spectral flow is the integral of one forms on the banach manifold of self adjoint fredholm operators abstract one may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 vancouver icm address of i m singer our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded breuer fredholm operators in a semifinite von neumann algebra these formulae have a geometric interpretation which derives from the proof namely we define a family of banach submanifolds of all bounded self adjoint breuer fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path we emphasise that our methods do not give a single globally defined one form on the self adjoint breuer fredholms whose integral along all paths is spectral flow rather as the choice of the plural forms in the title suggests we need a family of such one forms in order to confirm singer s idea the original context for this result concerned paths of unbounded self adjoint fredholm operators we therefore prove analogous formulae for spectral flow in the unbounded case as well the proof is a synthesis of key contributions by previous authors whom we acknowledge in detail in the introduction combined with an additional important recent advance in the differential calculus of functions of non commuting operators

tweedie s formula and selection bias we suppose that the statistician observes some large number of estimates zi each with its own unobserved expectation parameter i the largest few of the zi s are likely to substantially overestimate their corresponding i s this being an example of selection bias or regression to the mean tweedie s formula first reported by robbins in 1956 offers a simple empirical bayes approach for correcting selection bias this article investigates its merits and limitations in addition to the methodology tweedie s formula raises more general questions concerning empirical bayes theory discussed here as relevance and empirical bayes information there is a close connection between applications of the formula and james stein estimation

a note on an alternative outlier model summary this paper examines modelling a single outlier in the normal theory fixed effects linear model as arising from an unknown observation with inflated variance the maximum likelihood estimates are characterized in terms of standard least squares statistics the estimated position of the outlier does not necessarily agree with the estimated position under the usual mean slippage outlier model and an example where they differ is presented a sufficient and common condition for agreement is given leads to considering the studentized residuals since under normality and the possibility of each observation being the outlier the maximum likelihood estimate for the position of the possible outlier corresponds to the case with the largest absolute studentized residual also the likelihood ratio test statistic for the presence of an outlier in this model is a monotonic function of the largest absolute studentized residual srikantan 1961 tietjen et al 1973 ellenberg 1976 intuitively one might expect similar results to be true whenever the effect of a single outlier is modelled identically for all observations regardless of values of the independent variables or fixed effects we shall see that this is not necessarily so we consider here a single outlier model that assumes an outlier arises from an error term with an inflated variance box and tiao 1968 consider this model from a bayesian point of view in section 2 the model is defined and shown to be after a parameter transformation a special case of harville s general linear model harville 1977 the joint maximum likelihood estimates mle s are characterized in section 3 and an example where the estimated outlier does not correspond to the observation with the largest absolute studentized residual is presented in section 4 in the final section several issues pertinent to the modelling of outliers are discussed in terms of this model and the mean slippage model our intent here is not to propose this model as a common replacement for the mean slippage model but rather to examine the performance of the mean slippage model under a plausible alternative

local shrinkage rules levy processes and regularized regression we use l evy processes to generate joint prior distributions and therefore penalty functions for a location parameter b b1 bp as p grows large this generalizes the class of local global shrinkage rules based on scale mixtures of normals illuminates new connections among disparate methods and leads to new results for computing posterior means and modes under a wide class of priors we extend this framework to large scale regularized regression problems where p n and provide comparisons with other methodologies

elimination of nuisance parameters with reference priors summary the problem of eliminating nuisance parameters is tackled from different points of view standard likelihood techniques such as profile likelihood and its modifications are compared with a bayesian analysis based on reference priors berger bernardo 1992a examples are considered to illustrate what happens in multiparameter problems in particular the frequentist coverage probabilities of the credible sets derived from the reference priors are shown to be better than those computed from a likelihood approach advantages and disadvantages of bayesian and likelihood analyses are discussed

the banach saks p property we obtain general theorems which enable the calculation of the banach saks index set for rearrangement invariant function spaces in terms of their boyd indices for many important classes of rearrangement invariant spaces the index set is non trivial if and only if the boyd indices of the underlying space are non trivial although the latter equivalence does not hold in full generality we also study banach saks index sets defined for weakly null sequences of identically distributed independent random variables and their connections with the classical banach saks index set

weak type commutator and lipschitz estimates resolution of the nazarov peller conjecture let m be a semi finite von neumann algebra and let f r c be a lipschitz function if a b m are self adjoint operators such that a b l1 m then

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

the best constants for operator lipschitz functions on schatten classes suppose that f is a lipschitz function on r with kfklip 1 let a be a bounded self adjoint operator on a hilbert space h let p 1 and suppose that x b h is an operator such that the commutator a x is contained in the schatten class sp it is proved by the last two authors that then also f a x sp and there exists a constant cp independent of x and f such that k f a x kp cpk a x kp the main result of this paper is to give a sharp estimate for cp in terms of p namely we show that cp p 2 p 1 in particular this gives the best estimates for operator lipschitz inequalities we treat this result in a more general setting this involves commutators of n self adjoint operators a1 an for which we prove the analogous result the case described here in the abstract follows as a special case

gerstenhaber algebras and bv algebras in poisson geometry the purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well known algebraic structures such as gerstenhaber algebras and bv algebras some applications are discussed in particular we find an explicit connection between the koszul brylinski operator and the modular class of a poisson manifold as a consequence we prove that poisson homology is isomorphic to poisson cohomology for unimodular poisson structures

a dixmier trace formula for the density of states a version of connes trace formula allows to associate a measure on the essential spectrum of a schrodinger operator with bounded potential in solid state physics there is another celebrated measure associated with such operators the density of states in this paper we demonstrate that these two measures coincide we show how this equality can be used to give explicit formulae for the density of states in some circumstances

outlier models and prior distributions in bayesian linear regression summary bayesian inference in regression models is considered using heavy tailed error distri butions to accommodate outliers the particular class of distributions that can be con structed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in bayesian linear modelling includ ing simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters the modelling of outliers in nominally normal linear regression models using alternative error distributions which are heavy tailed relative to the normal provides an automatic means of both detecting and accommodating possibly aberrant observations such realistic models do however often lead to analytically intractable analyses with complex posterior distributions in several dimensions that are difficult to summarize and understand in this paper we consider a special yet rather wide class of heavy tailed unimodal and symmetric error distributions for which the analyses though apparently intractable can be examined in some depth by exploiting certain properties of the assumed error form the distributions concerned are those that can be con structed as scale mixtures of normal distributions in his paper concerning location parameters de finetti 1961 discusses such distributions and suggests the hypothetical interpretation that each observation is taken using an instrument with normal error but each time chosen at random from a collection of instruments of different precisions the distribution of the

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

default bayesian analysis with global local shrinkage priors we provide a framework for assessing the default nature of a prior distribution using the property of regular variation which we study for global local shrinkage priors in particular we show that the horseshoe priors originally designed to handle sparsity are regularly varying and thus are appropriate for default bayesian analysis to illustrate our methodology we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions in each case we show that global local horseshoe priors perform as required global local shrinkage priors can separate a low dimensional signal from high dimensional noise even for nonlinear functions

differentiation of operator functions in non commutative lp spaces the principal results in this paper are concerned with the description of differentiable operator functions in the non commutative lp spaces 1 p associated with semifinite von neumann algebras for example it is established that if f r r is a lipschitz function then the operator function f is g teaux differentiable in l2 m for any semifinite von neumann algebra m if and only if it has a continuous derivative furthermore if f r r has a continuous derivative which is of bounded variation then the operator function f is g teaux differentiable in any lp m 1 p

fedosov quantization on symplectic ringed spaces we expose the basics of the fedosov quantization procedure placed in the general framework of symplectic ringed spaces this framework also includes some poisson manifolds with nonregular poisson structures presymplectic manifolds complex analytic symplectic manifolds etc

the optimal range of the calderon operator and its applications abstract we identify the optimal range of the calderon operator and that of the classical hilbert transform in the class of symmetric quasi banach spaces further consequences of our approach concern the optimal range of the triangular truncation operator operator lipschitz functions and commutator estimates in ideals of compact operators

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

reference prior bayesian analysis for normal mean products abstract two reference priors for the product of means of n normal distributions with common known variance are developed one of them induces an improper posterior distribution and therefore is not of much interest the other is a generalized form of the n 2 case derived by berger and bernardo the latter is compared with the uniform prior the jeffreys prior in posterior inference and the optimal frequentist coverage criterion the reference prior is shown to be better than the uniform prior in the sense of correct frequentist coverage probability of the posterior quantile by numerical computation the computation was performed by gibbs sampling for n 3 and n 10 furthermore it is shown that the reference prior is among the asymptotic optimal frequentist coverage probability priors under a transformation of the parameter space such that the parameter of interest and the nuisance parameters are orthogonal in an example the bayesian credible interval for the product of normal means using the re

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

exact and approximate posterior moments for a normal location parameter the forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions exact and approximate illustrations are given where the prior distribution is double exponential or student t respectively

double operator integrals this paper is concerned with perturbation formulae of the form f a f b lp m k a b lp m with k 0 being a constant depending on p and f only where f is a real valued lipschitz function and a b are self adjoint measurable operators affiliated with a semifinite von neumann algebra m such that the difference a b belongs to lp m 1 p in order to treat the situation where the von neumann algebra m is not necessarily hyperfinite we first develop an integration theory with respect to finitely additive spectral measures in a banach space applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to birman and solomyak to describe the class of integrable functions we employ our recent study of multiplier theory in umd spaces our perturbation formulae extend those of davies and birman solomyak for the case when m is a hyperfinite i factor i e for the schatten p classes we also discuss analogous perturbation results in the setting of symmetric operator spaces associated with m

bmo estimates for non commutative vector valued lipschitz functions abstract we construct markov semi groups t and associated bmo spaces on a finite von neumann algebra m and obtain results for perturbations of commutators and non commutative lipschitz estimates in particular we prove that for any a m self adjoint and f r r lipschitz there is a markov semi group t such that for x m f a x bmo m t c a b s f a x we obtain an analogue of this result for more general von neumann valued functions f r n n by imposing hormander mikhlin type assumptions on f in establishing these result we show that markov dilations of markov semi groups have certain automatic continuity properties we also show that markov semi groups of double operator integrals admit standard and reversed markov dilations

outliers and credence for location parameter inference abstract heavy tailed distributions are important for modeling problems in which there may be outlying observations or parameters examples of their use are given in o hagan 1988 this article develops some general theory based on the notion of credence for inference about unknown location parameters in the case of known variances a density on the real line is defined to have credence c if it is bounded above and below by positive multiples of 1 x 2 c 2 for instance a t distribution with d degrees of freedom has credence 1 d i prove that the credence of a sum of independent random variables is the minimum of their individual credences and that the credence of a posterior density of a location parameter is the sum of the credences of the prior and the observations more generally when independent information sources are combined their credences add when groups of information sources conflict outlier rejection occurs with the group having the greatest total credence dominating all others

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

lipschitz continuity of the absolute value and riesz projections in symmetric operator spaces a principal result of the paper is that ifeis a symmetric banach function space on the positive half line with the fatou property then for all semifinite von neumann algebras m the absolute value mapping is lipschitz continuous on the associated symmetric operator spacee m with lipschitz constant depending only oneif and only ifehas non trivial boyd indices it follows that if m is any von neumann algebra then the absolute value map is lipschitz continuous on the corresponding haageruplp space provided 1 p

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

compatible symplectic connections on a cotangent bundle and the fedosov quantization a global construction of a family of symplectic connections on a cotangent bundle compatible with some linear symmetric connection on the base space is proposed examples of the compatible symplectic connections are given a detailed analysis of an abelian connection and of flat sections for this kind of symplectic connection in the fedosov algorithm are presented some properties of the product determined by the compatible symplectic connection are shown

on the arazy conjecture concerning schur multipliers on schatten ideals abstract we confirm in the affirmative the conjecture made by arazy in 2 suppose that f is a continuously differentiable function on 1 1 such that f t c t t 1 1 for some constant c 0 and 0 let j j 1 l r for some 0 r if 0 p and 1 q satisfy 1 p r 1 q then the matrix f j f k j k j k 1 is a schur multiplier from the schatten class s q into s p we also consider a general version of this conjecture for arbitrary quasi banach ideals of compact operators

flexible empirical bayes estimation for wavelets summary wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression although bayes estimators can provide excellent mean squared error mse properties the selection of an effective prior is a difficult task to address this problem we propose empirical bayes eb prior selection methods for various error distributions including the normal and the heavier tailed student t distributions under such eb prior distributions we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging these eb estimators are seen to be computationally competitive with standard classical thresholding methods and to be robust to outliers in both the data and wavelet domains simulated and real examples are used to illustrate the flexibility and improved mse performance of these methods in a wide variety of settings