evaluation of eeg localization methods using realistic simulations of interictal spikes performing an accurate localization of sources of interictal spikes from eeg scalp measurements is of particular interest during the presurgical investigation of epilepsy the purpose of this paper is to study the ability of six distributed source localization methods to recover extended sources of activated cortex due to the frequent lack of a gold standard to evaluate source localization methods our evaluation was performed in a controlled environment using realistic simulations of eeg interictal spikes involving several anatomical locations with several spatial extents simulated data were corrupted by physiological eeg noise simulations involving pairs of sources with the same amplitude were also studied in addition to standard validation criteria e g geodesic distance or mean square error we proposed an original criterion dedicated to assess detection accuracy based on receiver operating characteristic roc analysis six source localization methods were evaluated the minimum norm the minimum norm weighted by multivariate source prelocalization msp cortical loreta with or without additional minimum norm regularization and two derivations of the maximum entropy on the mean mem approach results showed that loreta based and mem based methods were able to accurately recover sources of different spatial extents with the exception of sources in temporo mesial and fronto mesial regions several spurious sources were generated by those methods however whereas methods using the msp always located very accurately the maximum of activity but not its spatial extent these findings suggest that one should always take into account the results from different localization methods when analyzing real interictal spikes

three dimensional vibrations of thick circular and annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick circular and annular plates the method is formulated in a manner which allows one to have any combination of free or fixed plate boundaries admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and axial co ordinates upper bound convergence of the non dimensional frequencies to at least four significant figures is demonstrated comparisons of results are made with ones obtained by others using 2 d mindlin thick plate theory and with other 3 d solutions extensive and accurate four significant figure frequencies are presented for completely free circular plates having thickness to diameter ratios of 0 2 0 3 0 4 and 0 5 for poisson s ratios 0 0 3 and 0 499 frequencies are also given for thick annular plates having a thickness to outer diameter of 0 2 inside to outside diameter ratios of 0 1 0 5 and 0 9 and 0 3 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional the circular and annular plate frequency data given is exact to at least four digits thus being benchmark data against which results from 2 d thick plate theories or other approximate methods e g finite elements may be compared

capacity results on multiple input single output wireless optical channels this paper derives upper and lower bounds on the capacity of the multiple input single output free space optical intensity channel with signal independent additive gaussian noise subject to both an average intensity and a peak intensity constraint in the limit where the signal to noise ratio snr tends to infinity the asymptotic capacity is specified while in the limit where the snr tends to zero the exact slope of the capacity is given

on the capacity of free space optical intensity channels upper and lower bounds are derived on the capacity of the free space optical intensity channel this channel has a nonnegative input representing the transmitted optical intensity which is corrupted by additive white gaussian noise to preserve the battery and for safety reasons the input is constrained in both its average and its peak power for a fixed ratio of the allowed average power to the allowed peak power the difference between the upper and the lower bound tends to zero as the average power tends to infinity and their ratio tends to one as the average power tends to zero when only an average power constraint is imposed on the input the difference between the bounds tends to zero as the allowed average power tends to infinity and their ratio tends to a constant as the allowed average power tends to zero

combined spatial and non spatial prior for inference on mri time series when modelling fmri and other mri time series data a bayesian approach based on adaptive spatial smoothness priors is a compelling alternative to using a standard generalized linear model glm on presmoothed data another benefit of the bayesian approach is that biophysical prior information can be incorporated in a principled manner however this requirement for a fixed non spatial prior on a parameter would normally preclude using spatial regularization on that same parameter we have developed a gaussian process based prior to apply adaptive spatial regularization while still ensuring that the fixed biophysical prior is correctly applied on each voxel a parameterized covariance matrix provides separate control over the variance the diagonal elements and the between voxel correlation due to off diagonal elements analysis proceeds using evidence optimization eo with variational bayes vb updates used for some parameters the method can also be applied to non linear forward models by using a linear taylor expansion centred on the latest parameter estimates applying the method to fmri with a constrained haemodynamic response function hrf shape model shows improved fits in simulations compared to using either the non spatial or spatial smoothness prior alone we also analyse multi inversion arterial spin labelling data using a non linear perfusion model to estimate cerebral blood flow and bolus arrival time by combining both types of prior information this new prior performs consistently well across a wider range of situations than either prior alone and provides better estimates when both types of prior information are relevant

bounds on the capacity region of the optical intensity multiple access channel this paper provides new inner and outer bounds on the capacity region of the optical intensity multiple access channel oimac with a per user average or peak power constraint for the average power constrained oimac our bounds at high power are asymptotically tight thereby characterizing the asymptotic capacity region the bounds are extended to the k user oimac with an average power constraint without loss of asymptotic optimality for the peak power constrained oimac at high power we bound the asymptotic capacity region to within 0 09 bits and determine the asymptotic capacity region in the symmetric case at moderate power for both types of constraints the capacity regions are bounded to within fairly small gaps

