥... | https://arxiv.org/abs/2503.18523v1 |
regression setting. First, the non-differentiable score function φ(·)utilized in the debiasing step (7)introduces technical difficulties. Second, unlike focusing solely on the conditional mean, achieving uniform convergence across a set of quantile levels requires more nuanced techniques, where sophisticated empirical ... | https://arxiv.org/abs/2503.18523v1 |
Donsker theorems for the weak convergence of finite dimensional quantile processes (Koenker and Xiao, 2002; Angrist et al., 2006) are not applicable. Third, the technical tool utilized here is based on Gaussian coupling, which is fundamentally different from the approaches used in existing works on the quantile process... | https://arxiv.org/abs/2503.18523v1 |
rate-optimal. Specifically, we say CI α(xnew,Z)∈ I α(Θ;xnew,Z) attains minimax rate optimality asymptotically, if it satisfies lim n→∞P(L{CIα(xnew,Z)} ≍L∗ α(Θ;xnew)) = 1 . The following theorem presents the minimax expected length for the confidence inter- vals with guaranteed coverage over the parameter space defined ... | https://arxiv.org/abs/2503.18523v1 |
the minimax lower bound particularly challenging. We address this by leveraging theoreti- cal results related to the asymmetric Laplace distribution to achieve the desired minimax optimality. As a consequence, our approach allows ∥xnew∥0to be of the same order as the dimension of the covariates pfor the lower bound der... | https://arxiv.org/abs/2503.18523v1 |
by Belloni et al. (2019). Finally, we adopt the five-fold cross-validation to construct the projection direction cMτ,kin order to reach the proposed debiased estimator b∆τ,new. The comparisons under heavy-tailed distributions are collected in Section B of the Supplementary Material. 4.2 Simulation Results We compare th... | https://arxiv.org/abs/2503.18523v1 |
0.072 0.525 0.082 0.142 0.8 >0 0.407 0.385 0.946 0.952 0.922 1.991 2.082 0.062 0.508 0.059 0.531 0.069 0.614 0.095 0.344 intervals and the rejection rates of their testing procedures, calculated as the proportion of rejected null hypotheses over 1,000 replications in both settings. Note that, under the alternative with... | https://arxiv.org/abs/2503.18523v1 |
5.2 Data Analysis We conduct the analysis based on the research of Patel et al. (2012), utilizing laboratory and questionnaire data sourced from NHANES surveys conducted between 1999 and 2006. Following their initial selection of 127 covariates, we further refine the dataset by excluding variables with missing values, ... | https://arxiv.org/abs/2503.18523v1 |
alized and quantile heterogeneities. Leveraging a novel variance-enhancement constraint, this estimator and its corresponding inference procedures accommodate structure-free loading vectors. We validate the proposed procedures by examining the coverage proba- bility of the confidence interval and the type I error in hy... | https://arxiv.org/abs/2503.18523v1 |
Stat. Methodol. , 83(4):669–719. Cai, T., Tian, L., Wong, P. H., and Wei, L. (2011). Analysis of randomized comparative clinical trial data for personalized treatment selections. Biostatistics , 12(2):270–282. Cai, T. T. and Guo, Z. (2017). Confidence intervals for high-dimensional linear regression: minimax rates and ... | https://arxiv.org/abs/2503.18523v1 |
B., Dawber, T. R., Kagan, A., Revotskie, N., and Stokes, J. (1961). Factors of risk in the development of coronary heart disease—six year follow-up experience: the Framingham study. Ann. Intern. Med. , 55:33–50. Koenker, R. (2005). Quantile regression . Cambridge University Press. Koenker, R. (2010). Rank tests for het... | https://arxiv.org/abs/2503.18523v1 |
