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to 1+∥v∥2−p 2(1+2ε)∥v∥≤{ 1−∥v∥(√ 0.5+ε−√ 0.5−ε)}2, (A.4) which is a quadratic in ∥v∥and can easily be verified to hold when ∥v∥<(0.5+ε)−1/2. Consequently in the sub-case (I), we have for Tnthe bound Tn≥ρn∥v∥2(√ 0.5+ε−√ 0.5−ε)2. What remains, is tackling the case where ∥Pnv∥2/∥v∥2>0.5+εand∥un+v∥>1. In this case it holds... | https://arxiv.org/abs/2502.04046v1 |
that Mis diagonal with strictly positive diagonal elements. Let next unbe the leading eigenvector of the matrix G−1/2 nMnG−1/2 nand let u=e1be the leading eigenvector of the matrix G−1/2MG−1/2. Then, Lemma 2 and Corollary 1 in [46] imply that u2 n1−1=Op(1/c2 n) and that un j=Op(1/cn), since we have assumed that the lea... | https://arxiv.org/abs/2502.04046v1 |
9 (2015) 32–105. [16] L. D ¨umbgen, D. E. Tyler, On the breakdown properties of some multivariate m-functionals, Scandinavian Journal of Statistics 32 (2005) 247–264. [17] W. Fu, K. Knight, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356–1378. [18] P. Georgiev, F. Theis, A. Cichocki, H. Bakar... | https://arxiv.org/abs/2502.04046v1 |
P. O. Hoyer, A. Hyv ¨arinen, A. Kerminen, M. Jordan, A linear non-Gaussian acyclic model for causal discovery, Journal of Machine Learning Research 7 (2006). [41] S. Sirki ¨a, S. Taskinen, H. Oja, Symmetrised M-estimators of multivariate scatter, Journal of Multivariate Analysis 98 (2007) 1611–1629. [42] S. Taskinen, S... | https://arxiv.org/abs/2502.04046v1 |
arXiv (2025), 0, 0,pp.1–?? Generalised Bayesian distance-based phylogenetics for the genomics era MATTHEW J PENN1∗, NEILSCHEIDWASSER2∗, MARK P K HURANA2, CHRISTL A D ONNELLY1,3, DAVID A DUCHÊNE2∗,AND SAMIR BHATT2,4∗ 1Department of Statistics, University of Oxford, Oxford, United Kingdom 2Section of Epidemiology, Univer... | https://arxiv.org/abs/2502.04067v1 |
Felsenstein’s likelihood (Felsenstein, 1981) is highly complex to compute for large datasets. Specifically, calculating the likelihood of a single tree isO(nNc2)fornleaves,Nunique site patterns and c character states. Even with the efficiency savings that occur when considering similar trees, the calculation remains hi... | https://arxiv.org/abs/2502.04067v1 |
framework has only been quantified via bootstrapping. To address the various shortcomings of phylogenetic methods for inference using large modern datasets, we develop a new likelihood function by considering the effect of using tree entropy as a prior distribution. Using simplifying approximations, we reduce this new ... | https://arxiv.org/abs/2502.04067v1 |
topology of the tree under consideration The classical maximum likelihood problem in phylogenetics involves the construction of 6 PENN M., SCHEIDWASSER N., ET AL. a weighted, binary tree with topology Uand branch lengths b, describing the evolutionary history of a set ofntaxa, each of which corresponds to a leaf node o... | https://arxiv.org/abs/2502.04067v1 |
of branch length estimation, the length of this tree (that is, the sum of its branch lengths) is minimised. To produce a similarly-motivated likelihood for each possible tree (comprised of a topology Uand branch lengths b), we develop a measure, E(U,b), of the entropy of this tree, defined explicitly in the subsequent ... | https://arxiv.org/abs/2502.04067v1 |
for some constant C2. Under these assumptions, we can evaluate the integral (3) to see that L becomes L(U) =ψ(U,b∗(U,D)) (7) and hence ℓ(U) =E(U,b∗(U,D)) (8) which is the expected likelihood of the tree with topology Uand branch lengths given by the BME estimates. In the remainder of this paper, we will derive an effic... | https://arxiv.org/abs/2502.04067v1 |
the A PHYLOGENETIC LIKELIHOOD FROM BME 11 BME approximations). Thus, we seek to find an entropic distance matrix dSsuch thatdS ijis the distance between taxa iandjin our entropy-weighted tree (for a single site). In this case, the objective function is the length of the BME tree with distance matrix dS, meaning ℓS(U) =... | https://arxiv.org/abs/2502.04067v1 |
optimisation process, as the distance matrix will remain the same throughout, independently of the topology under consideration. When the number of sites, Nis much larger than the number of taxa n(which is often the case, particularly in phylogenomics) this results in a substantial saving in computational cost, as afte... | https://arxiv.org/abs/2502.04067v1 |