bayesian fmri time series analysis with spatial priors we describe a bayesian estimation and inference procedure for fmri time series based on the use of general linear models glms importantly we use a spatial prior on regression coefficients which embodies our prior knowledge that evoked responses are spatially contiguous and locally homogeneous further using a computationally efficient variational bayes framework we are able to let the data determine the optimal amount of smoothing we assume an arbitrary order auto regressive ar model for the errors our model generalizes earlier work on voxel wise estimation of glm ar models and inference in glms using posterior probability maps ppms results are shown on simulated data and on data from an event related fmri experiment

prior knowledge on cortex organization in the reconstruction of source current densities from eeg abstract the reconstruction of the generators of electroencephalographic eeg signals is important for understanding brain processes since the inverse problem has no unique solution additional knowledge or assumptions are needed often results from other anatomical or functional measurement modalities are difficult to interpret directly in terms of eeg source strengths but they provide valuable information about the functional similarity between brain regions for example in form of parcellations we propose a novel approach to the incorporation of such parcellations as priors into the reconstruction of distributed source current densities from eeg two algorithms are described based on a surface constrained loreta low resolution electromagnetic tomography approach the first patchloreta1 uses both topological neighborhood and prior information to define smoothness while the second patchloreta2 neglects topological neighborhood computer simulations using a smooth reconstruction surface on the brain envelope reveal important aspects of the algorithms performance in particular the influences of noise and incongruence between measurements and prior information it turns out that patchloreta1 makes efficient use of the provided prior information and at the same time is quite robust towards faulty priors as well as noise the algorithms are also tested on the localization of the sources of event related potentials here both the smooth brain and folded cortical surfaces serve as reconstruction spaces we find that patchloreta1 becomes ineffective on the folded cortex while patchloreta2 yields plausible results we also discuss the extension of the proposed algorithms to other types of priors and propose ways to overcome shortcomings of the current implementation

hierarchical bayesian inference for the eeg inverse problem using realistic fe head models depth localization and source separation for focal primary currents the estimation of the activity related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill posed inverse problem this is especially true in the recovery of brain networks involving deep lying sources by means of eeg meg recordings which is still a challenging task for any inverse method recently hierarchical bayesian modeling hbm emerged as a unifying framework for current density reconstruction cdr approaches comprising most established methods as well as offering promising new methods our work examines the performance of fully bayesian inference methods for hbm for source configurations consisting of few focal sources when used with realistic high resolution finite element fe head models the main foci of interest are the correct depth localization a well known source of systematic error of many cdr methods and the separation of single sources in multiple source scenarios both aspects are very important in the analysis of neurophysiological data and in clinical applications for these tasks hbm provides a promising framework and is able to improve upon established cdr methods such as minimum norm estimation mne or sloreta in many aspects for challenging multiple source scenarios where the established methods show crucial errors promising results are attained additionally we introduce wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios

classical and bayesian inference in neuroimaging theory this paper reviews hierarchical observation models used in functional neuroimaging in a bayesian light it emphasizes the common ground shared by classical and bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical bayesian framework in particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between classical inference and parametric empirical bayes peb through covariance component estimation this estimation is based on an expectation maximization or em algorithm the key point is that hierarchical models not only provide for appropriate inference at the highest level but that one can revisit lower levels suitably equipped to make bayesian inferences bayesian inferences eschew many of the difficulties encountered with classical inference and characterize brain responses in a way that is more directly predicated on what one is interested in the motivation for bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods in particular restricted maximum likelihood reml this paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging these issues include i estimating nonsphericity or variance components in fmri time series that can arise from serial correlations within subject or are induced by multisubject i e hierarchical studies ii spatiotemporal bayesian models for imaging data in which voxels specific effects are constrained by responses in other voxels iii bayesian estimation of nonlinear models of hemodynamic responses and iv principled ways of mixing structural and functional priors in eeg source reconstruction although diverse all these estimation problems are accommodated by the peb framework described in this paper

arbitrarily varying channels with general alphabets and states previous results of csiszar and narayan 1988 1989 1991 on the capacity of discrete arbitrarily varying channels with input and state constraints are extended to the case of arbitrary alphabets and state sets for channels with scalar or vector inputs and additive interference consisting of a deterministic part and noise both arbitrarily varying subject to power constraints the capacity for deterministic codes is shown to be equal to the random coding capacity if the input power exceeds the power of the deterministic interference and zero otherwise explicit capacity formulas are also given