optimal personalized treatment rules in consideration of benefit and risk: with an application to treating type 2 diabetes patients with insulin therapies. J. Amer. Statist. Assoc. , 113(521):1–13. Williams, P. T. (2020). Quantile-specific heritability of high-density lipoproteins with implications for precision medici... | https://arxiv.org/abs/2503.18523v1 |
arXiv:2503.18608v2 [stat.ML] 25 Mar 20251–15 AutoBayes: A Compositional Framework for Generalized Variational Inference Toby St Clere Smithe VERSES Research Marco Perin VERSES Research Abstract We introduce a new compositional framework for generalized variat ional inference, clarify- ing the different parts of a model,... | https://arxiv.org/abs/2503.18608v2 |
families of distribution that they may yield; to separate sy ntax (model specification) from semantics (optimization); and to ensure that all of these co mponents interact well with each other. All the same, we have done our best to minimize the dema nds of novel mathematics in the main text of this paper. Their authors... | https://arxiv.org/abs/2503.18608v2 |
ld measures for each element of their domain, which we write in conditional probability sty le, ascpdy|xq. Thus, we denote sets and spaces with upper-case letters and elements of thos e spaces with corresponding 2 AutoBayes lower-case letters. We will assume that each measurable spa ceXis equipped with a canonical meas... | https://arxiv.org/abs/2503.18608v2 |
and later we will have a precise notion of parameter, t oo. Finally, we say “model” rather than “Bayesian network”, as our models will be more ge neral than Bayesian networks. 3 Smithe Perin Remark 3 We wrote in the introduction that our framework does not supp lant PPLs, even though it is intended to be computationall... | https://arxiv.org/abs/2503.18608v2 |
the right. A1BA p qA B p1A1 The string diagram depicts the model q˝ ¨ ppbp1q: 1Ñ˝ ¨B. Note that AbA1is latent. Because the latent space is just a factor of the codomain of th e kernel making up an open model, it can be revealed as part of the observed space by a purely formal manoeuvre; this correspondsto a 2-cell in t... | https://arxiv.org/abs/2503.18608v2 |
the set of kernelsYùX. Applying c:to a priorπPPXyields a kernel c: π:YùXin the opposite direction to c, defined canonically by c: πpdx|yq “cpdy|xqπpdxq pc˚πqpdyq for allyin the support of c˚π. This expression is Bayes’ law, and c: πis called the exact posterior with respect to π. The chain rule for Bayesian inference ( ... | https://arxiv.org/abs/2503.18608v2 |
the kernel defined by ` c1 π˝|d1 c˚π˘ pdx,da,dy,db|zq “c1 πpdx,da|yqd1 c˚πpdy,db|xq. Theorem 13 (Chain rule for open models) Define a function p´q:mapping open modelsc:XÑ˝ ¨Yto exact Bayesian lenses pcq::“ pc,c:q:XÞ ÑY. This function satisfies pd˝ ¨cq:“ pd,d:q ˛ pc,c:q “ pdq:˛ pcq:. That is to say, p´q:isfunctorial . Rema... | https://arxiv.org/abs/2503.18608v2 |
KLpd,d1qforpd,d1q:YÞ ÑZ. This leads one to wonder whether there is a chain rule for the r elative entropy, corresponding to the chain rule for exact inference, and, of course, there is: it is quite easy to show that KL` pd,d1q ˝|pc,c1q˘ pπ,zq “Epy,bq„d1c˚πpzq“ KLpc,c1qpπ,yq‰ `KLpd,d1qpc˚π,zq; (St Clere Smithe ,2023b,§5... | https://arxiv.org/abs/2503.18608v2 |
like the chain rule. Remark 19 Observations like these also seem to underlie the proposals ofKnoblauch et al. (2019) for ‘generalized’ variational inference, and the proposa ls ofKhan and Rue (2023) for a “Bayesian learning rule”. Both sets of authors observe that typical learning objectives (such as ELBOs or free ener... | https://arxiv.org/abs/2503.18608v2 |