our entropic likelihood can be used to approximate sampling under Felsenstein’s likelihood. We do this by deriving our likelihood directly from the expected Felsenstein’s likelihood, and analysing the impact of each required approximation. Setup Consider the expected Felsenstein’s likelihood of a single-site tree Uwith... | https://arxiv.org/abs/2502.04067v1 |
we essentially just change the likelihood scale, the approximation (29) does add meaningful error and means that the correlation between the entropic and Felsenstein’s likelihood is not as strong as, for example, the correlation between the entropic likelihood and the BME objective - which is virtually the same. Howeve... | https://arxiv.org/abs/2502.04067v1 |
likelihood. To show the limitations of our entropic likelihood, we simulate a biologically realistic alignment from a known tree. Our tree simulation follows a birth death process with 50 species and withλ= 0.5,µ= 0.1,ρ= 1and time since origin (TMRCA) of 65(reflecting many major radiations since the K-Pg transition eve... | https://arxiv.org/abs/2502.04067v1 |
all the way to entirely random trees. Next we show how the gradient, m, between the Felsenstein and entropic likelihoods varies between 0 and 1 across alignments with different rates. As we have noted from our discussion of the κ1scaling parameter in (28), our theory suggests with low rates we expect a gradient,m, of a... | https://arxiv.org/abs/2502.04067v1 |
Felsenstein’s likelihood and a linear model of entropic distance. The percentage error is on a log likelihood scale. Black dotted lines show where most empirical data exist (Klopfstein et al., 2017). In summary, the linearity of our entropic likelihood with Felsenstein’s likelihood is a useful property, and justifies t... | https://arxiv.org/abs/2502.04067v1 |
not of substantial concern. Certainly, in our examples (e.g. Fig. 3), we empirically observe that the error term e−ϵis much smaller than the average distance between trees, particularly when we restrict our attention to the trees topologically close to the optima. To explore this more precisely, we follow the method in... | https://arxiv.org/abs/2502.04067v1 |
specify the model. These simulations are not definitive, but a theoretical asymptotic analysis is not possible given the complexities of both likelihoods. This analysis simply provides empirical reassurance to the theory we have already outlined. Implementation procedures To estimate the entropic distance matrix, we fi... | https://arxiv.org/abs/2502.04067v1 |
0.009 0.009 0.008 0.018 0.019 0.018 0.060 0.120 0.180 1.400 6.640 14.990 750 0.020 0.020 0.032 0.040 0.019 0.038 0.070 0.150 0.260 2.040 11.670 23.310 1,000 0.039 0.045 0.046 0.051 0.077 0.079 0.080 0.180 0.330 2.730 15.520 31.340 2,500 0.266 0.313 0.255 0.263 0.427 0.395 0.170 0.405 0.770 6.840 42.330 83.860 Table 3. ... | https://arxiv.org/abs/2502.04067v1 |
ET AL. In DS2, the continuous tree search (Penn et al., 2023) found a better tree than the optimal FastME tree, which was the most similar to the RAxML-NG tree. In this dataset, only one RAxML-NG mode was found across all 100 runs, and the bootstrap (blue circles) is tighter than in DS1 (Fig. 5b). The entropic MCMC pos... | https://arxiv.org/abs/2502.04067v1 |
that aims to generate representative draft genome sequences from all extant bird species. Here we analyse the release of 363 genomes representing 92% of all bird families. We estimated pairwise genetic distance under a GTR+ Γsubstitution model (general time-reversible model with gamma-distributed rate variation among s... | https://arxiv.org/abs/2502.04067v1 |
been recognised as difficult to place in the avian tree of life (Reddy et al., 2017; Houde et al., 2019). These results are consistent with the genome-wide disagreement in the relationships early after the post-K-Pg transition. Critically, the entropic likelihood Bayesian approach provides evidence that those nodes wit... | https://arxiv.org/abs/2502.04067v1 |
only viable solutions along with parsimony-based approaches. Considering the inherent uncertainty in these distance-based phylogenetic inferences, the only viable approach until now has been bootstrapping, which provides limited meaningful information when the sample size of sites is very large. Our proposed entropic l... | https://arxiv.org/abs/2502.04067v1 |
views expressed are those of the author(s) and not necessarily those of the NIHR, UK Health Security Agency or the Department of Health and Social Care.” S.B. acknowledges support from the Novo Nordisk Foundation via The Novo Nordisk Young Investigator Award (NNF20OC0059309). S.B. acknowledges support from the Danish N... | https://arxiv.org/abs/2502.04067v1 |