graph partitioned spatial priors for functional magnetic resonance images abstract spatial models of functional magnetic resonance imaging fmri data allow one to estimate the spatial smoothness of general linear model glm parameters and eschew pre process smoothing of data entailed by conventional mass univariate analyses recently diffusion based spatial priors harrison l m penny w daunizeau j and friston k j 2008 diffusion based spatial priors for functional magnetic resonance images neuroimage were proposed which provide a way to formulate an adaptive spatial basis where the diffusion kernel of a weighted graph laplacian wgl is used as the prior covariance matrix over glm parameters an advantage of these is that they can be used to relax the assumption of isotropy and stationarity implicit in smoothing data with a fixed gaussian kernel the limitation of diffusion based models is purely computational due to the large number of voxels in a brain volume one solution is to partition a brain volume into slices using a spatial model for each slice this reduces computational burden by approximating the full wgl with a block diagonal form where each block can be analysed separately while fmri data are collected in slices the functional structures exhibiting spatial coherence and continuity are generally three dimensional calling for a more informed partition we address this using the graph laplacian to divide a brain volume into sub graphs whose shape can be arbitrary their shape depends crucially on edge weights of the graph which can be based on the euclidean distance between voxels isotropic or on glm parameters anisotropic encoding functional responses the result is an approximation the full wgl that retains its 3d form and also has potential for parallelism we applied the method to high resolution 1 mm3 fmri data and compared models where a volume was divided into either slices or graph partitions models were optimized using expectation maximization and the approximate log evidence computed to compare these different ways to partition a spatial prior the high resolution fmri data presented here had greatest evidence for the graph partitioned anisotropic model which was best able to preserve fine functional detail

a strong version of the redundancy capacity theorem of universal coding the capacity of the channel induced by a given class of sources is well known to be an attainable lower bound on the redundancy of universal codes with respect to this class both in the minimax sense and in the bayesian maximin sense we show that this capacity is essentially a lower bound also in a stronger sense that is for most sources in the class this result extends rissanen s 1984 1986 lower bound for parametric families we demonstrate the applicability of this result in several examples e g parametric families with growing dimensionality piecewise fixed sources arbitrarily varying sources and noisy samples of learnable functions finally we discuss implications of our results to statistical inference

a switching multi scale dynamical network model of eeg meg we introduce a new generative model of the encephalography eeg meg data the inversion of which allows for inferring the locations and temporal evolution of the underlying sources as well as their dynamical interactions the proposed switching mesostate space model smsm builds on the multi scale generative model for eeg meg by daunizeau and friston 2007 smsm inherits the assumptions that 1 bioelectromagnetic activity is generated by a set of distributed sources 2 the dynamics of these sources can be modelled as random fluctuations about a small number of mesostates and 3 the number of mesostates engaged by a cognitive task is small additionally four generalising assumptions are now included 4 the mesostates interact according to a full dynamical causal network dcn that can be estimated 5 the dynamics of the mesostates can switch between multiple approximately linear operating regimes 6 each operating regime remains stable over finite periods of time temporal clusters and 7 the total number of times the mesostates dynamics can switch is small the proposed model adds therefore a level of flexibility by accommodating complex brain processes that cannot be characterised by purely linear and stationary gaussian dynamics importantly the smsm furnishes a new interpretation of the eeg meg data in which the source activity may have multiple discrete modes of behaviour each with approximately linear dynamics this is modelled by assuming that the connection strengths of the underlying mesoscopic dcn are time dependent but piecewise constant i e they can undergo discrete changes over time a variational bayes inversion scheme is derived to estimate all the parameters of the model by maximising a negative free energy lower bound on the model evidence this bound is used to select among different model choices that are defined by the number of mesostates as well as by the number of stationary linear regimes the full model is compared to a simplified version that uses no dynamical assumptions as well as to a standard eeg inversion technique the comparison is carried out using an extensive set of simulations and the application of smsm to a real data set is also demonstrated our results show that for experimental situations in which we have some a priori belief that there are multiple approximately linear dynamical regimes the proposed smsm provides a natural modelling tool

a verifiable solution to the meg inverse problem magnetoencephalography meg is a non invasive brain imaging technique with the potential for very high temporal and spatial resolution of neuronal activity the main stumbling block for the technique has been that the estimation of a neuronal current distribution based on sensor data outside the head is an inverse problem with an infinity of possible solutions many inversion techniques exist all using different a priori assumptions in order to reduce the number of possible solutions although all techniques can be thoroughly tested in simulation implicit in the simulations are the experimenter s own assumptions about realistic brain function to date the only way to test the validity of inversions based on real meg data has been through direct surgical validation or through comparison with invasive primate data in this work we constructed a null hypothesis that the reconstruction of neuronal activity contains no information on the distribution of the cortical grey matter to test this we repeatedly compared rotated sections of grey matter with a beamformer estimate of neuronal activity to generate a distribution of mutual information values the significance of the comparison between the un rotated anatomical information and the electrical estimate was subsequently assessed against this distribution we found that there was significant p 0 05 anatomical information contained in the beamformer images across a number of frequency bands based on the limited data presented here we can say that the assumptions behind the beamformer algorithm are not unreasonable for the visual motor task investigated