programming. Remark 24 An important sanity check on the structure is that the compos ition of statistical games (as with open models and Bayesian lenses) yields a bica tegory. The identity game X/rightarrowtriangleXis given by the identity lens XÞ ÑXequipped with constantly 0energy and entropy. The reader may have note... | https://arxiv.org/abs/2503.18608v2 |
using only the local loss function data, which amount t o the ‘functoriality’ of gradient descent. First, we need to understand how the parameters int eract with the composition. 6. For Khan and Rue, Θ typically picks the natural parameter o f the posterior. But this means that c1is for them not a map Θ ˆPXˆYÑPXbut mer... | https://arxiv.org/abs/2503.18608v2 |
the statistical game data (so that an imp lementation may make use of it) requires annotating the games with predicates—which ca n be done compositionally, but laxly. The principal difficulties are thus in computing the (expecta tions involved in the) gradients, and different models call for different strategies. And just ... | https://arxiv.org/abs/2503.18608v2 |
its exact inversion c:, yielding a lens pc,c:q, and we let lcandHcbe given by the negative log-likelihood and entropy respectively. T o construct the game 1/rightarrowtriangleM, we note that the inversion (and thus entropy) are trivial, and l etlπagain be given by the negative log-likelihood. Under these circumstances,... | https://arxiv.org/abs/2503.18608v2 |
into a product over ΘandXindependently, then the cup trivializes the factor over X, leaving only a posterior over Θ. The classic situation in Bayesian deep learning then corresponds to optimizing the p arameters of this posterior — but note that, in this case, both cand the prior may themselves be complex models constr... | https://arxiv.org/abs/2503.18608v2 |
Differentially Private Joint Independence Test Xingwei Liu1, Yuexin Chen1, Wangli Xu1∗ 1Center for Applied Statistics and School of Statistics, Renmin University of China, Beijing, China Abstract Identification of joint dependence among more than two random vectors plays an important role in many statistical applicatio... | https://arxiv.org/abs/2503.18721v2 |
Pfister et al. (2018) embedded the joint distribution and the product of the marginals into a reproducing kernel Hilbert space (RKHS). They defined the d-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between these embeddings. Chakraborty and Zhang (2019) generalized the notion of Brown... | https://arxiv.org/abs/2503.18721v2 |
empirically. This work serves as an inspiration for our differentially private joint independence test. Another different line of work aims to obtain private versions of existing hypothesis tests in a black-box fashion. The main idea to this end is the subsample-and-aggregate framework (Nissim et al., 2007). This metho... | https://arxiv.org/abs/2503.18721v2 |
(Ferrando et al., 2022; Alabi and Vadhan, 2023) first constructs private parameters and then uses para- metric bootstrap to conduct inference. However, the parametric bootstrap is not ap- propriate for our problem since dHSIC is a non-parametric method. Inference based on non-parametric bootstrap such as Wang et al. (2... | https://arxiv.org/abs/2503.18721v2 |
have P(M(Xn;w)∈S| Xn, w)≤eϵP(M(eXn;w)∈S|eXn, w) +δ. Theϵandδin Definition 2.1 are referred to as privacy parameters. The smaller ϵandδ indicate the stricter privacy guarantee, where the probability distributions of M(Xn;w) and M(eXn;w) are forced to be similar, so the output of Mis not very sensitive to small changes i... | https://arxiv.org/abs/2503.18721v2 |