7:607–633. Ho, S. Y . W., S. Duchêne, and D. Duchêne. 2015. Simulating and detecting autocorrelation of molecular evolutionary rates among lineages. Mol. Ecol. Resour. 15:688–696. Hoang, D. T., O. Chernomor, A. von Haeseler, B. Q. Minh, and L. S. Vinh. 2018. UFBoot2: Improving the ultrafast bootstrap approximation. Mol... | https://arxiv.org/abs/2502.04067v1 |
2011. Reconciling molecular phylogenies with the fossil record. Proc. Natl. Acad. Sci. U.S.A. 108:16327–16332. Nguyen, L.-T., H. A. Schmidt, A. V on Haeseler, and B. Q. Minh. 2015. Iq-tree: a fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies. Mol. Biol. Evol. 32:268–274. Paradis, E.,... | https://arxiv.org/abs/2502.04067v1 |
and B. Rannala. 2012. Molecular phylogenetics: principles and practice. Nat. Rev. Genet. 13:303–314. Yang, Z. and A. D. Yoder. 2003. Comparison of likelihood and Bayesian methods for estimating divergence times using multiple gene loci and calibration points, with application to a radiation of cute-looking mouse lemur ... | https://arxiv.org/abs/2502.04067v1 |
subset of calibrations from the original study TABLE CAPTIONS 1. Evaluation datasets 2. Comparison of run times (seconds) for balanced minimum evolution and Felstensteins likelihood for different numbers of taxa and sites 40 APPENDIX APPENDIX SUPPLEMENTARY INFORMATION Connection of entropic likelihood to BME In this se... | https://arxiv.org/abs/2502.04067v1 |
small as X→0. Moreover, in the region τ <<X , we haveKτ θ<<1and so, using Lemma A.9, S′′(t)∼K t. Thus, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayX kS′′/parenleftiggτ θ/parenrightigg e−τdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∼/vextendsingle/v... | https://arxiv.org/abs/2502.04067v1 |
and so, P(Vx=a,Vy=b) =/summationdisplay d∈VP(Vx=a,Vy=b|Vm=d)πd (A.31) Now, the substitution process on the path from mtoais independent of the substitution process on the path from mtob. Thus, if the distance from mtoaistaand the distance from mtobistb, P(Vx=a,Vy=b) =/summationdisplay d∈VπdPda(ta)Pdb(tb) (A.32) APPENDI... | https://arxiv.org/abs/2502.04067v1 |
(A.48) and so, using Lemma A.9 to show that the singularity of S′(t)att= 0is logarithmic, and therefore integrable h(τ) =/integraldisplayτ 0/bracketleftigg e−θt(S′(t) +θS(t)) +/integraldisplayt 0S(s)θ2e−θsds/bracketrightigg dt (A.49) Maximum likelihood estimation of θfor a Markovian branching process LEMMA A.6 Given ... | https://arxiv.org/abs/2502.04067v1 |
multiplying our equation bypi qi, and summing over i, we have /summationdisplay ip2 i qi=λ/summationdisplay ipilog(qi) + (1−Cλ) (A.72) We suppose that our trees are close so that/summationtext ip2 i qi=O(1)(which follows aspi qi≈1and hence we are left with approximately/summationtext ipi= 1). Ignoring this term then yi... | https://arxiv.org/abs/2502.04067v1 |
MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF MATRIX VARIATE SYMMETRIC LAPLACE DISTRIBUTION POOJA YADAV1. TANUJA SRIVASTAVA2 Abstract. This paper considers an extension of the multivariate symmetric Laplace distribution to matrix variate case. The symmetric Laplace distribu- tion is a scale mixture of normal distr... | https://arxiv.org/abs/2502.04118v1 |
readers are referred to [1], [17], [20] for the definition and properties of the modified Bessel function of the third kind. One characterization of the multivariate symmetric Laplace distribution is a scale mixture of the multivariate normal distribution, with random scale fac- tor having an exponential distribution. ... | https://arxiv.org/abs/2502.04118v1 |
of the Kronecker product of the estimators are shown using the simulation. Section 6contains the conclusion of the paper. Notation .The following notations are used throughout the paper: Np(0,Σ)denotes the multivariate normal distribution with 0a vector with zero entries, and Σis ap×ppositive definite matrix. tr(A)and ... | https://arxiv.org/abs/2502.04118v1 |
normal distribution [3] Z∼ MN p,q(0,Σ1,Σ2)⇐⇒ vec(Z)∼ N pq(vec(0),Σ2⊗Σ1). From the representation (1.2) of multivariate symmetric Laplace distribution, √ Wvec (Z)∼ SL pq(Σ2⊗Σ1). So, by definition 2.1,√ WZ∼ MSL p,q(Σ1,Σ2). □ Theorem 2.3 (Characteristic function ).IfX∼ MSL p,q(Σ1,Σ2), then the characteristic function of X... | https://arxiv.org/abs/2502.04118v1 |
maximization is (3.9) Q(Σ) =−N 2log Σ −1 2NX i=11 Wi Yi⊤Σ−1Yi . Since Wis a latent variable, which is not observable, it is replaced with its conditional expectation given Y1, Y2,···, YNand the current estimate of Σ, (say ˆΣ). After taking the conditional expectation, the function to be maximized is (3.10) Q1(Σ) =−N ... | https://arxiv.org/abs/2502.04118v1 |