proper complex random processes with applications to information theory the covariance of complex random variables and processes when defined consistently with the corresponding notion for real random variables is shown to be determined by the usual complex covariance together with a quantity called the pseudo covariance a characterization of uncorrelatedness and wide sense stationarity in terms of covariance and pseudo covariance is given complex random variables and processes with a vanishing pseudo covariance are called proper it is shown that properness is preserved under affine transformations and that the complex multivariate gaussian density assumes a natural form only for proper random variables the maximum entropy theorem is generalized to the complex multivariate case the differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum if and only if the random vector is proper gaussian and zero mean the notion of circular stationarity is introduced for the class of proper complex random processes a discrete fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain an application of the theory is presented

bayesian model averaging in eeg meg imaging in this paper the bayesian theory is used to formulate the inverse problem ip of the eeg meg this formulation offers a comparison framework for the wide range of inverse methods available and allows us to address the problem of model uncertainty that arises when dealing with different solutions for a single data in this case each model is defined by the set of assumptions of the inverse method used as well as by the functional dependence between the data and the primary current density pcd inside the brain the key point is that the bayesian theory not only provides for posterior estimates of the parameters of interest the pcd for a given model but also gives the possibility of finding posterior expected utilities unconditional on the models assumed in the present work this is achieved by considering a third level of inference that has been systematically omitted by previous bayesian formulations of the ip this level is known as bayesian model averaging bma the new approach is illustrated in the case of considering different anatomical constraints for solving the ip of the eeg in the frequency domain this methodology allows us to address two of the main problems that affect linear inverse solutions lis a the existence of ghost sources and b the tendency to underestimate deep activity both simulated and real experimental data are used to demonstrate the capabilities of the bma approach and some of the results are compared with the solutions obtained using the popular low resolution electromagnetic tomography loreta and its anatomically constraint version cloreta

capacity achieving probability measure for conditionally gaussian channels with bounded inputs a conditionally gaussian channel is a vector channel in which the channel output given the channel input has a gaussian distribution with well behaved input dependent mean and covariance we study the capacity achieving probability measure for conditionally gaussian channels subject to bounded input constraints and average cost constraints many practical communication systems including additive gaussian noise channels certain optical channels fading channels and interference channels fall within this framework subject to bounded input constraint and average cost constraints we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving under certain conditions the capacity achieving measure is proved to be discrete

multi subject analyses with dynamic causal modeling currently most studies that employ dynamic causal modeling dcm use random effects rfx analysis to make group inferences applying a second level frequentist test to subjects parameter estimates in some instances however fixed effects ffx analysis can be more appropriate such analyses can be implemented by combining the subjects posterior densities according to bayes theorem either on a multivariate bayesian parameter averaging or bpa or univariate basis posterior variance weighted averaging or pvwa or by applying dcm to time series averaged across subjects beforehand temporal averaging or ta while all these ffx approaches have the advantage of allowing for bayesian inferences on parameters a systematic comparison of their statistical properties has been lacking so far based on simulated data generated from a two region network we examined the effects of signal to noise ratio snr and population heterogeneity on group level parameter estimates data sets were simulated assuming either a homogeneous large population n 60 with constant connectivities across subjects or a heterogeneous population with varying parameters ta showed advantages at lower snr but is limited in its applicability because bpa and pvwa take into account posterior co variance structure they can yield non intuitive results when only considering posterior means this problem is relevant for high snr data pronounced parameter interdependencies and when ffx assumptions are violated i e inhomogeneous groups it diminishes with decreasing snr and is absent for models with independent parameters or when ffx assumptions are appropriate group results obtained with these ffx approaches should therefore be interpreted carefully by considering estimates of dependencies among model parameters

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

fading channels how perfect need perfect side information be the analysis of flat fading channels is often performed under the assumption that the additive noise is white and gaussian and that the receiver has precise knowledge of the realization of the fading process these assumptions imply the optimality of gaussian codebooks and of scaled nearest neighbor decoding here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process we quantify the degradation in performance that results from such estimation errors and demonstrate the lack of robustness of this scheme for some situations we suggest the rule of thumb that in order to avoid degradation the estimation error should be negligible compared to the reciprocal of the signal to noise ratio snr

on the capacity of the gaussian channel with a finite number of input levels an analytic confirmation of an observation made by g ungerboeck 1965 that approximately optimal performance on a gaussian channel with capacity c can be obtained with about 2 sup c 1 levels is given in particular the results imply that by using about 2 sup c 1 levels a rate of nearly c 1 bit can be achieved and that by using 2 sup c 1 levels a rate of about c 0 4 bits can be achieved

