large sample learning of bayesian networks is np hard in this paper we provide new complexity results for algorithms that learn discrete variable bayesian networks from data our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest structure for which the model is able to represent the generative distribution exactly our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset we show that identifying high scoring structures is np hard even when any combination of one or more of the following hold the generative distribution is perfect with respect to some dag containing hidden variables we are given an independence oracle we are given an inference oracle we are given an information oracle we restrict potential solutions to structures in which each node has at most k parents for all k 3 our proof relies on a new technical result that we establish in the appendices in particular we provide a method for constructing the local distributions in a bayesian network such that the resulting joint distribution is provably perfect with respect to the structure of the network

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

edge augmentation of hypergraphs let g be a connected hypergraph we give a min max relation for the minimum number of edges that are required to increase the connectivity by 1 we also show that the corresponding algorithmic problem can be solved in polynomial time

sufficient conditions for hypergraphs to be maximally edge connected abstract let h be a connected hypergraph h is said to be linear if any two edges of h share at most one vertex if all edges of h have the same cardinality then h is uniform we call h maximally edge connected if the edge connectivity of h attains its minimum degree in this paper we present some sufficient conditions for linear uniform hypergraphs to be maximally edge connected that generalize the corresponding well known results for graphs

augmenting hypergraphs by edges of size two we give a good characterization for the minimum number of edges of size two whose addition to a given hypergraph h makes it k edge connected our result extends a previous theorem by e cheng 4 for the case when h is already k 1 edge connected we also describe a strongly polynomial algorithm to find both a minimum cardinality augmentation and a minimum cost augmentation where the cost of an edge is equal to the sum of the weights of its endvertices for some given weight function on v h our proof is based on a new splitting theorem for hypergraphs for the case when the special vertex s is only incident with edges of size two this theorem generalizes a splitting result of lovasz for graphs

local existence for the non resistive mhd equations in besov spaces in this paper we prove the existence of solutions to the viscous non resistive magnetohydrodynamics mhd equations on the whole of rn n 2 3 for divergence free initial data in certain besov spaces namely u0 bn 2 1 2 1 and b0 bn 2 2 1 the a priori estimates include the term t 0 u s 2 hn 2 ds on the right hand side which thus requires an auxiliary bound in hn 2 1 in 2d this is simply achieved using the standard energy inequality but in 3d an auxiliary estimate in h1 2 is required which we prove using the splitting method of calderon 1990 2 by contrast our proof that such solutions are unique only applies to the 3d case

computation causation and discovery in science business and policymaking anywhere data are used in prediction two sorts of problems requiring very different methods of analysis often arise the first problems of recognition and classification concerns learning how to use some features of a system to accurately predict other features of that system the second problems of causal discovery concerns learning how to predict those changes to some features of a system that will result if an intervention changes other features this book is about the second more difficult type of problem typical problems of causal discovery are how will a change in commission rates affect the total sales of a company how will a reduction in cigarette smoking among older smokers affect their life expectancy how will a change in the formula a college uses to award scholarships affect its dropout rate these sorts of changes are interventions that directly alter some features of the system and perhaps and this is the question indirectly alter others the contributors discuss recent research and applications using bayes nets or directed graphic representations including representations of feedback of recursive systems the book contains a thorough discussion of foundational issues algorithms proof techniques and applications to economics physics biology educational research and other areas

rigorous numerical verification of uniqueness and smoothness in a surface growth model abstract based on numerical data and a posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3d navier stokes equations for certain initial data for which analytical approaches fail the key point is the derivation of a scalar ode controlling the norm of the solution whose coefficients depend on the numerical data instead of solving this ode explicitly we explore three different numerical methods that provide rigorous upper bounds for its solution

optimal structure identification with greedy search in this paper we prove the so called meek conjecture in particular we show that if a dag h is an independence map of another dag g then there exists a finite sequence of edge additions and covered edge reversals in g such that 1 after each edge modification h remains an independence map of g and 2 after all modifications g h as shown by meek 1997 this result has an important consequence for bayesian approaches to learning bayesian networks from data in the limit of large sample size there exists a two phase greedy search algorithm that when applied to a particular sparsely connected search space provably identifies a perfect map of the generative distribution if that perfect map is a dag we provide a new implementation of the search space using equivalence classes as states for which all operators used in the greedy search can be scored efficiently using local functions of the nodes in the domain finally using both synthetic and real world datasets we demonstrate that the two phase greedy approach leads to good solutions when learning with finite sample sizes