Rddue to the continuity of the kernel function. In addition, we define product kernels on X1×···× Xd. For j∈[d], let kj:Xj×Xj→R be a continuous, bounded, positive semi-definite kernel and denote by Hjthe corresponding RKHS. Throughout the paper, a superscript j∈[d] always denotes an index rather than an 6 exponent. Ass... | https://arxiv.org/abs/2503.18721v2 |
instead of the resampling p-valuebpfor the α-level test. With this in mind, the undesirable dependence on Bis removed via representing the test function with the quantile, where both the statistic T(Xn) and the quantile of T(Xφin) have sensitivity ∆ T. This representation removes the unpleasant factor Band introduces t... | https://arxiv.org/abs/2503.18721v2 |
in implementation. In this section, we use dHSIC (Pfister et al., 2018) along with the methodology in Section 3 to construct a DP joint independence test. Our goal is to determine whether the components of a random vector X= (X1, . . . , Xd) defined on X1×···× Xdare joint independent in private regime, based on an i.i.... | https://arxiv.org/abs/2503.18721v2 |
domain can be a subset of the natural domain. For example, when investigating the associations between single nucleotide polymorphisms (SNPs), the value of xis restricted to {0,1,2}, leading to linear kernel bounded between 0 and 4. •For dHSIC, the differentially private bootstrap may only has asymptotic level by Pfist... | https://arxiv.org/abs/2503.18721v2 |
x, x′∈Xjandj∈[d]. Then ϕdpdHSIC satisfies (i) (Privacy Guarantee) For ϵ >0andδ∈[0,1),ϕdpdHSIC is(ϵ, δ)-differentially private. (ii) (Validity) The type I error of ϕdpdHSIC is controlled at level αnon-asymptotically. (iii) (Pointwise Consistency) Assume that n−1ξ−1 ϵ,δ→0asn→ ∞ . Then for any sequence Bnsuch that min n≥1... | https://arxiv.org/abs/2503.18721v2 |
any α∈(0,1),β∈(0,1−α),ϵ >0, δ∈[0,1]andB≥6α−1log(2β−1), the minimum separation for ϕdpdHSIC satisfies ρϕdpdHSIC (α, β, ϵ, δ, n )≤CKmaxr max{log(1/α),log(1/β)} n,max{log(1/α),log(1/β)} nξϵ,δ , where CKis a positive constant that only depends on Kjforj∈[d]. 12 Assume αandβare fixed to simplify our discussion, and a few ... | https://arxiv.org/abs/2503.18721v2 |
for any fixed c <1/2,ϵ >0andδ∈[0,1). Assume that the kernel functions kjare translation invariant on Rpjforj∈[d]. In particular, there exist some functions κj such that kj(x, x′) =κj(x−x′)for all x, x′∈Rpj. Moreover, assume that there exist positive constants ηjsuch that κj(0)−κj(x0)≥ηjfor some xj 0∈Rpj. Then the minim... | https://arxiv.org/abs/2503.18721v2 |
the collection distributions PX1,...,XdonRp1× ··· × Rpd, where PX1,...,Xdequipped with the Lebesgue density function pX1,...,Xdand the product of the marginals pX1. . . p Xdsuch that ∥pX1,...,Xd−pX1. . . p Xd∥L2≥ρ. Here ∥ · ∥ L2refers to the L2norms on Rdwith the Lebesgue measure. Consider a subset of PL2(ρ) defined as... | https://arxiv.org/abs/2503.18721v2 |
(III), respectively, in mid and high privacy regimes. Therefore, term (IV) has no impact on the rate for balanced vector dimensions. In the low privacy regime where the first term (I) dominates (II) and (III), by setting λj,i=n−2/(4s+p), we can achieve the optimal separation rate n−2s/(4s+p)over the Sobolev ball. This ... | https://arxiv.org/abs/2503.18721v2 |