effect of satisfying stress boundary conditions in the axisymmetric vibration analysis of circular and annular plates abstract in the present study effect of satisfying stress boundary conditions in addition to displacement boundary conditions in the axisymmetric vibration analysis of circular and annular plates is investigated a new axisymmetric finite element which is based on a combination of the conventional displacement type variational principle and reissner s principle is proposed with this formulation stresses like displacements are primary variables and both displacement and stress boundary conditions can be easily and exactly imposed axisymmetric vibration frequencies of some typical circular and annular plates are then obtained with the present approach and are compared with those by the displacement type axisymmetric finite element based on the results of the present method it is found that the conventional finite element though not satisfying stress boundary conditions can still obtain sufficiently accurate vibration frequencies of circular and annular plates

on the discreteness of capacity achieving distributions we consider a scalar additive channel x spl rarr x n whose input is amplitude constrained by extending smith s 1969 argument we derive a sufficient condition on noise probability density functions pdf that guarantee finite support for the associated capacity achieving distribution s

the capacity of average and peak power limited quadrature gaussian channels the capacity c spl rho sub a spl rho sub p of the discrete time quadrature additive gaussian channel qagc with inputs subjected to normalized average and peak power constraints spl rho sub a and spl rho sub p respectively is considered by generalizing smith s results for the scalar average and peak power constrained gaussian channel it is shown that the capacity achieving distribution is discrete in amplitude envelope having a finite number of mass points with a uniformly distributed independent phase and it is geometrically described by concentric circles it is shown that with peak power being solely the effective constraint a constant envelope with uniformly distributed phase input is capacity achieving for spl rho sub p spl les 7 8 db 4 8 db per dimension the capacity under a peak power constraint is evaluated for a wide range of spl rho sub p by incorporating the theoretical observations into a nonlinear dynamic programming procedure closed form expressions for the asymptotic low and large spl rho sub a and spl rho sub p capacity and the corresponding capacity achieving distribution and for lower and upper bounds on the capacity c spl rho sub a spl rho sub p are developed the capacity c spl rho sub a spl rho sub p provides an improved ultimate upper bound on the reliable information rates transmitted over the qagc with any communication systems subjected to both average and peak power limitations when compared to the classical shannon formula for the capacity of the qagc which does not account for the peak power constraint this is in particular important for systems that operate with restrictive close to 1 average to peak power ratio spl rho sub a spl rho sub p and at moderate power values

finite element analysis of three dimensional vibrations of thick circular and annular plates abstract three dimensional vibrations of thick circular and annular plates are analyzed by a finite element method which with a properly assumed set of displacement field is different from the traditional 3 d finite element analysis and is reduced to a sequence of 2 d analyses one for each circumferential wave number the present approach has several unique features 1 it can obtain vibration frequencies which are comparable to or as accurate as those by other three dimensional approaches whenever comparisons between them are possible and yet the formulation of the present method is simpler and its application is straightforward 2 different simply supported boundary conditions can be exactly imposed with the present approach the latter feature may be quite difficult if not impossible for other methods in the present analysis vibration frequencies of a circular and an annular plate under different combinations of boundary conditions wave numbers and finite element meshes are calculated vibration characteristics are examined and some typical mode shapes are shown graphically then the results of some example problems by the present method are compared to those available in the literature to show its validity and accuracy

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

the capacity achieving distribution for the amplitude constrained additive gaussian channel an upper bound on the number of mass points this paper studies an n dimensional additive gaussian noise channel with a peak power constrained input it is well known that in this case when n 1 the capacity achieving input distribution is discrete with finitely many mass points and when n 1 the capacity achieving input distribution is supported on finitely many concentric shells however due to the previous proof technique not even a bound on the exact number of mass points shells was available this paper provides an alternative proof of the finiteness of the number mass points shells of the capacity achieving input distribution while producing the first firm bounds on the number of mass points and shells paving an alternative way for approaching many such problems the first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity achieving output probability density function next this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution an o mathsf a 2 upper bound where mathsf a denotes the constraint on the input amplitude the second main result of this paper generalizes the first one to the case when n 1 showing that for each and every dimension n ge 1 the number of shells that the optimal input distribution contains is o mathsf a 2 finally the third main result of this paper reconsiders the case n 1 with an additional average power constraint demonstrating a similar o mathsf a 2 bound