a scoring function for learning bayesian networks based on mutual information and conditional independence tests we propose a new scoring function for learning bayesian networks from data using score search algorithms this is based on the concept of mutual information and exploits some well known properties of this measure in a novel way essentially a statistical independence test based on the chi square distribution associated with the mutual information measure together with a property of additive decomposition of this measure are combined in order to measure the degree of interaction between each variable and its parent variables in the network the result is a non bayesian scoring function called mit mutual information tests which belongs to the family of scores based on information theory the mit score also represents a penalization of the kullback leibler divergence between the joint probability distributions associated with a candidate network and with the available data set detailed results of a complete experimental evaluation of the proposed scoring function and its comparison with the well known k2 bdeu and bic mdl scores are also presented

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

mind change optimal learning of bayes net structure from dependency and independency data this paper analyzes the problem of learning the structure of a bayes net in the theoretical framework of gold s learning paradigm bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning we follow constraint based approaches to learning bayes net structure where learning is based on observed conditional dependencies and independencies between variables of interest e g the data are of the form x is dependent on y given any assignment to variables s or of the form x is independent of y given any assignment to variables s applying learning criteria in this model leads to the following results 1 the mind change complexity of identifying a bayes net graph over variables v from either dependency data or from independency data are v 2 the maximum number of edges 2 there is a unique fastest mind change optimal bayes net learner for either data type convergence speed is evaluated using gold s dominance notion of uniformly faster convergence for dependency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a minimum number of edges and outputs no guess otherwise for independency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a maximum number of edges and outputs no guess otherwise we investigate the complexity of computing the output of the fastest mind change optimal learner for either data type and show that each of these two problems is np hard assuming p rp to our knowledge these are the first np hardness results concerning the existence of a uniquely optimal bayes net structure

the 3d incompressible magnetohydrodynamic equations with fractional partial dissipation abstract the magnetohydrodynamic mhd equations have played pivotal roles in the study of many phenomena in geophysics astrophysics cosmology and engineering the fundamental problem of whether or not classical solutions of the 3d mhd equations can develop finite time singularities remains an outstanding open problem mathematically this problem is supercritical in the sense that the 3d mhd equations do not have enough dissipation if we replace the standard velocity dissipation u and the magnetic diffusion b by u and b respectively the resulting equations with 5 4 and 5 2 then always have global classical solutions an immediate issue is whether or not the hyperdissipation can be further reduced this paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion h 5 4 b where h 1 2 2 2

strong ill posedness of the incompressible euler equation in borderline sobolev spaces for the d dimensional incompressible euler equation the standard energy method gives local wellposedness for initial velocity in sobolev space h s mathbb r d s s c d 2 1 the borderline case s s c was a folklore open problem in this paper we consider the physical dimension d 2 and show that if we perturb any given smooth initial data in h s c norm then the corresponding solution can have infinite h s c norm instantaneously at t 0 in a companion paper 1 we settle the 3d and more general cases the constructed solutions are unique and even c infty smooth in some cases to prove these results we introduce a new strategy large lagrangian deformation induces critical norm inflation as an application we also settle several closely related open problems

the resistive magnetohydrodynamic equation near an equilibrium abstract this work intends to understand the stability and large time behavior of perturbations near a stationary solution of the 2d resistive magnetohydrodynamic mhd equation the stationary solution is taken to be a background magnetic field parallel to the horizontal axis we obtain three main results the first result assesses the stability and the precise large time asymptotic behavior for solutions to the linearzied system satisfied by the perturbation due to the lack of viscosity the standard energy estimates do not work and the proof is achieved by constructing a suitable lyapunov function the second result makes use of the special wave structure of the linearization to establish the linear stability and decay rates the third result obtains the h 1 stability for the full nonlinear system and shows that the l q norm q 2 of the magnetic field perturbation b and the l 2 norm of the gradient of b approach zero as t