2log(Wi)! . Since the last term does not contain any unknown parameter, it can be ignored formaximizationof ℓc(Σ1,Σ2)withrespectto Σ1andΣ2. Therefore, thefunction considered for maximization is (3.15) Q(Σ1,Σ2) =−qN 2log Σ1 −pN 2log Σ2 −1 2NX i=11 Witr Σ−1 2Xi⊤Σ−1 1Xi . Since Wis a latent variable, which is not obser... | https://arxiv.org/abs/2502.04118v1 |
ℓ(Σ1,Σ2)is 13 ℓ(Σ1,Σ2) =−qN 2log Σ1 −pN 2log Σ2 +ν 2NX i=1log tr Σ2−1Xi⊤Σ1−1Xi +NX i=1logKνq 2 tr(Σ2−1Xi⊤Σ1−1Xi) . Inthenextsection, theexistenceandstabilityoftheproposedMLEarediscussed. 4.Existence and stability of estimators It is claimed that maximum likelihood estimators exist for the parameters Σ1,Σ2of matri... | https://arxiv.org/abs/2502.04118v1 |
of Σof the multivariate symmetric Laplace distribution, ˆΣ2⊗ˆΣ1= Σ∗ 2⊗Σ∗ 1. The equivalence of (4.24) is obtained using the definition 2.1 and the above result. The stability of maximum likelihood estimates ˆΣ1andˆΣ2measured by∥Σ∗ 2⊗Σ∗ 1−ˆΣ2⊗ˆΣ1∥2is validated in the next section. 5.The performance of the proposed MLE I... | https://arxiv.org/abs/2502.04118v1 |
129 20100116 114 123 131 3050119 116 126 133 5030121 121 128 136 10020125 123 131 140 Table 1. Mean number of iterations required to meet the stop- ping criterion of this algorithm, for all the Cases (Case 1-4), with ϵ= 10−11. For all Cases p= 5, q= 3(Cases 1-4 are as given above in this section). •The initial estimate... | https://arxiv.org/abs/2502.04118v1 |
estimators. Thus, the proposed algorithm for the matrix variate symmetric Laplace distribu- tion estimate Σ2⊗Σ1for all four structures considered in nominal iterations. The proposed algorithm for the matrix variate symmetric Laplace distribution can be applied to estimate the parameter Σof multivariate symmetric Laplac... | https://arxiv.org/abs/2502.04118v1 |
and Hall/CRC (2018). [10] S. Kotz, T. Kozubowski, and K. Podgórski, The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance , Springer Science and Business Media, 183(2001). [11] T.Kozubowski, S. Mazur, and K. Podgórski, Matrix variate generalized... | https://arxiv.org/abs/2502.04118v1 |
Cyclic quantum causal modelling with a graph separation theorem Carla Ferradini1, Victor Gitton1, and V. Vilasini1,2 1Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland 2Université Grenoble Alpes, Inria, 38000 Grenoble, France Causal modelling frameworks link observable correlations to causal expla... | https://arxiv.org/abs/2502.04168v2 |
. . . . . . . 17 4.3 General probability rule for cyclic quantum causal models . . . . . . . . . . 21 4.4 Examples of cyclic causal graphs . . . . . . . . . . . . . . . . . . . . . . . . 23 5 A sound and complete graph-separation property for cyclic QCMs 25 5.1 Previous results for acyclic case: d-separation . . . . . ... | https://arxiv.org/abs/2502.04168v2 |
of non-classical causal modelling frameworks that encompass quantum and broader operational theories [HLP14, BLO20], enabling the causal explanation of quan- tum correlations without invoking fine-tuning or adjustments to the operational causal structure. Causal models are typically represented as directed graphs, with... | https://arxiv.org/abs/2502.04168v2 |
models enforce conditions like factorization of unitary channels (corresponding to requiring valid process operators [BLO21]). Although these approaches capture meaningful subclasses of cyclic causal models and provide valu- able techniques for studying them, the soundness of d-separation already fails within such clas... | https://arxiv.org/abs/2502.04168v2 |
relativity, with a causal modelling and graph-separation semantic. Furthermore, in a companion paper [FGV25], we develop the framework and results for the classical causal modelling community by introducing the concept of classical post- selected teleportation. This includes an alternative formulation of p-separation f... | https://arxiv.org/abs/2502.04168v2 |
our causal models and proving consistency with the classical formulation of the cyclic probability rule and p-separation given in [FGV25]. Finally, section 8 summarizes the main contributions and discusses directions for future research. 1.2 Notation We denote withHa finite-dimensional Hilbert space, i.e., a finite-dim... | https://arxiv.org/abs/2502.04168v2 |