hierarchical bayesian estimates of distributed meg sources theoretical aspects and comparison of variational and mcmc methods magnetoencephalography meg provides millisecond scale temporal resolution for noninvasive mapping of human brain functions but the problem of reconstructing the underlying source currents from the extracranial data has no unique solution several distributed source estimation methods based on different prior assumptions have been suggested for the resolution of this inverse problem recently a hierarchical bayesian generalization of the traditional minimum norm estimate mne was proposed in which the variance of distributed current at each cortical location is considered as a random variable and estimated from the data using the variational bayesian vb framework here we introduce an alternative scheme for performing bayesian inference in the context of this hierarchical model by using markov chain monte carlo mcmc strategies in principle the mcmc method is capable of numerically representing the true posterior distribution of the currents whereas the vb approach is inherently approximative we point out some potential problems related to hyperprior selection in the previous work and study some possible solutions a hyperprior sensitivity analysis is then performed and the structure of the posterior distribution as revealed by the mcmc method is investigated we show that the structure of the true posterior is rather complex with multiple modes corresponding to different possible solutions to the source reconstruction problem we compare the results from the vb algorithm to those obtained from the mcmc simulation under different hyperparameter settings the difficulties in using a unimodal variational distribution as a proxy for a truly multimodal distribution are also discussed simulated meg data with realistic sensor and source geometries are used in performing the analyses

dynamical causal modelling for m eeg spatial and temporal symmetry constraints we describe the use of spatial and temporal constraints in dynamic causal modelling dcm of magneto and electroencephalography m eeg data dcm for m eeg is based on a spatiotemporal generative model of electromagnetic brain activity the temporal dynamics are described by neural mass models of equivalent current dipole ecd sources and their spatial expression is modelled by parameterized lead field functions often in classical ecd models symmetry constraints are used to model homologous pairs of dipoles in both hemispheres these constraints are motivated by assumptions about symmetric activation of bilateral sensory sources in classical approaches these constraints are hard i e the parameters of homologous dipoles are shared here in the context of dcm we illustrate the use of informed bayesian priors to implement soft symmetry constraints that are expressed in the posterior estimates only when supported by the data critically with dcm one can deploy symmetry constraints in either the temporal or spatial components of the model this enables one to test for symmetry in temporal neural mass parameters in the presence of non symmetric spatial expressions of homologous sources and vice versa furthermore we demonstrate that bayesian model comparison can be used to identify the best models among a range of symmetric and non symmetric variants our main finding is that the use of soft symmetry priors is recommended for evoked responses to bilateral sensory input we illustrate the use of symmetry constraints in dcm on synthetic and real eeg data

bayesian estimation of dynamical systems an application to fmri this paper presents a method for estimating the conditional or posterior distribution of the parameters of deterministic dynamical systems the procedure conforms to an em implementation of a gauss newton search for the maximum of the conditional or posterior density the inclusion of priors in the estimation procedure ensures robust and rapid convergence and the resulting conditional densities enable bayesian inference about the model parameters the method is demonstrated using an input state output model of the hemodynamic coupling between experimentally designed causes or factors in fmri studies and the ensuing bold response this example represents a generalization of current fmri analysis models that accommodates nonlinearities and in which the parameters have an explicit physical interpretation second the approach extends classical inference based on the likelihood of the data given a null hypothesis about the parameters to more plausible inferences about the parameters of the model given the data this inference provides for confidence intervals based on the conditional density 2002 elsevier science usa

on the capacity of mimo optical wireless channels this paper studies the capacity of a general multiple input multiple output mimo free space optical intensity channel under a per input antenna peak power constraint and a total average power constraint over all input antennas the focus is on the scenario with more transmit than receive antennas in this scenario different input vectors can yield identical distributions at the output when they result in the same image vector under multiplication by the channel matrix we first determine the most energy efficient input vectors that attain each of these image vectors based on this we derive an equivalent capacity expression in terms of the image vector and establish new lower and upper bounds on the capacity of this channel the bounds match when the signal to noise ratio snr tends to infinity establishing the high snr asymptotic capacity we also characterize the low snr slope of the capacity of this channel

nonlinear wavelet shrinkage with bayes rules and bayes factors abstract wavelet shrinkage the method proposed by the seminal work of donoho and johnstone is a disarmingly simple and efficient way of denoising data shrinking wavelet coefficients was proposed from several optimality criteria in this article a wavelet shrinkage by coherent bayesian inference in the wavelet domain is proposed the methods are tested on standard donoho johnstone test functions