local existence for the non resistive mhd equations in nearly optimal sobolev spaces this paper establishes the local in time existence and uniqueness of solutions to the viscous non resistive magnetohydrodynamics mhd equations in rd where d 2 3 with initial data b0 hs rd and u0 hs 1 e rd for s d 2 and any 0 e 1 the proof relies on maximal regularity estimates for the stokes equation the obstruction to taking e 0 is explained by the failure of solutions of the heat equation with initial data u0 hs 1 to satisfy u l1 0 t hs 1 we provide an explicit example of this phenomenon

coverings and structure of crossing families the second problem we consider is to find a compact representation of f we prove that there exists a so called hypercactus k of size o v consisting of cycles and hyper edges arranged in a tree like manner and a mapping from v to v k in such a way that there is a bijection between the minimum cuts of k and the members of f if f corresponds to the minimum cuts of a k edge connected graph then k reduces to the well known cactus representation of minimum cuts due to dinitz et al

on approximate solutions of the equations of incompressible magnetohydrodynamics abstract inspired by an approach proposed previously for the incompressible navier stokes ns equations we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics mhd on a torus of arbitrary dimension d this setting involves a sobolev space of infinite order made of c vector fields with vanishing divergence and mean on the torus given any approximate solution of the mhd cauchy problem its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution and to bound from above the sobolev distance of any order between the exact and the approximate solution in certain cases the above mentioned lower bound on the time of existence is found to be infinite so one infers the global existence of the exact mhd solution we present some applications of this general scheme the most sophisticated one lives in dimension d 3 with the abc flow perturbed magnetically as an initial datum and uses for the cauchy problem a galerkin approximate solution in 124 fourier modes we illustrate the conclusions arising in this case from the a posteriori analysis of the galerkin approximant these include the derivation of global existence of the exact mhd solution with the abc datum when the dimensionless viscosity and resistivity are equal and stay above a critical value

mind change complexity of learning logic programs the present paper motivates the study of mind change complexity for learning minimal models of length bounded logic programs it establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts building on angluin s notion of finite thickness and wright s work on finite elasticity shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data this paper shows that an effective version of shinohara s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound both in the context of learnability from positive data and for learnability from complete both positive and negative data let be a notation for the first limit ordinal then it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness then for each natural number m 0 the class of languages defined by formal systems of length m is identifiable in the limit from positive data with a mind change bound of m is identifiable in the limit from both positive and negative data with an ordinal mind change bound of m the above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of length bounded prolog programs including shapiro s linear programs arimura and shinohara s depth bounded linearly covering programs and krishna rao s depth bounded linearly moded programs it is also noted that the bound for learning from positive data is tight for the example classes considered

local existence and uniqueness for the non resistive mhd equations in homogeneous besov spaces abstract in the paper we consider the cauchy problem of the non resistive mhd equations in homogeneous besov spaces we prove the local existence and uniqueness of the solution to the non resistive mhd equations by using the iterative scheme and compactness arguments our obtained result improves considerably the recent results in 2 11

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

global small solutions of 2 d incompressible mhd system abstract in this paper we consider the global wellposedness of 2 d incompressible magneto hydrodynamical system with smooth initial data which is close to some non trivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system to achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation we first reformulate our system 1 1 in the lagrangian coordinates 2 19 then we employ anisotropic littlewood paley analysis to establish the key a priori l 1 r lip r 2 estimate for the lagrangian velocity field y t with this estimate we can prove the global wellposedness of 2 19 with smooth and small initial data by using the energy method we emphasize that the algebraic structure of 2 19 is crucial for the proofs to work the global wellposedness of the original system 1 1 then follows by a suitable change of variables

mind change efficient learning this paper studies efficient learning with respect to mind changes our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem but also locally in subproblems after receiving some evidence formalizing this idea leads to the notion of strong mind change optimality we characterize the structure of language classes that can be identified with at most mind changes by some learner not necessarily effective a language class l is identifiable with mind changes iff the accumulation order of l is at most accumulation order is a classic concept from point set topology we show that accumulation order is related to other established notions of structural complexity such as thickness and intrinsic complexity to aid the construction of learning algorithms we show that the characteristic property of strongly mind change optimal learners is that they output conjectures languages with maximal accumulation order we illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces one variable patterns and fixed length patterns