the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

computing bayes factors using a generalization of the savage dickey density ratio abstract we present a simple method for computing bayes factors the method derives from observing that in general a bayes factor can be written as the product of a quantity called the savage dickey density ratio and a correction factor both terms are easily estimated from posterior simulation in some cases it is possible to do these computations without ever evaluating the likelihood

natural conjugate priors for the instrumental variables regression model applied to the angrist krueger data we propose a natural conjugate prior for the instrumental variables regression model the prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior the jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator we construct informative priors for the angrist krueger 1991 does compulsory school attendance affect schooling and earnings quarterly journal of economics 106 979 1014 data and show that the marginal posterior of the return on education in the us coincides with the marginal posterior from the southern region when we use the jeffreys prior this result occurs since the instruments are the strongest in the southern region and the posterior using the jeffreys prior identical to maximum likelihood focusses on the strongest available instruments we construct informative priors for the other regions that make their posteriors of the return on education similar to that of the us and the southern region these priors show the amount of prior information needed to obtain comparable results for all regions

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

asymptotics of the teichmuller harmonic map flow abstract the teichmuller harmonic map flow introduced by rupflin and topping 2012 11 evolves both a map from a closed riemann surface to an arbitrary compact riemannian manifold and a constant curvature metric on the domain in order to reduce its harmonic map energy as quickly as possible in this paper we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time we obtain a branched minimal immersion from the degenerate domain

hammer hierarchical attribute matching mechanism for elastic registration a new approach is presented for elastic registration of medical images and is applied to magnetic resonance images of the brain experimental results demonstrate very high accuracy in superposition of images from different subjects there are two major novelties in the proposed algorithm first it uses an attribute vector i e a set of geometric moment invariants gmis that are defined on each voxel in an image and are calculated from the tissue maps to reflect the underlying anatomy at different scales the attribute vector if rich enough can distinguish between different parts of an image which helps establish anatomical correspondences in the deformation procedure it also helps reduce local minima by reducing ambiguity in potential matches this is a fundamental deviation of our method referred to as the hierarchical attribute matching mechanism for elastic registration hammer from other volumetric deformation methods which are typically based on maximizing image similarity second in order to avoid being trapped by local minima i e suboptimal poor matches hammer uses a successive approximation of the energy function being optimized by lower dimensional smooth energy functions which are constructed to have significantly fewer local minima this is achieved by hierarchically selecting the driving features that have distinct attribute vectors thus drastically reducing ambiguity in finding correspondence a number of experiments demonstrate that the proposed algorithm results in accurate superposition of image data from individuals with significant anatomical differences

synthesizing average 3d anatomical shapes abstract average shape estimates are often used to characterize normal morphological variation and discriminate dysmorphology in populations the purpose of this paper is to estimate average or the most representative shapes in populations by using high resolution medical images as input the average shape is computed from high dimensional spatial transformations used to co register each subject in the population rather than the image intensities inverse consistent image registration is used to help minimize correspondence errors and produce better population average estimates testing the method was done using a population of adult mr brain scans from 22 individuals with no know structural abnormalities population averages were computed using the spatial transformation method and local changes in morphology were mapped results suggest that this method is a feasible means for robust estimation of population average shape it is also shown that using inverse consistent transformations produces average shape estimates with less error compared to those produced with transformations with nontrivial inverse consistent errors

implicit reference based group wise image registration and its application to structural and functional mri abstract in this study an implicit reference group wise irg registration with a small deformation linear elastic model was used to jointly estimate correspondences between a set of mri images the performance of pair wise and group wise registration algorithms was evaluated for spatial normalization of structural and functional mri data traditional spatial normalization is accomplished by group to reference g2r registration in which a group of images are registered pair wise to a reference image g2r registration is limited due to bias associated with selecting a reference image in contrast implicit reference group wise irg registration estimates correspondences between a group of images by jointly registering the images to an implicit reference corresponding to the group average the implicit reference is estimated during irg registration eliminating the bias associated with selecting a specific reference image registration performance was evaluated using segmented t1 weighted magnetic resonance images from the nonrigid image registration evaluation project nirep dti and fmri images implicit reference pair wise irp registration a special case of irg registration for two images is shown to produce better relative overlap than irg for pair wise registration using the same small deformation linear elastic registration model however irp g2r registration is shown to have significant transitivity error i e significant inconsistencies between correspondences defined by different pair wise transformations in contrast irg registration produces consistent correspondence between images in a group at the cost of slightly reduced pair wise ro accuracy compared to irp g2r irg spatial normalization of the fractional anisotropy fa maps of dti is shown to have smaller fa variance compared with g2r methods using the same elastic registration model analyses of fmri data sets with sensorimotor and visual tasks show that irg registration on average increases the statistical detectability of brain activation compared to g2r registration