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

data driven parceling and entropic inference in meg in amblard et al amblard c lapalme e lina j m 2004 biomagnetic source detection by maximum entropy and graphical models ieee trans biomed eng 55 3 427 442 the authors introduced the maximum entropy on the mean mem as a methodological framework for solving the magnetoencephalography meg inverse problem the main component of the mem is a reference probability density that enables one to include all kind of prior information on the source intensity distribution to be estimated this reference law also encompasses the definition of a model we consider a distributed source model together with a clustering hypothesis that assumes functionally coherent dipoles the reference probability distribution is defined as a prior parceling of the cortical surface in this paper we present a data driven approach for parceling out the cortex into regions that are functionally coherent based on the recently developed multivariate source pre localization msp principle mattout j pelegrini issac m garnero l benali h 2005 multivariate source pre localization msp use of functionally informed basis functions for better conditioning the meg inverse problem neuroimage 26 2 356 373 the data driven clustering ddc of the dipoles provides an efficient parceling of the sources as well as an estimate of parameters of the initial reference probability distribution on meg simulated data the ddc is shown to further improve the mem inverse approach as evaluated considering two different iterative algorithms and using classical error metrics as well as roc receiver operating characteristic curve analysis the mem solution is also compared to a loreta like inverse approach the data driven clustering allows to take most advantage of the mem formalism its main trumps lie in the flexible probabilistic way of introducing priors and in the notion of spatial coherent regions of activation the latter reduces the dimensionality of the problem in so doing it narrows down the gap between the two types of inverse methods the popular dipolar approaches and the distributed ones

multiple sparse priors for the m eeg inverse problem this paper describes an application of hierarchical or empirical bayes to the distributed source reconstruction problem in electro and magnetoencephalography eeg and meg the key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors this obviates the need to use priors with a specific form e g smoothness or minimum norm or with spatial structure e g priors based on depth constraints or functional magnetic resonance imaging results furthermore the inversion scheme allows for a sparse solution for distributed sources of the sort enforced by equivalent current dipole ecd models this means the approach automatically selects either a sparse or a distributed model depending on the data the scheme is compared with conventional applications of bayesian solutions to quantify the improvement in performance

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

dem a variational treatment of dynamic systems this paper presents a variational treatment of dynamic models that furnishes time dependent conditional densities on the path or trajectory of a system s states and the time independent densities of its parameters these are obtained by maximising a variational action with respect to conditional densities under a fixed form assumption about their form the action or path integral of free energy represents a lower bound on the model s log evidence or marginal likelihood required for model selection and averaging this approach rests on formulating the optimisation dynamically in generalised coordinates of motion the resulting scheme can be used for online bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches such as kalman and particle filtering furthermore it provides for dual and triple inferences on a system s states parameters and hyperparameters using exactly the same principles we refer to this approach as dynamic expectation maximisation dem

capacity bounds for discrete time amplitude constrained additive white gaussian noise channels the capacity achieving input distribution of the discrete time additive white gaussian noise awgn channel with an amplitude constraint is discrete and seems difficult to characterize explicitly a dual capacity expression is used to derive analytic capacity upper bounds for scalar and vector awgn channels the scalar bound improves on mckellips bound and is within 0 1 bit of capacity for all signal to noise ratios snrs the 2 d bound is within 0 15 bits of capacity provably up to 4 5 db numerical evidence suggests a similar gap for all snrs as the snr tends to infinity these bounds are accurate and match with a volume based lower bound for the 2 d complex case an analytic lower bound is derived by using a concentric constellation and is shown to be within 1 bit of capacity

simple proof of the concavity of the entropy power with respect to added gaussian noise a very simple proof of m h costa s result see ibid vol it 31 p 751 60 1985 that the entropy power of x sub t x n o ti is concave in t is derived as an immediate consequence of an inequality concerning fisher information this relationship between fisher information and entropy is found to be useful for proving the central limit theorem thus one who seeks new entropy inequalities should try first to find new equalities about fisher information or at least to exploit the existing ones in new ways

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

bayesian model reduction and empirical bayes for group dcm studies this technical note describes some bayesian procedures for the analysis of group studies that use nonlinear models at the first within subject level e g dynamic causal models and linear models at subsequent between subject levels its focus is on using bayesian model reduction to finesse the inversion of multiple models of a single dataset or a single hierarchical or empirical bayes model of multiple datasets these applications of bayesian model reduction allow one to consider parametric random effects and make inferences about group effects very efficiently in a few seconds we provide the relatively straightforward theoretical background to these procedures and illustrate their application using a worked example this example uses a simulated mismatch negativity study of schizophrenia we illustrate the robustness of bayesian model reduction to violations of the commonly used laplace assumption in dynamic causal modelling and show how its recursive application can facilitate both classical and bayesian inference about group differences finally we consider the application of these empirical bayesian procedures to classification and prediction