, Xj−1) for all j∈ {2, . . . , d }. This allows us to reduce the complicated joint independence test to a series of d−1 pairwise independence tests. As such, we can conduct dpHSIC for d−1 times and then combine the results by Bonferroni correction. We summarize the procedure as Algorithm 2. This algorithm is ( ϵ, δ)-DP... | https://arxiv.org/abs/2503.18721v2 |
set level α= 0.05 and permutation times B= 200 throughout the simulations. We fix the privacy parameter δ= 0 and vary the parameter ϵto control the level of privacy. 18 Simulation 1 (testing level). Consider the data X= (X1, . . . , Xd) are generated as X∼N(0, Id). Then it holds that PX1,...,Xd=⊗d j=1PXj∈H0, in (4.1). ... | https://arxiv.org/abs/2503.18721v2 |
which may be attributed to its efficient utilization of the optimal differential privacy mechanism tailored for binomial variables. Nevertheless, the power of TOT dHSIC is inherently limited for a fixed sample size, primarily due to the unavoidable impact of data splitting. Consequently, TOT dHSIC excels in maintaining... | https://arxiv.org/abs/2503.18721v2 |
method to privately integrate the public results for subsample-and-aggregate methodology, it demonstrates even lower statistical power than SAR dHSIC in low privacy regime. This discrepancy may stem from the differing recommendations for parameter selection as outlined in the respective original papers. For the second ... | https://arxiv.org/abs/2503.18721v2 |
G, consider the following procedure. •Use generalized additive model regression Wood and Augustin (2002) to regress each node Xjon all its parents PAjand denote the resulting vector of residuals by resj. •Perform dHSIC to test whether (res1, . . . , resd) is jointly independent. •If (res1, . . . , resd) is jointly inde... | https://arxiv.org/abs/2503.18721v2 |
Glucose Concentration on Diastolic Blood Pressure. Our aim is to test whether the candidate DAGs are rejected by the DAG verification method in privacy context. Given that the randomness of permutation and the added noise, we perform 1000 repetitions to study the empirical rejection rate under the level α= 0.05. The pe... | https://arxiv.org/abs/2503.18721v2 |
Cam. In Conference on Algorithmic Learning Theory , pages 48–78. Alabi, D. and Vadhan, S. P. (2023). Differentially private hypothesis testing for linear re- gression. Journal of Machine Learning Research , 24:1–50. Albert, M., Laurent, B., Marrel, A., and Meynaoui, A. (2022). Adaptive test of independence based on HSI... | https://arxiv.org/abs/2503.18721v2 |
, 35:651–670. Dong, J., Roth, A., and Su, W. (2022). Gaussian differential privacy. Journal of the Royal Statistical Society Series B: Statistical Methodology , 84:3–37. Drton, M., Han, F., and Shi, H. (2020). High-dimensional consistent independence testing with maxima of rank correlations. The Annals of Statistics , ... | https://arxiv.org/abs/2503.18721v2 |
Distribution-free joint independence testing and robust independent component analysis using optimal transport. Arxiv , arXiv:2211.15639. Pena, V. and Barrientos, A. F. (2025). Differentially private hypothesis testing with the subsampled and aggregated randomized response mechanism. Statistica Sinica , 35:671– 691. Pf... | https://arxiv.org/abs/2503.18721v2 |
dHSIC (Appendix E.2). Appendix B Theoretical Results on the U-statistic The U-statistics are often enjoys the optimal separation rate in non-private regimes (Albert et al., 2022; Kim et al., 2022). In this section, we explore the private dHSIC framework based on a U-statistic, which exhibits sub-optimal separation perf... | https://arxiv.org/abs/2503.18721v2 |