teichmuller harmonic map flow into nonpositively curved targets the teichmuller harmonic map flow deforms both a map from an oriented closed surface m into an arbitrary closed riemannian manifold and a constant curvature metric on m so as to reduce the energy of the map as quickly as possible 16 the flow then tries to converge to a branched minimal immersion when it can 16 18 the only thing that can stop the flow is a finite time degeneration of the metric on m where one or more collars are pinched in this paper we show that finite time degeneration cannot happen in the case that the target has nonpositive sectional curvature and indeed more generally in the case that the target supports no bubbles in particular when combined with 16 18 9 this shows that the flow will decompose an arbitrary such map into a collection of branched minimal immersions

bayesian and classical approaches to instrumental variable regression we establish relationships between certain bayesian and classical approaches to instrumental variable regression we determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2sls and liml and in doing so provide some new insight especially in the context of weak instruments to the small sample behavior of bayesian and classical procedures using a reduced rank restriction on a multivariate linear model we determine the priors that give rise to posteriors that are identical in functional form to the sampling densities of the classical 2sls and liml estimators

consistent image registration presents a new method for image registration based on jointly estimating the forward and reverse transformations between two images while constraining these transforms to be inverses of one another this approach produces a consistent set of transformations that have less pairwise registration error i e better correspondence than traditional methods that estimate the forward and reverse transformations independently the transformations are estimated iteratively and are restricted to preserve topology by constraining them to obey the laws of continuum mechanics the transformations are parameterized by a fourier series to diagonalize the covariance structure imposed by the continuum mechanics constraints and to provide a computationally efficient numerical implementation results using a linear elastic material constraint are presented using both magnetic resonance and x ray computed tomography image data the results show that the joint estimation of a consistent set of forward and reverse transformations constrained by linear elasticity give better registration results than using either constraint alone or none at all

posterior distributions in limited information analysis of the simultaneous equations model using the jeffreys prior abstract this paper studies the use of the jeffreys prior in bayesian analysis of the simultaneous equations model sem exact representations are obtained for the posterior density of the structural coefficient in canonical sems with two endogenous variables for the general case with m endogenous variables and an unknown covariance matrix the laplace approximation is used to derive an analytic formula for the same posterior density both the exact and the approximate formulas we derive are found to exhibit cauchy like tails analogous to comparable results in the classical literature on liml estimation moreover in the special case of a two equation just identified sem in canonical form the posterior density of is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding liml estimator this paper also examines the occurrence first documented in kleibergen and van dijk 1994a of a nonintegrable asymptotic cusp in the posterior distribution of the coefficient matrix of the reduced form equations for the included endogenous regressors an explanation for this phenomenon is provided in terms of the jacobian of the mapping from the structural model to the reduced form this interpretation assists in understanding the success of the jeffreys prior in resolving this problem

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

bubbling of the heat flows for harmonic maps from surfaces in this article we prove that any palais smale sequence of the energy functional on surfaces with uniformly l2 bounded tension fields converges pointwise by taking a subsequence if necessary to a map from connected possibly singular surfaces which consist of the original surfaces and finitely many bubble trees we therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time 1997 john wiley sons inc

flowing maps to minimal surfaces we introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow but is more effective at finding minimal surfaces in the genus 0 case our flow is just the harmonic map flow and it tries to find branched minimal 2 spheres as in sacks uhlenbeck 1981 and struwe 1985 etc in the genus 1 case we show that our flow is exactly equivalent to that considered by ding li liu 2006 in general we recover the result of schoen yau 1979 and sacks uhlenbeck 1982 that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on pi 1 and this minimal immersion will be homotopic to the original map in the case that pi 2 0