the impact of meg source reconstruction method on source space connectivity estimation a comparison between minimum norm solution and beamforming despite numerous important contributions the investigation of brain connectivity with magnetoencephalography meg still faces multiple challenges one critical aspect of source level connectivity largely overlooked in the literature is the putative effect of the choice of the inverse method on the subsequent cortico cortical coupling analysis we set out to investigate the impact of three inverse methods on source coherence detection using simulated meg data to this end thousands of randomly located pairs of sources were created several parameters were manipulated including inter and intra source correlation strength source size and spatial configuration the simulated pairs of sources were then used to generate sensor level meg measurements at varying signal to noise ratios snr next the source level power and coherence maps were calculated using three methods a l2 minimum norm estimate mne b linearly constrained minimum variance lcmv beamforming and c dynamic imaging of coherent sources dics beamforming the performances of the methods were evaluated using receiver operating characteristic roc curves the results indicate that beamformers perform better than mne for coherence reconstructions if the interacting cortical sources consist of point like sources on the other hand mne provides better connectivity estimation than beamformers if the interacting sources are simulated as extended cortical patches where each patch consists of dipoles with identical time series high intra patch coherence however the performance of the beamformers for interacting patches improves substantially if each patch of active cortex is simulated with only partly coherent time series partial intra patch coherence these results demonstrate that the choice of the inverse method impacts the results of meg source space coherence analysis and that the optimal choice of the inverse solution depends on the spatial and synchronization profile of the interacting cortical sources the insights revealed here can guide method selection and help improve data interpretation regarding meg connectivity estimation

model selection and accounting for model uncertainty in graphical models using occam s window abstract we consider the problem of model selection and accounting for model uncertainty in high dimensional contingency tables motivated by expert system applications the approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic p values leading to the selection of a single model inference is then conditional on the selected model the sampling properties of such a strategy are complex and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest in principle a panacea is provided by the standard bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models weighted by their posterior model probabilities furthermore this approach is optimal in the sense of maximizing predictive ability but this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large often greater than 1

the capacity of discrete time memoryless rayleigh fading channels we consider transmission over a discrete time rayleigh fading channel in which successive symbols face independent fading and where neither the transmitter nor the receiver has channel state information subject to an average power constraint we study the capacity achieving distribution of this channel and prove it to be discrete with a finite number of mass points one of them located at the origin we numerically compute the capacity and the corresponding optimal distribution as a function of the signal to noise ratio snr the behavior of the channel at low snr is studied and finally a comparison is drawn with the ideal additive white gaussian noise channel

a general minimax result for relative entropy suppose nature picks a probability measure p sub spl theta on a complete separable metric space x at random from a measurable set p sub spl theta p spl theta spl theta spl isin spl theta then without knowing spl theta a statistician picks a measure q on s finally the statistician suffers a loss d p sub 0 spl par q the relative entropy between p sub spl theta and q we show that the minimax and maximin values of this game are always equal and there is always a minimax strategy in the closure of the set of all bayes strategies this generalizes previous results of gallager 1979 and davisson and leon garcia 1980

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

meg source localization under multiple constraints an extended bayesian framework reconstructing the sources of magnetoencephalographic meg measurements is an ill posed inverse problem which does not admit a unique solution forward modeling of the observed signal and prior information or constraints on the solution are needed many types of constraints have been considered however a general framework is required that can accommodate for multiple priors and enables the evaluation of their contribution with this in mind we developed an empirical bayesian framework based on hierarchical linear models our approach entails the estimation of both the expected source distribution and its conditional variance the latter being constrained by an empirically determined mixture of prior variance components corresponding to the constraints one wants to introduce finally a second level of inference based on bayesian model selection enables one to compare different combinations of priors we describe here the methodological framework the estimation scheme and illustrate its application on a single subject meg data set

peak to average power ratio of good codes for gaussian channel consider a problem of forward error correction for the additive white gaussian noise awgn channel for finite blocklength codes the backoff from the channel capacity is inversely proportional to the square root of the blocklength in this paper it is shown that the codes achieving this tradeoff must necessarily have peak to average power ratio papr proportional to logarithm of the blocklength this is extended to codes approaching capacity slower and to papr measured at the output of an orthogonal frequency division multiplex ing modulator as a by product the convergence of smith s amplitude constrained awgn capacity to shannon s classical formula is characterized in the regime of large amplitudes this converse type result builds upon recent contributions in the study of empirical output distributions of good channel codes

multiresolution imaging of meg cortical sources using an explicit piecewise model imaging neural generators from meg magnetic fields is often considered as a compromise between computationally reasonable methodology that usually yields poor spatial resolution on the one hand and more sophisticated approaches on the other hand potentially leading to intractable computational costs we approach the problem of obtaining well resolved source images with unexcessive computation load with a multiresolution image model selection mims technique the building blocks of the mims source model are parcels of the cortical surface which can be designed at multiple spatial resolutions with the combination of anatomical and functional priors computation charge is reduced owing to 1 compact parametric models of the activation of extended brain parcels using current multipole expansions and 2 the optimization of the generalized cross validation error on image models which is closed form for the broad class of linear estimators of neural currents model selection can be complemented by any conventional imaging approach of neural currents restricted to the optimal image support obtained from mims the estimation of the location and spatial extent of brain activations is discussed and evaluated using extensive monte carlo simulations an experimental evaluation was conducted with meg data from a somatotopic paradigm results show that mims is an efficient image model selection technique with robust performances at realistic noise levels

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold