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that investigated the clinical efficacy of AI-based healthcare interventions. Among these, only 3 trials reported statistically significant primary outcomes, including: • An AI-powered blood glucose reminder system (Nayak et al. [2023]), • The Wysa AI chatbot for mental health support (MacNeill et al. [2024]), • An AI ... | https://arxiv.org/abs/2503.21138v5 |
developed to address general causal inference problems—such as CausalML (Zhao and Liu [2023]), EconML (Battocchi et al. [2019]), DoWhy (Blöbaum et al. [2024], Sharma and Kiciman [2020]), CauseBox (paras2612), CausalNex (quantumblacklabs), Causal Curve (Kobrosly [2020]), CausalDiscovery (Kalainathan et al. [2020]), pcal... | https://arxiv.org/abs/2503.21138v5 |
+r 1 2naln(2 σ)), Eemp(b) +r 1 2nbln(2 σ)})≥1−σ , where ∆ˆf=ˆf(c, sa)−ˆf(c, sb),naandnbare number of independently, randomly, identically sampled error measurements (IIDE) where s=saands=sb,0<1−σ < 1is confidence, evaluation error function Lis mean square error ranged from 0 to 1. Proof. Because evaluation error functi... | https://arxiv.org/abs/2503.21138v5 |
Website and license of the mentioned asset. The heterogeneous mini agent space is configured as a combination of a linear mini agent space (logistic regression or linear regression) and a non-linear mini agent space (MLP Classifier or MLP Regressor). The agent’s input dimension is the number of features, and output dim... | https://arxiv.org/abs/2503.21138v5 |
Trade 160∗200∗30 = 960000 40∗200∗20 = 160000 Table 5: Evaluation sample numbers in different scenes. 25 D.1 Scenes to test evaluation model For 11 scenes to test evaluation model, 2000 agents was randomly sampling from heterogeneous agent space. 20% data was used to build the real systems to generate true evaluation me... | https://arxiv.org/abs/2503.21138v5 |
learning models for observation data as base learners of our evaluation models, such as CEV AE (Louizos et al. [2017]), BNR (Li and Fu [2017]), BNN (Johansson et al. [2016]), SITE (Yao et al. [2018]), GANITE (Yoon et al. [2018]), DeepMatch (Kallus [2020]), Dragonnet (Shi et al. [2019]), TARNet/CFR (Shalit et al. [2017]... | https://arxiv.org/abs/2503.21138v5 |
base learners. The name of each row is “EvaluationModelName”-“TestName”- “MetricName”. The name of each column is the scene name. 27 Ratio p >= 0.05 withdraw higgs grid insurance climate alert Het(Linear)-IID-ROCAUC 0.96 0.93 0.93 0.93 1.0 0.97 Het(Linear)-IID-ACC 0.93 0.97 0.9 0.9 0.97 0.97 Het(Linear)-ID-ROCAUC 1.0 0... | https://arxiv.org/abs/2503.21138v5 |
our ID check and Bias check, the dramatically empirical error reduction of our evaluation model is still believed as valuable if upper bound existed. Of course it can be not used to calculate the upper bound of the two evaluation models by theorem 4 directly in this scene. Ratio p≥0.05 A-Share Trade Het(Linear)-IID 1.0... | https://arxiv.org/abs/2503.21138v5 |
{} } We also use the Grid search for the hyperparameter searching in quantum trade scene, the searched range is listed as following, the cross validation for the searching is 3-fold cross validation. "CatBoost" = { ‘iterations’: [100, 200], ‘depth’: [6, 8], ‘learning_rate’: [0.01, 0.1], ‘l2_leaf_reg’: [1, 3], ‘border_c... | https://arxiv.org/abs/2503.21138v5 |
0.17976 0.10362 0.02825 Het-Linear 0.01850 0.01438 0.00696 0.01887 0.03357 0.01391 Het-MLP 0.04886 0.03994 0.03924 0.05166 0.06719 0.04581 Het-SVR 0.07261 0.03029 0.03749 0.07211 0.06634 0.07091 Het-RF 0.03444 0.01641 0.00973 0.03068 0.05216 0.02255 Het-LGBM 0.02929 0.01526 0.00972 0.02610 0.04359 0.02255 Het-XGBoost 0... | https://arxiv.org/abs/2503.21138v5 |
0.000589 E+ 0.00122 E+ 0.00152 E+ 0.00186 10000000 E+ 0.000186 E+ 0.000387 E+ 0.000480 E+ 0.000588 100000000 E+ 0.0000589 E+ 0.000122 E+ 0.000152 E+ 0.000186 1000000000 E+ 0.0000186 E+ 0.0000387 E+ 0.0000480 E+ 0.0000588 Table 14: Upper bound of normalized generalized error. E is the empirical prediction error. Samples... | https://arxiv.org/abs/2503.21138v5 |
arXiv:2503.21275v1 [math.ST] 27 Mar 2025Use of stochastic orders and statistical dependence in error analysis for multi-component system Subarna Bhattacharjee1∗, Aninda Kumar Nanda2, Subhashree Patra3 1,3Department of Mathematics,Ravenshaw University, Cuttack -753003, Odisha, India 2Indian Statistical Institute, Delhi ... | https://arxiv.org/abs/2503.21275v1 |
s of deviations from inde- pendence when the lifetimes of the components of a series sys tem are exponentially distributed. They have assumed the joint distribution to follow a Gumbel b ivariate exponential model. Klein and Moeschberger (1986) studied the magnitude of the e rrors under a similar as- sumption about the ... | https://arxiv.org/abs/2503.21275v1 |
error bo und, we could decide if we assume independence and prefer mathematical simplicity (i f we know the error is within our allowed limit), or keep the mathematical complexity and get accurate results without assuming independence. We know that errors occur in different functions, viz. SF,FR,RFR,MRL, andAI due to th... | https://arxiv.org/abs/2503.21275v1 |
consideration. Definition 2.2 LetT1,T2,...,T nbe random variables. T1,T2,...,T nare said to be positively (negatively) lower orthant dependent PLOD (NLOD) if P(T1≤t1,T2≤t2,...,T n≤tn)≥(≤)n/productdisplay i=1P(Ti≤ti), i.e.,FT1,T2,...,Tn(t1,t2,...,tn)≥(≤)/producttextn i=1FTi(ti)where(t1,t2,...,tn)∈Rn. The next proposition... | https://arxiv.org/abs/2503.21275v1 |
and Sh anthikumar (2007)) by the following definition. Definition 2.4 LetXandYbe two random variables with SF,FR,RFR,MRL,AI denoted by¯FX,rX,µX,MX,LXand¯FY,rY,µY,mY,LYrespectively. Then (i)Xis said to be less (greater) than Yin usual stochastic order ( ST), denoted by X≤ST(≥ST)Yif¯FX(t)≤(≥)¯FY(t)for allt≥0. (ii)Xis said ... | https://arxiv.org/abs/2503.21275v1 |
faster order, respectively. If there is lrorder between systems that have dependent and independent c omponents, then one can accordingly conclude about the assessment of ot her functions. The next theorem helps us to determine the sign of relative er ror of survival function if the sign of failure rate function is kno... | https://arxiv.org/abs/2503.21275v1 |
are all non-negativ e. The SF,FR,RFR andAI function of a series system with lifetime, denoted by TDformed out of ndependent components havingMOME distribution are obtained using (3.5) and (2.1), FD(t) = exp(−λt),rD(t) =λ,µD(t) =λ exp/parenleftbig λt/parenrightbig −1,LD(t) = 1, where λ=/parenleftign/summationdisplay i=... | https://arxiv.org/abs/2503.21275v1 |
θ(t) with respect to t. InMGImodel, relative error in SFis decreasing in t,forFR, it is increasing in t. The relative error inAIfunction is decreasing in t,An upperboundof relative error in AIfunction is ( n−1), (by applying Lemma 3.2). A similar study can be extended for other multivariate expon ential distributions. ... | https://arxiv.org/abs/2503.21275v1 |
parameter. TheSF,FR,RFR andAIof a series system formed out of ncomponents having MCW are respectively given by FD(t) = exp/braceleftig γl−/parenleftbig γ+n/summationdisplay i=1λitαi/parenrightbigl/bracerightig ,rD(t) =l/parenleftig γ+n/summationdisplay i=1λitαi/parenrightigl−1/parenleftign/summationdisplay i=1λiαi... | https://arxiv.org/abs/2503.21275v1 |
I Lu and Bhattacharyya I As an application, Hougaard(1986) used the above Weibull mo del on a data on tumour appear- ance in 50 litters of female rats. Each litter contained one d rug treated and two control rats. The data was studied to find estimate of failures in marginal d istribution, maximized likelihood function ... | https://arxiv.org/abs/2503.21275v1 |
−/summationtext/summationtext/summationtext 1≤i<j<k≤n¯F(−∞,···,−∞, ti/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ith,−∞,···,−∞, xj/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright jth,−∞,···,−∞, xk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright kth,... | https://arxiv.org/abs/2503.21275v1 |
is given by F(x1,x2,...,x n) = exp/braceleftig −n/summationdisplay i=1λitαi i/bracerightig . Hence, ¯FT(t) =n/summationdisplay k=1(−1)k−1/summationdisplay ···/summationdisplay 1≤i1<···<ik≤nexp −k/summationdisplay p=1λiptαip , fT(t) =n/summationdisplay k=1(−1)k−1/summationdisplay ···/summationdisplay 1≤i1<···<... | https://arxiv.org/abs/2503.21275v1 |
arXiv:2503.21286v1 [math.ST] 27 Mar 2025Use of copula functions in error assessment due to deviation from dependence assumption Subarna Bhattacharjee1∗, Aninda Kumar Nanda2,Subhashree Patra3 1,3Department of Mathematics, Ravenshaw University, Cuttack -753003, Odisha, India 2Indian Statistical Institute, Delhi Centre, I... | https://arxiv.org/abs/2503.21286v1 |
can be inferred on the basis of copula functions. The magnitude of error or relative er ror can be expressed in terms of copula functions. The role of stochastic orders in knowing the erro r assessment is discussed in Table 2 and we demonstrate it for some well-known copula functions. The fin dings of this article are d... | https://arxiv.org/abs/2503.21286v1 |