setsX1andX2and POVMs E(1)=/braceleftig E(1) x1∈L/parenleftig H/parenleftig X(3,1)/parenrightig/parenrightig/bracerightig x1∈X1andE(2)=/braceleftig E(2) x2∈L/parenleftig H/parenleftig X(4,2)/parenrightig/parenrightig/bracerightig x2∈X2.(7) In the special case where the causal graph is acyclic, the probabilit... | https://arxiv.org/abs/2502.04168v2 |
post-selected teleportation protocol (definitions 4 and 6), e.g., since (v4,v3)is missing in G′we get Gtp= R v1 v3 v4 v2 T . (10) 3. Define a causal model on the acyclic graph Gtpby keeping the same associations of the original causal model to all vertices and edges that are preserved from Gin 8 Gtpand associating a po... | https://arxiv.org/abs/2502.04168v2 |
in [VP90, GVP90] and in the non-classical case in [HLP14], both for acyclic graphs. One could hope that an anal- ogous theorem holds for cyclic graphs with some definition for probability distributions compatible with that graph. However, this typically fails [Pea09, Nea00] (section 5.2). For example, consider the foll... | https://arxiv.org/abs/2502.04168v2 |
(deferring the discussion of probabilities in the cyclic case to the next section). Finally, we provide examples of causal models. 3.1 Causal models and acyclic probabilities A causal graph is a decorated directed graph which is allowed to be cyclic. The graph specifies causation relations between systems or variables,... | https://arxiv.org/abs/2502.04168v2 |
order: •ThedecoherenceconditionrequiredforCPTPmapsassociatedtounobservedvertices ensures that the output of Evon systems defined over classical edges is classical, i.e., diagonal in the computational basis. The analogous condition for classical input edgese= (v′,v)∈Eclis already satisfied due to the decoherence conditi... | https://arxiv.org/abs/2502.04168v2 |
that each map acts on the correct subsystems. 13 Prepare-and-measure scenarios. Consider the following causal graphs: Gq= A L B andGc= A L B . (22) Let us analyze a general causal model for Gq. The vertex Ais observed, thus, it has associated a POVM {EA a∈L(HIn(A))}a∈XA. Since the vertex Ahas no incoming edges, we have... | https://arxiv.org/abs/2502.04168v2 |
respectively with POVMs {EA a∈L(HX⊗H L1)}a∈XAand{EB b∈L(HL2⊗H Y)}b∈XB. Since these vertices have no children, the associated CP maps act on states of the form ρL1⊗|x⟩⟨x|Xfor allρL1∈L(HL1)andρL2⊗|x⟩⟨x|Yfor allρL2∈L(HL2)as MA a(ρL1⊗|x⟩⟨x|X) = Tr L1X/bracketleftig EA aρL1⊗|x⟩⟨x|X/bracketrightig =: Tr L1/bracketleftig E... | https://arxiv.org/abs/2502.04168v2 |
have the following feature: it can be that for a specific outcome of the ABmeasurement, there is no correction to apply on the system C. If that is the case, this allows for a post-selected teleportation protocol: upon conditioning on 16 this specific outcome of the ABmeasurement (i.e., discarding all rounds where a di... | https://arxiv.org/abs/2502.04168v2 |
our framework. This makes Gtp identical to Gup to replacing each split edge (vi,v′ i)∈Es(Gtp)⊆Ewith the following structure: viv′ i Ri Ti (Ri,Ti)(vi,Ti) (Ri,v′ i) (47) We will refer to every Gtp∈Gtp(G)as ateleportation graph , as we will later associate teleportation protocols to such graphs. It will be useful to denot... | https://arxiv.org/abs/2502.04168v2 |
is an acyclic causal model, we can readily compute probabilities within such models using the acyclic probability rule of definition 3. Specifically, note that all observed vertices Vo⊆VofGare preserved in everyGtp∈Gtp(G). For a given teleportation graph Gtp, denotingVpostas the set of all post-selection vertices of th... | https://arxiv.org/abs/2502.04168v2 |
If p✓>0, the probability associated with the joint observed event x:={xv∈Xv}v∈VoinCmGis defined as Pr (x)G:= Pr acyc/parenleftbigx/vextendsingle/vextendsingle{ti=✓}Ti∈Vpost/parenrightbig Gtp=Pracyc/parenleftbigx,{ti=✓}Ti∈Vpost/parenrightbig Gtp p✓.(53) Ifp✓= 0, we say that the causal model CmGis inconsistent and the pr... | https://arxiv.org/abs/2502.04168v2 |
the causal mechanisms {Mv xv}v∈Voand{Ev}v∈Vuof the causal model as in proposition 14, is the same independently of the choice of teleportation graphGtp∈Gtp(G)used in its construction. Finally, we obtain the following corollary showing that definition 12 does not depend on the implementation of teleportation protocol (d... | https://arxiv.org/abs/2502.04168v2 |
m)E}m∈XMand{(EN n)F}n∈XNforMandN, and collections of channels {EL1,x AE|D}x∈XXand{EL2,y DF|A}y∈XYforL1andL2. In section 3.2, we applied definition 2 to examples of acyclic graphs, a similar procedure applies here noting that Gtpis also acyclic. To define the causal model on Gtpassociated to the above causal model on G,... | https://arxiv.org/abs/2502.04168v2 |
three types of simple directed graphs. Although d-separation is a purely graph-theoretic property defined for any directed graph, for explaining the motivation behind this concept, it is useful to consider classical causal models and probabilities on the associated vertices. We therefore stylize all vertices as observe... | https://arxiv.org/abs/2502.04168v2 |
the vertices of GwithV1andV2being non-empty. Then, the following holds: (Soundness) For any causal model CmGonGwhere the sets Viare observed, we have thatd-separation between the vertex sets Viimplies conditional independence for the corresponding sets of random variables Xi:={Xv}v∈Viwherei∈{1,2,3}, i.e., (V1⊥dV2|V3)G=... | https://arxiv.org/abs/2502.04168v2 |
as a collider for the exogenous vertices v3andv4. Recall that conditioning on a collider can d-connect previously d-separated vertices. Here, the collider is not explicitly conditioned upon in the original cyclic graph, however the consistency conditions of the model impose an effective post-selection on the values of ... | https://arxiv.org/abs/2502.04168v2 |
theorem) .Consider a directed graph Gand letV1,V2andV3 be any three disjoint sets of the vertices of GwithV1andV2being non-empty. Then, the following holds: (Soundness) For any causal model CmGonGwhere the sets Viare observed, we have thatp-separation between the vertex sets Viimplies conditional independence for the c... | https://arxiv.org/abs/2502.04168v2 |
other words, our definition only implies a p-connection in Gwhen that connection is reflected in allthe graphs of the graph family Gtp(G). Indeed, we have shown (see proposition 10 and definition 12) that the observed probabilities for the causal model are independent of the representative of Gtp(G)chosen in computing ... | https://arxiv.org/abs/2502.04168v2 |
property, which corresponds to the following factorisation of this distribution Prf,acyc({xv}v∈V)G=/productdisplay v∈VPrf,acyc(xv|Pa(xv))G, (84) where Pa(xv)denotes the outcome set of all parents of the vertex v. If some of the vertices are unobserved, one simply marginalises over the outcomes of V\Voon both sides of t... | https://arxiv.org/abs/2502.04168v2 |
cycle(Cx), (86) wherexin the self-cycle expression is short hand for {xv}v∈Vo, as we had in proposition 14. While equation (20) and equation (85) are not defined in the cyclic case, the above equation is, since the cycleoperation is defined generally. This expression is independent of the choice of teleportation graph ... | https://arxiv.org/abs/2502.04168v2 |
causal mechanisms {Mv xv}v∈Voand{Ev}v∈Vuof the causal model. Causal models leading to Markovian distributions are therefore linear in the causal mechanisms of the model. Note, however, that for linearity, it suffices that/summationtext xcycle(Cx)equals a constant valuecnot necessarily equal to 1, while Markovianity req... | https://arxiv.org/abs/2502.04168v2 |
be faithfully mapped to ours is an open question (represented by a blue arrow) that relies on another prominent open question in field relating to causal decompositions of quantum channels (see appendix B.1.2 for further discussion). The green boxes highlight the graph-separation property (annotated above the box) whos... | https://arxiv.org/abs/2502.04168v2 |
approaches, there are notable differences in the defi- 12Unique solvability corresponds to having a unique solution of the functional dependences for every valuation of the exogenous vertices. 36 nition of causal influence conditions and the associated concept of causal structure between these two approaches (see also ... | https://arxiv.org/abs/2502.04168v2 |
our work generalizes this by allowing for arbitrary post-selected teleportation protocols that do not necessarily involve maximally entangled pre- and post-selections (definition 4). 37 Notably, we have demonstrated in section 4.3 that the operationally accessible prob- abilities can always be computed via a cyclic com... | https://arxiv.org/abs/2502.04168v2 |
causal structure supports a certifiable gap between sets of correlations realizable via classical vs non-classical causal models (see e.g., [HLP14]). Our results create possibilities for exploring analogous questions in cyclic graphs, such as whether meaningful separations between classical and non-classical correlatio... | https://arxiv.org/abs/2502.04168v2 |
graph and causal model) can we have different conditional independence constraints imposed by the soundness of two graph separa- tion properties (among d,σandpseparation), where a causal model violates one of the constraints and satisfies the other? The results and further examples of [FGV25] present relevant insights ... | https://arxiv.org/abs/2502.04168v2 |
whether certain properties of spatio-temporal cau- sation can “emerge” from information-theoretic models of causation, and can we thus understand them from more basic operational principles. With this motivation, in a follow-up work based on the master’s thesis [Fer23], we link the present causal modelling framework to... | https://arxiv.org/abs/2502.04168v2 |
, doi:10.48550/arXiv.1906.10726, 2020. [BLO21] J. Barrett, R. Lorenz, O. Oreshkov, “Cyclic quantum causal models”, Nature Communications , 12(1), doi:10.1038/s41467-020-20456-x, 2021. [BW16a] A. Baumeler, S. Wolf, “Device-independent test of causal order and relations to fixed-points”, New Journal of Physics , 18(3):03... | https://arxiv.org/abs/2502.04168v2 |
of causal inference for biomed- ical informatics”, Journal of biomedical informatics , 44:1102–12, doi:10.1016/j.jbi.2011.07.001, 2011. [LB21] R. Lorenz, J. Barrett, “Causal and compositional structure of unitary trans- formations”, Quantum , 5:511, doi:10.22331/q-2021-07-28-511, 2021. [LDLL90] S. L. Lauritzen, A. P. D... | https://arxiv.org/abs/2502.04168v2 |
R. Colbeck, “Impossibility of superluminal signaling in minkowski spacetime does not rule out causal loops”, Physical Review Let- ters, 129(11), doi:10.1103/physrevlett.129.110401, 2022. [VP90] T. Verma, J. Pearl, “Causal networks: Semantics and expressiveness”, in R. D. Shachter, T. S. Levitt, L. N. Kanal, J. F. Lemme... | https://arxiv.org/abs/2502.04168v2 |
the teleportation probability is independent of the input state (as shown in lemma 25), by linearity, we must have that R◦O=ptpI, where I:L(HA)∝⇕⊣√∫⊔≀→L(HC)is the identity map, and O:L(HA)∝⇕⊣√∫⊔≀→L(HB),R:L(HB)∝⇕⊣√∫⊔≀→L(HC) are defined as O(ρA) = Tr A[EABρA], R (ρB) = Tr B[φBCρB]. (101) Since the rank of the identity ma... | https://arxiv.org/abs/2502.04168v2 |
in the form of lemma 26: we have φk=1√dA, and the pure state |E⟩BB′C=|Φ+⟩BB′C defining the POVM element of the implementation can be written as |E⟩BB′C=/radicaligg 1 d2 AdA/summationdisplay k=1/radicalbig dA|k⟩BB′⊗|k⟩C, (113) 48 thus showing that for this protocol, ptp= 1/d2 A. We now wish to show that any teleportati... | https://arxiv.org/abs/2502.04168v2 |
preserving (CPTP), we have TrB(ρB|A) = 1A∗. The inverse isomorphism is given as follows and allows us to compute the action of the channel EA∝⇕⊣√∫⊔≀→Bon an input state ρA: EA∝⇕⊣√∫⊔≀→B(ρA) = Tr[τid AA∗ρB|AρA], whereτid AA∗=/summationtext i,j|i⟩⟨j|A⊗|i⟩⟨j|A∗is known as the link- ing operator. This equation has a striking... | https://arxiv.org/abs/2502.04168v2 |
HAin i⊗ /circlemultiplydisplay Ak∈Pa(Ai)H∗ Aout k where Pa(Ai)denotes the set of all parents of AiinG, such that [σAi|Pa(Ai),σAj|Pa(Aj)] = 0 for alli,jandσA1,...,A N=/producttextN i=1σAi|Pa(Ai)is a valid process operator. Note that the original papers [BLO20, BLO21] use ρAi|Pa(Ai)rather than σAi|Pa(Ai)for the ... | https://arxiv.org/abs/2502.04168v2 |
three vertices as three labs and using trivial in/output spaces for exogenous or childless vertices, we notice that the BLO framework associates a single global channel from the output space AoutofAto the inputs BinandCinwhose CJ operator is σBC|A∈L(HBin⊗HCin⊗H∗ Aout)(this would in fact be the process operator, if we i... | https://arxiv.org/abs/2502.04168v2 |
(BLO-QCM⊗).We define a tensor product restricted BLO-QCM, denoted BLO-QCM⊗as being specified by the following. 1. A causal structure which corresponds to directed graph Gwith vertices A1,...,A N. 2. For each Ai, a tensor factorisation of its output space as HAout i=/circlemultiplytext Ak∈Ch(Ai)HAout ik where Ch(Ai)deno... | https://arxiv.org/abs/2502.04168v2 |
CJ operators are ex- pressed. Each of the σivertices is assigned the quantum channel ˆσi:L(HAin i)∝⇕⊣√∫⊔≀→ L(/circlemultiplytext Ak∈Pa(Ai)HAout ki)whose CJ operator is the operator σAi|Pa(Ai)of the BLO frame- work18. Each of the Aivertices is assigned the corresponding local quantum instru- ment,Maiwith a slight modifi... | https://arxiv.org/abs/2502.04168v2 |
since we have a choice 55 over the pre and post-selection mechanisms (El,φl)that result in a post-selected telepor- tation protocol (definition 4). Using the acyclic probability rule, the required probability Pr(x1,...,x N|a1,..,a N)G′of the original causal model on G′is then given as follows. Pr(x1,...,x N|a1,..,a N)G... | https://arxiv.org/abs/2502.04168v2 |
σB|AandσC|A. It is also not immediate how the overall channel EA∝⇕⊣√∫⊔≀→BCcan be written as a composition of the marginal channels EA∝⇕⊣√∫⊔≀→BandEA∝⇕⊣√∫⊔≀→C. In an earlier paper [ABH+17], it is shown that whenever the overall channel factorises into a product of its (commuting) marginals σBC|A=σB|AσC|A, thenHAout=/circ... | https://arxiv.org/abs/2502.04168v2 |
note that unfaithful map- pings into our formalism still exist for all process matrices and therefore for BLO-QCMs (based on unitary process matrices), and discuss this in the next subsection. 57 Process matrix protocols within our framework. In usual quantum protocols, op- erationsoccurinawell-definedandacyclicorder. ... | https://arxiv.org/abs/2502.04168v2 |
procedure as we used the previous section for BLO-QCM ⊗s. The generic graph for all processes would be the one where we have a directed cycle (σ,A i) and(Ai,σ)for eachAi, along with an edge (Ai,Xi)for every agent (see figure 2 for an example). The generic graph constructed above has a high connectivity, a process where... | https://arxiv.org/abs/2502.04168v2 |
Then, we define a mapping fromfunctionalcausalmodelstocausalmodels(definition2). Finally, weproveconsistency between the results of [FGV25] and the quantum formalism presented here. C.1 Review of functional models Classical causal models [Pea09], also known as functional or structural causal models, de- scribe how vari... | https://arxiv.org/abs/2502.04168v2 |
edges of the graph are represented as , and all vertices are represented as v. Notice that the outgoing edges of an observed vertex are necessarily classical, since all edges are classical, which proves that definition 34 gives a well-defined causal graph. Causal model. Mapping a functional causal model to a causal mod... | https://arxiv.org/abs/2502.04168v2 |
associated to observed vertices). Thus, the input of all maps is always of the form considered in equation (140), up to a constant factor. Since the graph is finite and acyclic, the output of the maps composition will eventually be in C. By linearity and because of trace-preservation, such composition will equal the pr... | https://arxiv.org/abs/2502.04168v2 |
in Gtpall the vertices and edges of the subgraph G′associated with the same vertex types (observed or unobserved) and edge types (classical or quantum) as the original causal graph G. 3. Denoting the set of so-called split edges Es(Gtp) :=E\E′, for each edge (vi,v′ i)∈ Es(Gtp), include in Gtp, two vertices TiandRiand t... | https://arxiv.org/abs/2502.04168v2 |
both definitions involve considering an acyclic subgraph of G. The subgraphs that are used to constructdefinition38areasubsetofthoseconsideredfordefinition6. Indeed, inthelatter, one can remove any subset of edges from Gto make graph acyclic, while in definition 38 one removes all outgoing edges from a subset of vertic... | https://arxiv.org/abs/2502.04168v2 |
(V1⊥p,cV2|V3)Gimplies that∃Gc,tp∈Gc,tp(G) (definition38)suchthat (V1⊥dV2|V3∪Vpost)Gc,tp, whereVpostisthesetofallpost-selection vertices in Gc,tp. We now define a teleportation graph Gtp∈Gtp(G)(definition 6) by choosingEs(Gtp) := Out(Vs(Gc,tp)). Then the graphs GtpandGc,tpare equivalent up to the following replacement (... | https://arxiv.org/abs/2502.04168v2 |
tp), this means that v∈In(W)is also a split vertex, v∈Vs(Gc,tp). Therefore, there exists e= (v,v′)∈Es(Gtp). InG′ tpwe have UA B ...... v WW′and inGtpUA B ...... v ...W′. (154) Hence,W′∈V3∪Vpostand it is a descendent of U. This is not possible by hypothesis, meaning that even in G′ tp,Uhas no descendents in the conditio... | https://arxiv.org/abs/2502.04168v2 |
cyclic graph G, namely fCm G, and those of its image causal model through the embedding definition 36, namely Cm(fCm G)(which, for brevity, we will simply denote as CmGin what follows). To achieve this, we define auxiliary functional models (definition 32) and causal models (definition 2). Although the idea behind the ... | https://arxiv.org/abs/2502.04168v2 |
a specific teleportation graph Gtpinduced byCmG:CmGtp(definition 6 and definition 8). Specifically, we choose Gtpwith split edges correspond to the outgoing edges of split vertices of Gc,tp(seeStep 1), i.e., Es(Gtp) =∪v∈Vs(Gc,tp)Out(v). Letusdenotethesetofpost-selectionverticesof Gtpas˜Vpost. Theacyclicprobability dist... | https://arxiv.org/abs/2502.04168v2 |
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