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

econometric model determination our general subject is model determination methods and their use in the prediction of economic time series the methods suggested are bayesian in spirit but they can be justified by classical as well as bayesian arguments the main part of the paper is concerned with model determination forecast evaluation and the construction of evolving sequences of models that can adapt in dimension and form including the way in which any nonstationarity in the data is modelled as new characteristics in the data become evident the paper continues some recent work on bayesian asymptotics by the author and werner ploberger 1995 develops embedding techniques for vector martingales that justify the role of a class of exponential densities in model selection and forecast evaluation and implements the modelling ideas in a multivariate regression framework that includes bayesian vector autoregressions bvar s and reduced rank regressions rrr s it is shown how the theory in the paper can be used i to construct optimized bvar s with data determined hyperparameters ii to compare models such as bvar s optimized bvar s and rrr s iii to perform joint order selection of cointegrating rank lag length and trend degree in a var and iv to discard data that may be irrelevant and thereby reset the initial conditions of a model

invariant bayesian inference in regression models that is robust against the jeffreys lindley s paradox we obtain the prior and posterior probability of a nested regression model as the hausdorff integral of the prior and posterior on the parameters of an encompassing linear regression model over a lower dimensional set that represents the nested model the hausdorff integral is invariant and therefore avoids the borel kolmogorov paradox basing priors and prior probabilities of nested regression models on the prior on the parameters of an encompassing linear regression model reduces the discrepancies between classical and bayesian inference like the jeffreys lindley s paradox we illustrate the analysis with examples of linear restrictions i e a linear regression model and non linear restrictions i e a cointegration and an autoregressive moving average model on the parameters of an encompassing linear regression model

bayes factors for nonlinear hypotheses and likelihood distributions summary new methods are proposed which allow bayes factors to be computed for hypotheses which impose nonlinear restrictions on the parameters projection methods are used to induce the prior distribution over the restricted parameter space which is required for computation of the bayes factor various distance metrics are introduced to define the projection including a utility based metric which gives kullback leibler divergence as a special case draws from the restricted and unrestricted prior distributions are used to construct marginal distributions of the likelihood which is shown to have additional diagnostic value over and above the bayes factor these methods are applied to hypotheses statisticians are frequently asked to entertain hypotheses which are nonlinear in the model parameters nonlinear hypotheses arise both from subject matter theory as well as from the process of model selection to address nonlinear hypotheses researchers use frequentist methods which are usually implemented using asymptotic approximations berger 1987 zellner 1971 1978 1984 and others have discussed the weaknesses of the classical significance testing approach and suggest bayesian alternatives such as odds ratios the odds ratio is the ratio of posterior probabilities of two competing hypotheses and is the product of the prior odds ratio and the bayes factor the odds ratio has the conceptual advantages of providing a natural probability metric for assessing the strength of data and prior beliefs because of the difficulties of assessing priors under the nonlinear restrictions and the computation of the required integrals few bayes factors have been computed for nonlinear hypotheses see lindley 1988 for an exception in the present paper we outline a general method for computing bayes factors which is flexible conceptually appealing and computationally tractable the literature on bayes factors starts with the work of jeffreys 1961 other references include berger delampady 1987 dickey 1971 good 1985 kass vaidyanathan 1992 smith spiegelhalter 1980 and zellner 1971 1978 1984 these papers deal with the testing of linear restrictions on the parameter space often in the context of linear models even the widely used schwarz 1978 model selection criterion considers only linear restrictions in models of the exponential family form the key difficulty which has kept researchers from developing bayes factors for nonlinear hypotheses is the

computing average shaped tissue probability templates this note presents a framework for generating tissue probability maps that represent the average shape of a number of subjects brain images the procedure is formulated as finding maximum a posteriori estimates within a probabilistic generative model estimating the parameters involves alternating between estimating the deformations that match tissue class images of individual subjects to template and updating the template according to the latest estimates of the deformations a multinomial matching criterion is used such that multiple tissue class images e g grey and white matter are registered simultaneously with the current template estimate in order to generalise the resulting template to a broader range of subjects a template blurriness prior is included within the model

twisted harmonic maps and the self duality equations si m est une surface de riemann compacte on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d auto dualite associees a une representation irreductible p 1 m psl 2 c