as Theorem B.2. Theorem B.2 (Minimum Separation of ϕdpdHSIC overPs L2).Assume that α∈(0,1),β∈ (0,1−α)are fixed, and ϵ >0,δ∈[0,1),B≥6α−1log(2β−1),λj≤1forj∈[d]. The 32 minimum separation of ϕu dpdHSIC with the Gaussian kernels over Ps L2is upper bounded as ρ2 ϕu dpdHSIC,L2≤Cα,β,s,R,M,p 1,...,pddX j=1pjX i=1λ2s j,i+1 n√λ... | https://arxiv.org/abs/2503.18721v2 |
ln1−α 1−p′>0 asp′>(α+p)/2, and cis some constant not depending on nandp′. Hence the probability is less than 1 as p′> αandB+ 1> α−1. The probability above along with (C.1) yields PP1 B+ 1BX i=11(M0≤Mi) + 1 ≤α < cµ(An) +µ(Ac n)<1. Taking n→ ∞ yields the desired result. 34 C.2 Proof for Proposition 4.1 and Examples W... | https://arxiv.org/abs/2503.18721v2 |
completes the induction foundation. Second, suppose (C.3) holds for a certain d≥2, then we are to prove it also holds for d+ 1. In fact, simple calculation leads to S2=1 nd+1d+1Y j=1nX i=1ψj(Xj i) =1 nddY j=1nX i=1ψj(Xj i)·1 nnX i=1ψd+1(Xd+1 i) =1 nddY j=1nX i=1ψj(Xj i)·1 nnX i=1{ψd+1(eXd+1 i) + (ψd+1(Xd+1 i)−ψd+1(eXd+... | https://arxiv.org/abs/2503.18721v2 |
now, we have completed the proof of the upper bound the permutation dHSIC. It worth remarking that, a sharper inequality in step (iv) can be derived by discussing the second term more meticulously. One can reformulate the expression and expand the averages to identify and cancel out same terms, leading to a sharper bou... | https://arxiv.org/abs/2503.18721v2 |
n3 =16(n−2)2 n4K1K2+C1ε1ε2+C2ε1+C3ε2, where C1, C2, C3are some constants that only depend on K1, K2, n. A similar calculation shows \dHSIC2(eXπ n) =4 n4K1K2+C′ 1ε1ε2+C′ 2ε1+C′ 3ε2. Combining them together yields that |\dHSIC( Xπ n)−\dHSIC( eXπ n)| ≥4(n−2.5) n2√ K1K2+h(ε1, ε2, K1, K2, n), where h(ε1, ε2, K1, K2, n) is ... | https://arxiv.org/abs/2503.18721v2 |
research. C.2.4 Inconsistency for the DP Bootstrap dHSIC In this subsection, we first demonstrate that the DP Bootstrap dHSIC is inconsistent against some fixed alternatives, and next further show that it is inconsistent against all alterna- tives for some α, ξϵ,δ. Denote the sensitivity of \dHSIC( Xb n) as ∆b T≥3√K0/8... | https://arxiv.org/abs/2503.18721v2 |
completes the proof. C.4 Proof for Theorem 4.2 Proof. With a little abuse of notations, C1, C2, . . .refer to universal constants only associated with the upper bounds for the kernels. By the quantile representation of resampling test, the type II error can be expressed as E[1−ϕdpdHSIC ] =P(bpdp> α) =P(M0≤q1−α,B), 42 w... | https://arxiv.org/abs/2503.18721v2 |
inf ϕ∈Φα,∞sup P∈PdHSICk(ρ)EP[1−ϕ]≤β ≥Cηmin{p log(1/(α+β)/n),1} by the definition of the minimax separation. For this direction, it is sufficient to pick a joint distribution PX1,...,Xd,0fromPdHSIC k(eρ) with eρ=Cηminr log(1/(α+β) n,1 (C.7) 44 such that the type II error rate of any test is higher than β. On the othe... | https://arxiv.org/abs/2503.18721v2 |
between P⊗n X1,...,Xd,0and⊗d j=1P⊗n Xj,0with D= 10E[dham(Xn,X′ n)]. Then for α+β < c for any fixed c <1/2, we have inf ϕ∈Φα,ϵ,δsup P∈PdHSICk(ρ)EP[1−ϕ]≥inf ϕ∈Φα,ϵ,δEPX1,...,X d,0[1−ϕ] (i) ≥e−Dϵ(0.9−α)−Dδ >(0.4 +β)e−Dϵ−Dδ ≥(0.2 +β)e−Dϵ, 46 provided that the condition Dδ≤0.2e−Dϵholds. Here step (i) holds by the proof of T... | https://arxiv.org/abs/2503.18721v2 |
equivalent to the permutation distribution of Uπ,dHSIC +D1+D2,πwhere D2,πhas the same form of D2but computed based on permuted data Xπ n. Denoting the sensitivity ofp Vπ,dHSIC as ∆ V1/2, Lemma E.1 yields that dpdHSIC test rejects the null if and only if p Vπ,dHSIC +2∆V1/2 ξϵ,δζ0> q1−α,B, where q1−α,Bis 1−αquantile of {... | https://arxiv.org/abs/2503.18721v2 |
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