consider FI P(·) andFD P(·).The error assessment of a series (parallel) system corresponding to a particular function is all about analyzing the difference between dependent and independent counterpart. In upcoming theorem, we prove that for any multivariate probability distribution, there is stochastic ordering between... | https://arxiv.org/abs/2503.21286v1 |
copu la for a given copula and vice-versa. Interestingly, for radially symmetric copulas, for which C≡ˆC,replacing u(ˆu) by (1−u)(u) in a copula C(u)(ˆC(ˆu)), one may get ˆC(ˆu)(C(u)). Proposition 2.1 The relation between joint survival and joint distribution function of random variables X1,X2,...,X n,is given by P(X1>... | https://arxiv.org/abs/2503.21286v1 |
rate of a series system, we have µD S(t)−µI S(t) =d dtln/parenleftBig 1−F(t,t,...,t)/parenrightBig −d dtln/parenleftBig 1−n/productdisplay i=1FTi(t)/parenrightBig =d dtln/parenleftBig1−F(t,t,...,t) 1−/producttextn i=1FTi(t)/parenrightBig =d dtln/parenleftBig1−ˆC(ˆu) 1−/producttextn i=1ˆui(t)/parenrightBig , whereµD S(t... | https://arxiv.org/abs/2503.21286v1 |
assessment arisin g due to the one of the biased assumption that components work independently, though they are not, so as to make the computations easy and simple. All through out the upcoming sections, we use notations viz., C1,C2,...,C 11so as to comprehend the present study involving several copulas. In each case, ... | https://arxiv.org/abs/2503.21286v1 |
we take up Gumbel-II distribution, and note that it can be obtained from FGM survival copula for a particular choice of ˆ ui. Example 3.1 The survival function of bivariate Gumbel II is ¯F(t1,t2) =/braceleftBig 1+α/parenleftbig 1−e−λ1t1/parenrightbig/parenleftbig 1−e−λ2t2/parenrightbig/bracerightBig e−λ1t1−λ2t2,|α|<1,λ... | https://arxiv.org/abs/2503.21286v1 |
its importance in esta blishing stochastic orderings between lifetime of parallel and series systems. Remark 3.1 IfC5represents the distribution copula of Gumbel-Barnet copul a then i.e., TD P≤rhrTI P, givingTD P≤stTI P.On the other hand, if C5represents the survival copula of Gumbel-Barnet copula the n TD S≥hrTI S.The... | https://arxiv.org/abs/2503.21286v1 |
that the minimum of two increasing functions is increasing. Now, consider the in creasing functions f1,f2,...,f n. Let us define gk(x) = min {F1(x),F2(x),...,F k(x)},for allk∈ {2,...,n}.Clearly, g2is increas- ing. That implies g3(x) = min {g2(x),F3(x)}is also increasing. In fact, if gkis increasing, then gk+1(x) = min{g... | https://arxiv.org/abs/2503.21286v1 |
C10(u) =/bracketleftBig/braceleftBig αminiui/bracerightBigm +/braceleftBig (1−α)/producttextn i=1ui/bracerightBigm/bracketrightBig1/m Theorem 3.10 for0≤θ≤1,ˆC10 ˆC1↓,i.e.,TD P≤rhrTI P; for−1≤θ≤0,ˆC10 ˆC1is constant ( TD PandTI Phave PRHR) until u1+u2<1, ✷ Theorem 3.11C10(u) C1(u)is decreasing in t,sinceminiui(t)/produc... | https://arxiv.org/abs/2503.21286v1 |
P≤st(≥st)TI Pifθ >(<)0, Farlie-Gumbel-MorgensternˆC2 ˆC1↑(↓) ifα >(<)0,i.e.,TD S≥hr(≤hr)TI S. So,TD S≥mrl(≤mrl)TI S,TD S≥st(≤st)TI Sifα >(<)0, C3(u) =/parenleftBig/producttextniui/parenrightBig/braceleftBig 1 +α/producttextni=1/parenleftBig 1−u1/r i/parenrightBig/bracerightBigr C3 C1↓(↑) ifα >(<)0, TD P≤rhrTI P⇒TD P≤st... | https://arxiv.org/abs/2503.21286v1 |
(UA) α <0 : hr,sf,mrl:UA Fischer and Kock C4(u) =/braceleftBig/summationtextni=1u−α i−1/bracerightBig−1 α Clayton C5(u) = exp/bracketleftBig/braceleftBig −/summationtextni=1/parenleftbig −lnui/parenrightbigα/bracerightBig1 α/bracketrightBig Gumbel-Hougaard C6(u) =/parenleftBig/producttextni=1ui/parenrightBig exp/parenl... | https://arxiv.org/abs/2503.21286v1 |
Génération de Matrices de Corrélation avec des Structures de Graphe par Optimisation Convexe Ali F AHKAR1Kévin P OLISANO1Irène G ANNAZ2Sophie A CHARD1 1Univ. Grenoble Alpes, CNRS, Grenoble INP, Inria, LJK, F-38000 Grenoble, France 2Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000 Grenoble, France Résumé – Ce tra... | https://arxiv.org/abs/2503.21298v1 |
cadre du programme France 2030, référence ANR-23-IACL-0006.revue les travaux connexes. La Section 4 décrit l’approche proposée, et la Section 5 présente les résultats ainsi qu’une comparaison avec d’autres approches. 2 Notations Une matrice réelle symétrique Ade dimension p×pest semi- définie positive (SDP) si x⊤Ax≥0po... | https://arxiv.org/abs/2503.21298v1 |
de probabilité à adopter de telle sorte que l’échantillonnage des matrices résultantes soit uniforme sur l’ensemble C. En utilisant cette paramétrisation polaire, il est facile d’incorporer la contrainte uij= 0pour tous (i, j)∈E pour obtenir U∈ U(G). Ceci assure que C=UU⊤∈ C(G) pour les graphes cordaux uniquement. Dans... | https://arxiv.org/abs/2503.21298v1 |
l’intervalle [−1,1]. Pour le motif E, nous utili- sons les différents modèles de graphes aléatoires mentionnés à la Section 2. Le problème d’optimisation (5)est résolu en utilisant la bibliothèque Python CVXPY [9]. Nous l’avons appliqué à ces modèles de graphes sur 50 exécutions2. Dans certains cas, la résolution de (5... | https://arxiv.org/abs/2503.21298v1 |
3Pour être plus précis, avec cette étape de post-traitement, la valeur moyenne change, et donc la contrainte (4)peut ne pas être satisfaite. Augmen- terbàb(1 +ϵ)permet d’atteindre notre objectif.matrices de corrélation observées, ce qui peut ne pas toujours être disponible en pratique. Dans notre contexte, nous sommes ... | https://arxiv.org/abs/2503.21298v1 |
0.00 0.25 0.50 0.75 Valeur seuil (b)0.00.20.40.60.81.0Densité des liens (d) 0.20.40.60.81.0 Proportion de solutions trouvées(a)Erd˝ os-Rényi 0.75 0.50 0.25 0.00 0.25 0.50 0.75 Valeur seuil (b)0.20.40.60.81.0Densité des liens (d) 0.20.40.60.81.0 Proportion de solutions trouvées (b)cordaux 0.75 0.50 0.25 0.00 0.25 0.50 0... | https://arxiv.org/abs/2503.21298v1 |
volume 733. John Wiley & Sons, 2012. [16] Harry JOE: Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis , 97(10): 2177–2189, 2006. [17] Angelika KIMMIG , Lilyana MIHALKOVA et Lise GETOOR : Lifted graphical models : a survey. Machine Learning , 99:1–45, 2015. [18] Daph... | https://arxiv.org/abs/2503.21298v1 |
arXiv:2503.21334v4 [math.ST] 23 May 2025Safety of particle filters: Some results on the time evolutio n of particle filter estimates Mathieu Gerber University of Bristol Abstract Particle filters (PFs) is a class of Monte Carlo algorithms th at propagate over time a set of N∈N particles which can be used to estimate, in a... | https://arxiv.org/abs/2503.21334v4 |
localization of the vehicle during its whole life time. I n this context, instead of ( 1) 1 a more relevant theoretical guarantee for PFs would be that, in so me sense, supt≥1|ˆηN t(f)−ˆηt(f)| →0 asN→ ∞and for all functions fbelonging to some class of functions. 1.2 Contributions of the paper In this paper we first show... | https://arxiv.org/abs/2503.21334v4 |
t−ˆηt∝bardbl ≥κ)≤q for allN≥N′ κ,T,q:=C′ κ(log(T+e)(1 + log(1 /q)) and for some finite constant C′ κ. We also note that it has not been realized (and therefore proved) in this reference t hatNmustincrease with Tto ensure thatP(supt∈{1,...,T}|ˆηN t(f)−ˆηt(f)| ≥κ) remains bounded by qasTincreases (and thus that for any fix... | https://arxiv.org/abs/2503.21334v4 |
in Se ction4and Section 5 concludes. All the proof are gathered in the appendix. 2 Model and filtering algorithms 2.1 The Model We let (Yt)t≥1be a sequenceof R-valued randomvariablesand throughoutthis workwe considerthe SSM which assumes that, for some constants ρ∈Rand (σ,c)∈(0,∞)2, and some distribution η1∈ P(R), the s... | https://arxiv.org/abs/2503.21334v4 |
with s tratified resampling in the de- randomization process of the vanilla PF (i.e. of the PF with multinomial r esampling) by also introducing, at each time t≥1, some dependence in the U(0,1) random numbers {Un t,2}N n=1used to perform the mutation step. In addition, for all t≥1 theNdependent U(0,1)2random variables {... | https://arxiv.org/abs/2503.21334v4 |
1and with δN,T= supt∈{1,...,T}D∗({un t}N n=1). We stress that, in the lemma, we did not aim at optimizing the assumptio ns on ({un t}N n=1)≥1but rather at obtaining a result that holds under weak assumptions on t his sequence of sets. In particular, when{un t}N n=1is the realization of an RQMC point set the upper bound... | https://arxiv.org/abs/2503.21334v4 |
intuitive reason. On the one hand, since for all t≥1 the uniform random variables {Un t,2}N n=1 used by a PF are independent, each mutation step of the algorithm m oves all the particle outside any given interval [ a,b]⊂Rwith a constant positive probability. On the other hand, because th e different mutation steps of a ... | https://arxiv.org/abs/2503.21334v4 |
t=1beTindependent sets ofNindependent U(0,1)srandom variables. Then, P/parenleftBig sup t∈{1,...,T}D∗ s/parenleftbig {Un t}N n=1/parenrightbig ≤δ/parenrightBig ≥1−q,∀N≥Ns,T,q:=160s+33log( T)−33log(q) δ2. By combining Lemma 1and Lemma 3we easily obtain the following result for PF with multinomial resampling: Theorem 1. ... | https://arxiv.org/abs/2503.21334v4 |
In these two examples the function fis unbounded and thus, informally speaking, even a small estimation error of ˆ ηtin a region R⊂Rwhere inf x∈R|f(x)|is large will have a significant impact on the overall estimation error |ˆηN t(f)−ˆηt(f)|. 4 Almost sure behaviour of sequential quasi-Monte Carlo 4.1 Remainder on scramb... | https://arxiv.org/abs/2503.21334v4 |
using Lemma 2and Theorem 2we can easily obtain almost sure time uniform bounds for the SQMC estimates of the filtering and predictive expectations (ˆ ηt(f))t≥1and (ηt(f))t≥1for a large class of functions. For instance, for SQMC the conclusion of Proposition 4holds with T=∞,q= 1 and with δN,T,qreplaced by δN. 4.3 Discuss... | https://arxiv.org/abs/2503.21334v4 |
a bound for/vextenddouble/vextenddouble1 N/summationtextN n=1δ{Xn t}−ˆηN t−1M/vextenddouble/vextenddouble (Lemma 8in the appendix) and a stability result for the particles generated by the algorithm (Lemma 9 in the appendix). The first of these three building blocks is standard , in the sense that in the literature on P... | https://arxiv.org/abs/2503.21334v4 |
J. (2001). Interacting particle filtering with discrete-time observa tions: Asymp- totic behaviour in the Gaussian Case , pages 101–122. Birkh¨ auser Boston, Boston, MA. Dick, J. and Pillichshammer, F. (2010). Digital nets and sequences: discrepancy theory and quasi-M onte Carlo integration . Cambridge University Press.... | https://arxiv.org/abs/2503.21334v4 |
we let Mr,s2(z,dx) =N1(rz,s2) andGs(z) =e−sz2/2, and for any bounded function H:R→(0,∞) we let ΨH:P(R)→ P(R) be defined by ΨH(π) =H(x)π(dx) π(H), π∈ P(R). Finally, we let ϕ(·) and Φ( ·) denote the probability and cumulative density function of the N1(0,1) distribution, respectively, and for all ( a,r)∈R2ands∈(0,∞) we le... | https://arxiv.org/abs/2503.21334v4 |
On the other hand, since fis continuous on [0 ,∞) we have c∞= lim k→∞ck+1= lim k→∞f(ck) =f( lim k→∞ck) =f(c∞) showing that c∞=c⋆, and thus lim k→∞ck=c⋆. Together with ( 16), this implies that there exists a k⋆∈Nsuch that ck⋆>|ρ|−(1+cσ2) σ2 14 and thus, using ( 15) and the fact that the sequence ( ck)k≥0is non-decreasin... | https://arxiv.org/abs/2503.21334v4 |
of th e lemma let t≥2,b∈[2,∞), and let ( ck)k≥0, (ρk)k≥0and (σk)k≥0be as defined in Lemma 5. Then, by using ( 4) and Lemma 5, for any function f:R→Rwe have ˆηt/parenleftbig f /BD[−b,b]c/parenrightbig =/integraltext RGct−1(x)Mρt−1,σ2 t−1(x,f /BD[−b,b]c)ˆη1(dx) ˆη1(Gct−1). (24) By Lemma 5, we have ct−1≥c1,|ρt−1| ≤ |ρ|andσ... | https://arxiv.org/abs/2503.21334v4 |
importance sam pling From Lemma 7we readily obtain the following corollary: Corollary 1. LetH∈ C1(R)be such that V(H)<∞and such that H(x)≥0for allx∈R, and let µ,π∈ P(R). Then, ∝bardblΨH(µ)−ΨH(π)∝bardbl ≤∝bardblH∝bardbl∞+2V(H) π(H)∝bardblµ−π∝bardbl. Proof.The result follows from Lemma 7and the fact that, for all B∈ B1, ... | https://arxiv.org/abs/2503.21334v4 |
2/parenleftbig {un}N n=1/parenrightbig +L−2/parenrightBig (35) where the second inequality holds by ( 34). Using (31) and the fact that, by assumption, the function {x1,...,x M} ∋x∝ma√sto→Fµx(b2) is either non-increasing or non-decreasing, it is readily checked that |U2| ≤2L. Together with ( 30) and (32)-(35), this imp... | https://arxiv.org/abs/2503.21334v4 |
To proceed further remark that if ( ˆXn t,Xn t+1) is as in ( 38) then |Xn t+1| ≤max/braceleftbig |ξ+σΦ−1(2/8)|,σ|Φ−1(1/8)|/bracerightbig = max/braceleftbig |ξ−σΦ−1(6/8)|,σΦ−1(7/8)/bracerightbig ≤ξ−δ where the inequality holds with δ=σΦ−1(6/8) and uses the condition imposed on ξin the statement of the lemma. By combinin... | https://arxiv.org/abs/2503.21334v4 |
noting that under the assumptions of the lemma there exists a constant ι′∈(0,1] such that inf k∈N0inf x∈IˆηN kM(Gc+c⋆)≥ι′, (48) it follows that /vextenddouble/vextenddoubleΨGc+ct−p(ηN p)−ΨGc+ct−p/parenleftbig ˆηN p−1M/parenrightbig/vextenddouble/vextenddouble/vextenddouble≤5∝bardblηN p−1−ˆηN p−1M∝bardbl ι′. (49) By Lem... | https://arxiv.org/abs/2503.21334v4 |
proof of the lemma is complete. A.3 Proof of Lemmas 1-3, Theorems 1-2and Propositions 1-3 Lemma1is a direct consequence of Lemmas 9-10. Lemma 2is a direct consequence of Lemmas 1,6and 7. Theorem 2directly follows from ( 9) and Lemma 1. Proposition 2is a direct consequence of Lemma 1 and Theorem 1. Consequently, below w... | https://arxiv.org/abs/2503.21334v4 |
Theorem 1 Proof.Westartwithtwopreliminaryresults. Tostatethefirstonelet πM(dx) =/summationtextM m=1Wmδ{xm}∈ P(R) for some M∈N, for allx∈ {x1,...,x M}letµx∈ P(R), letµ=/summationtextM m=1Wmµxm∈ P(R),{un}N n=1be a point set in (0 ,1) andµN=N−1/summationtextN n=1δF−1 µ(un). Then, it is direct to see that ∝bardblµN−µ∝bardbl... | https://arxiv.org/abs/2503.21334v4 |
arXiv:2503.21421v1 [math.OC] 27 Mar 2025Robust Mean Estimation for Optimization: The Impact of Heavy Tails Bart P.G. Van Parys∗1and Bert Zwart†1,2 1CWI Amsterdam 2TU Eindhoven March 28, 2025 Abstract We consider the problem of constructing a least conservativ e estimator of the expected value µof a non- negative heavy-... | https://arxiv.org/abs/2503.21421v1 |
serve as confidence set for the unknown distribution Delage and Ye (2010), Bertsimas et al. (2018) with a desired p robability level. That is, the probability1 P[D(P,Pn)≤r] (2) is required to be sufficiently large which is typically ensure d by appealing to an appropriate concentration inequality. Indeed, it is straightfo... | https://arxiv.org/abs/2503.21421v1 |
(Lam 2019, Duchi et al. 2021, Bennouna a nd Van Parys 2021). However, as we have pointed out in this introduction, the statistical analysis of moder n DRO formulations requires data with bounded support, light-tails or finite variance. In contrast, in many practic al situations, data is heavy-tailed, and sometimes even ... | https://arxiv.org/abs/2503.21421v1 |
Wasserstein DRO, truncated mean, and variance regularization) behave. We focus primarily on the probability of being too optimistic (namely that our estima te exceeds µ), but also examine the probability of being too pessimistic (the probability that our estimator is smal ler than µ−bfor some b >0). It turns out that n... | https://arxiv.org/abs/2503.21421v1 |
and the nature of the results potentially diffe rent. In fact, at a minimum it would require imposing a structural assumption on the distribution Psuch as the existence of a finite moment of order a >1 as considered in related work on heavy-tailed bandits Agrawal et al. (2021 ). Our focus on losses being bounded in one d... | https://arxiv.org/abs/2503.21421v1 |
Bennouna and Van Parys (2021) sublinear and super linear scalings were considered. It was shown in Bennouna and Van Parys (2021) that superlinear scalings o fλ(n) require en→maxζ=∞and is hence not of interest here. Consequently, we will concern ourselves wit h the case that lim n→∞λ(n)/n < ∞. Finally, Duchi et al. (202... | https://arxiv.org/abs/2503.21421v1 |
consider the Wasserstein mean estimator whic h is defined as the solution to the optimization problem ˆeW n,r:=/braceleftbiggmin/integraltext udQ(u) s.t.Q∈ P,W(Pn,Q)≤r. where Wdenotes the 1-Wasserstein metric, c.f., Villani et al. (200 8) for a detailed definition. The Wasserstein distance has the distance of choice in d... | https://arxiv.org/abs/2503.21421v1 |
general the variance is potent ially not defined, there are still two issues with the presented approach. The first issue is that even when the vari ance is finite, as discussed in Section 2.2, the informal line of argument fails to hold for Pregularly varying with index ρand disappointment rates satisfying lim inf n→∞λ(n... | https://arxiv.org/abs/2503.21421v1 |
section reviewed different methods to estimate µ, focusing on the (disappointment) probability of overestimating the reward and the probability of being ineffi cient by a term b > 0. None of the methods we discussed were able to reach both objectives in a purely data -driven way. In this section, we will introduce the KL-... | https://arxiv.org/abs/2503.21421v1 |
(20) That is, the same guarantee (18) as the Kullback-Leibler estimator for some λ(n)→ ∞ . Then, for any regularly varying ζwe have lim inf n→∞P[ˆen≤ˆeKL r′(n)(Pn)] = 1 , for any radius r′(n)so that limn→∞r′(n)n/λ(n)∈(0,1). Informally, the previous result says that any competing est imator must be at least as conservat... | https://arxiv.org/abs/2503.21421v1 |
concave function φwe have for any (1 −γ(n))l≤α≤(1−γ(n))l−1that φ((1−γ(n))l)≥(1−γ(n))l αφ(α) +/parenleftbigg 1−(1−γ(n))l α/parenrightbigg φ(0) =(1−γ(n))l αφ(α). If the event max (1−γ(n))l≤α≤(1−γ(n))l−1φ(α)≥0 occurs, then clearly we have for (1 −γ(n))l≤α≤(1−γ(n))l−1 the inequality φ((1−γ(n))l)≥(1−γ(n))l αφ(α)≥(1−γ(n))φ(α... | https://arxiv.org/abs/2503.21421v1 |
dP(u)/parenrightbigg/integraldisplay1 αr+udP(u) +qr s= 0 (28) for some qr s≥0 satisfying the complementarity condition αrqr s= 0. We can rewrite the previous optimality condition as (28) ⇐ ⇒/integraldisplayνr αr+udP(u) +qr s≤1⇐ ⇒/integraldisplay dQr c(u) +qr s= 1. We claim now that an optimal primal solution in (16) ca... | https://arxiv.org/abs/2503.21421v1 |
tion in Problem (27) it also follows immediately that the function ˜ ris nondecreasing. Furthermore, using the sharpened Jensen ’s inequality of Liao and Berg (2019) we get ˜r(˜α) = log/integraldisplay1 1 + ˜αudP(u)−/integraldisplay log/parenleftbigg1 1 + ˜αu/parenrightbigg dP(u)≥1 2VP/bracketleftbigg1 1 + ˜αζ/bracketr... | https://arxiv.org/abs/2503.21421v1 |
(1+z)2P[˜αζ≥z]dz =O(˜α1+ρL(1/˜α)). Furthermore, we have for any K≥1 that /integraldisplay∞ 0z (1 + z)2P[˜αζ≥z]dz =/integraldisplay1 0z (1 + z)2P[˜αζ≥z]dz+/integraldisplay∞ 1z (1 + z)2L(z/˜α) (z/˜α)ρdz ≥/integraldisplay1 0z (1 + z)2P[˜αζ≥z]dz+ ˜αρL(1/˜α)/integraldisplayK 1z (1 + z)2zρL(z/˜α) L(1/˜α)dz. Recall that by... | https://arxiv.org/abs/2503.21421v1 |
such a result no longer applies to heavy-tai led distributions. Proposition 5.5. Letζbe bounded. Then, lim r→0VP[ηr(ζ)] r≤2. Proof. Let 0 ≤ζ≤B < ∞which implies VP[ζ]≤B2/4<∞. Equation (31) specializes to ˜r(˜α)≥1 2VP/bracketleftbigg1 1 + ˜αζ/bracketrightbigg ≥1 2(1 + ˜ αB)2VP[˜αζ] where we applied Lemma A.4 with the fun... | https://arxiv.org/abs/2503.21421v1 |
Equation (32) to guarantee that the event ˆ en> µr(n)occurs with nonzero probability. To do so observe for that we have KLinf(Pn, µ) =/braceleftbiggminQKL(Pn,Q) s.t./integraltext udQ(u)≤µr(n),Q∈ P 17 ≥/braceleftbiggminQ/integraltext ηr(n)(u)dPn(u)−Λ(ηr(n),Q) s.t./integraltext udQ(u)≤µr(n),Q∈ P =/integraldisplay ηr(n)(u... | https://arxiv.org/abs/2503.21421v1 |
Samorodnitsky. Nearly optima l Catoni’s M-estimator for infinite variance. In International Conference on Machine Learning , pages 1925–1944. PMLR, 2022. N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation . Number 27 in Encyclopedia of Mathematics and its Applications. Cambridge university press, 1989. J.... | https://arxiv.org/abs/2503.21421v1 |
robust optimization: Theory and applications in machine learning. In Operations research & management science in the age of analytics , pages 130–166. Informs, 2019. H. Lam. Recovering best statistical guarantees via the empi rical divergence-based distributionally robust optimiza tion. Operations Research , 67(4):1090... | https://arxiv.org/abs/2503.21421v1 |
bound and using log(1 + x)≤x, we get P[/summationtextn i=1ζi∧r(n)−1/a> cnr (n)(a−1)/a+µn]≤exp{−n[scr(n)(a−1)/a−saC]} = exp {−nr(n)[uc−Cua]}. (41) We now choose c=ca= (a−1)−(a−1)/aaC1/a, u= (c/(aC))1/(a−1). Below, we show that these choices yield uc−Cua= 1 and u≤awhich was required in Equation (40), so that the proof fo... | https://arxiv.org/abs/2503.21421v1 |
reflection yields that an optimal solution Qnin (45) is a discrete measure on [0 ,∞) which removes mass s(n) =/radicalbig r(n)/2 of the largest revenues scenarios and reassigns it to the zero reward scenario. Cons equently, ˆ eTV,u(n) n ≤1 n/summationtextn i=1ζi∧u(n)−u(n)s(n). Combining these observations we obtain P/br... | https://arxiv.org/abs/2503.21421v1 |
Sparse Bayesian Learning for Label Efficiency in Cardiac Real-Time MRI Felix Terhag1,2, Philipp Knechtges1, Achim Basermann1, Anja Bach4, Darius Gerlach4, Jens Tank4, Ra´ ul Tempone2,3 1Institute of Software Technology, High-Performance Computing, German Aerospace Center (DLR), Cologne, Germany. 2Chair of Mathematics f... | https://arxiv.org/abs/2503.21443v1 |
altering the geometry of the heart. A more detailed description of the study that acquired the data can be found in [9]. This disease complicates training even further. While U-Nets can seg- ment inner slices with high accuracy, they tend to struggle with the outer slices a phenomenon also observed in classical cine MR... | https://arxiv.org/abs/2503.21443v1 |
by a superposition of frequencies, comprising a subset of F={f0= 0, f1, f2, . . . , f M}, where the main frequencies are the unknown heart and respiratory rates. This work proposes the following linear model: Y=AX+N, (1) with •Y∈RN×L, the real-valued measurement points yk,ℓ; •X∈R2M+1×L, the real and imaginary parts of ... | https://arxiv.org/abs/2503.21443v1 |
by data (i.e., irrelevant to the model). When a variance becomes small, the associated prior distribution effectively becomes a delta function centered at zero, ”switching off” those parameters from the model (for a comprehensive explanation of this method, see [17, Sec 15.2.8]). This work aims to determine the sparse ... | https://arxiv.org/abs/2503.21443v1 |
yam+aT m+MΣ−1 yam+M =||YTΣ−1 y(am+am+M)||2 2 −L aT mΣ−1 yam+aT m+MΣ−1 yam+M ,(11) where amdenotes the mth column vector of A(cf. [12]). The M-step uses the MacKay update rule introduced in [19], obtained by setting the derivative (11) to zero and using a fixed-point equation. The update rule αnew m=||(µx)m+ (µx)m+M|... | https://arxiv.org/abs/2503.21443v1 |
matrix, as proposed in [20], offers an alternative but may encounter numerical problems, particularly as det(Σ) approaches 0 in high-dimensional settings with small eigenvalues. A third option is to employ the largest eigenvalue of Σ, cor- responding to the variance along the first principal component. This approach is... | https://arxiv.org/abs/2503.21443v1 |
and i∈Ω the marginal gain of the objective function (18) simplifies to fi(J) =f(J∪ {i})−f(J) =tr Σ(J) posty −tr Σ(J∪{i}) posty .(22) Moreover, we can prove the following lemma. Lemma 5. Forfidefined in (18), it holds fi(J)>0for all J⊂Ωandi∈Ω\J. Proof. IfJ⊂Ω and i∈Ω\J, it holds that fi(J) =tr Σ(J) posty −tr Σ(J∪{... | https://arxiv.org/abs/2503.21443v1 |
set a threshold is ϵthresh = 0.01·max m∈{1,...,M}(αm), as frequencies with amplitudes orders of magnitudes smaller than the most significant frequency have a negligible effect on the final volume. A prior that assigns weight only to a few frequencies is beneficial when predicting slices without information about the vo... | https://arxiv.org/abs/2503.21443v1 |
work The labeling effort on the other slices can be minimized after empirically determining the prior on the good intermediate slices. We used a greedy algorithm to search for 13 0 2 4 6 8 10 time [s]0.20.40.60.81.0Normalized Ventricle Volume ySlice 1 0 2 4 6 8 10 time [s]Slice 2 Ground truth 0.60.750.95 0.4 0.25 0.05P... | https://arxiv.org/abs/2503.21443v1 |
greedy det greedy 1 greedy Tr drawing random (a) Trace Heart 1 2 4 6 8 10 12 14 Labeled Images0.00250.00300.00350.00400.0045det(posty) greedy det greedy 1 greedy Tr drawing random (b) Determinant Heart 1 2 4 6 8 10 12 14 Labeled Images90100110120130140150160Negative Log-Likelihood greedy det greedy 1 greedy Tr drawing ... | https://arxiv.org/abs/2503.21443v1 |
avoiding poor sequences can also be visually observed. The posterior distribution captures the ground truth for only five sample points selected with the greedy trace approach (see Figure 6a and 6b). In contrast , when selecting the sample images with a greedy worst approach, the posterior does not fit the data. Howeve... | https://arxiv.org/abs/2503.21443v1 |
ground-truth data . The measured data is marked with a green x. (c) Results of 15 labeled images selected by a greedy worst approach on Heart 2. Compared to the result of five selected images by a greedy algorithm minimizing the trace on Heart 1 (a) or Heart 2 (b), the posterior variance is significantly smaller in (a)... | https://arxiv.org/abs/2503.21443v1 |
F., Karakas, M., C ¸avu¸ s, E., Petersen, S.E., Escalera, S., Segu´ ı, S., Rodr´ ıguez-Palomares, J.F., Lekadir, K.: Multi-centre, multi-vendor and multi-disease cardiac segmentation: The M&Ms challenge. IEEE Transactions on Medical Imaging 40(12), 3543–3554 (2021) https://doi.org/10.1109/TMI.2021. 3090082 [3] Isensee,... | https://arxiv.org/abs/2503.21443v1 |
[17] Murphy, K.P.: Probabilistic Machine Learning: Advanced Topics. MIT Press, 23 Cambridge, MA (2023). http://probml.github.io/book2 [18] Wipf, D.P., Rao, B.D.: An empirical bayesian strategy for solving the simulta- neous sparse approximation problem. IEEE Transactions on Signal Processing 55(7), 3704–3716 (2007) htt... | https://arxiv.org/abs/2503.21443v1 |
Proceedings of Machine Learning Research 275:1–31, 2025 4th Conference on Causal Learning and Reasoning Constraint-based causal discovery with tiered background knowledge and latent variables in single or overlapping datasets Christine W Bang BANG @LEIBNIZ -BIPS .COM Vanessa Didelez DIDELEZ @LEIBNIZ -BIPS .COM Leibniz ... | https://arxiv.org/abs/2503.21526v2 |
FCI and build on this to propose an IOD algorithm exploiting tiered back- ground knowledge; we refer to these as the ‘tiered’ FCI/IOD, or ‘tFCI/tIOD’ algorithms. While the tFCI algorithm has been implemented (Scheines et al., 1998; Chen and Malinsky, 2023; Petersen, 2023) and applied (Lee et al., 2022) before, to our k... | https://arxiv.org/abs/2503.21526v2 |
Sections 4, 5 and 6 we describe the tFCI and tIOD algorithms, and investigate their properties. In Section 7 we discuss the assumptions and limitations of the proposed approaches. We include pseudo-algorithms in Appendix B and proofs in Appendix D. 2 CAUSAL DISCOVERY WITH TIERED BACKGROUND KNOWLEDGE AND OVERLAPPING DAT... | https://arxiv.org/abs/2503.21526v2 |
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