branch and bound search for experimental designs based on d optimality and other criteria this article presents a branch and bound algorithm that constructs a catalog of all d optimal n point designs for specified design region linear model and number of observations n while the primary design criterion is d optimality the algorithm may also be used to find designs performing well by other secondary criteria if a small sacrifice in ellkiency as measured by d optimahty is accepted finally some designs are supplied for a quadratic response surface model

estimation of nonstationary spatial covariance structure summary we introduce a method for estimating nonstationary spatial covariance structure from space time data and apply the method to an analysis of sydney wind patterns our method constructs a process honouring a given spatial covariance matrix at observing stations and uses one or more stationary processes to describe conditional behaviour given observing site values the stationary processes give a localised description of the spatial covariance structure the method is computationally attractive and can be extended to the assessment of covariance for multivariate processes the technique is illustrated for data describing the east west component of sydney winds for this example our own methods are contrasted with a geometrically appealing though computationally intensive technique which describes spatial correlation via an isotropic process and a deformation of the geographical space

practical codes for photon communication in a recent paper pierce studied the problems of communicating at optical frequencies using photon counting techniques and concluded that at low temperatures we encounter insuperable problems of encoding long before we approach channel capacity in this paper it is shown that even assuming a noiseless model for photon communication for which capacity measured in nats photon is infinite it is unlikely that a signaling efficiency of even 10 nats photon could be achieved practically on the positive side it is shown that pulse position modulation plus reed solomon coding yields practical results in the range of 2 to 3 nats photon

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

maximum entropy sampling and optimal bayesian experimental design when shannon entropy is used as a criterion in the optimal design of experiments advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters following previous work in which this idea was used in spatial sampling the method is applied to standard parameterized bayesian optimal experimental design under suitable conditions which include non linear as well as linear regression models it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy the usual bayesian criterion thus avoiding the use of conditional distributions it is shown using this marginal formulation that under normality assumptions every standard model which has a two point prior distribution on the parameters gives an optimal design supported on a single point other results include a new asymptotic formula which applies as the error variance is large and bounds on support size

multiplicity estimating algorithm for zeros of a complex polynomial and its applications abstract miyakoda t multiplicity estimating algorithm for zeros of a complex polynomial and its applications journal of computational and applied mathematics 46 1993 357 368 this paper presents a method for estimating the multiplicity of zeros of a complex polynomial when the durand kerner method is applied to solve it the method is based on previous this journal 1989 1992 work of the author numerical examples are given which show the efficiency of our algorithm compared with another method and we propose further applications for accelerating convergence and improving the accuracy of approximations for multiple zeros

information capacity of stochastic pooling networks is achieved by discrete inputs stochastic pooling networks spn are sensor networks where multiple sensors make independently noisy and compressed measurements of the same information source which are combined via pooling examples of spns range from nanoelectronics to biological sensory neurons here it is shown that optimal information transmission in spns with nodes that quantize to a finite number of states requires the input signal distribution to be discrete this is illustrated numerically for a simple spn consisting of n binary quantizing sensors the resultant information capacity is shown to be independent of the noise distribution when the signal distribution can be freely chosen but to imply an optimal noise distribution if the signal distribution is fixed while larger than the best performance of previously studied continuously valued input signals the capacity does not scale faster than the previous best result of text log 2 sqrt n bits per channel use it is also shown that a plot of the optimal input distribution contains bifurcations as n increases and that suprathreshold stochastic resonance occurs when the mutual information is determined for a suboptimal noise distribution

information capacity in the weak signal approximation we derive an approximate expression for mutual information in a broad class of discrete time stationary channels with continuous input under the constraint of vanishing input amplitude or power the approximation describes the input by its covariance matrix while the channel properties are described by the fisher information matrix this separation of input and channel properties allows us to analyze the optimality conditions in a convenient way we show that input correlations in memoryless channels do not affect channel capacity since their effect decreases fast with vanishing input amplitude or power on the other hand for channels with memory properly matching the input covariances to the dependence structure of the noise may lead to almost noiseless information transfer even for intermediate values of the noise correlations since many model systems described in mathematical neuroscience and biophysics operate in the high noise regime and weak signal conditions we believe that the described results are of potential interest also to researchers in these areas

approximation schemes for covering and packing problems in image processing and vlsi a unified and powerful approach is presented for devising polynomial approximation schemes for many strongly np complete problems such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e o with running time that is polynomial when e is fixed though the polynomiality of these algorithms depends on the degree of approximation e being fixed they cannot be improved owing to a negative result stating that there are no fully polynomial approximation schemes for strongly np complete problems unless np p the unified technique that is introduced here referred to as the shifting strategy is applicable to numerous geometric covering and packing problems the method of using the technique and how it varies with problem parameters are illustrated a similar technique independently devised by b s baker was shown to be applicable for covering and packing problems on planar graphs

iterative methods for multiple zeros of a polynomial by clustering abstract for the numerical determination of zeros of a complex polynomial by simultaneous iterative methods we show theorems on the fact that the approximations for a multiple zero gather around the exact zero when they become close approximations application of our theorems makes it possible to estimate the multiplicity of zeros dynamically as well as to improve the convergence to multiple zeros for the known methods e g durand kerner s or tanabe s methods

traub gander s family for the simultaneous determination of multiple zeros of polynomials abstract by combining traub gander s family of third order for finding a multiple zero and suitable corrective approximations od schroder s and halley s type a new family of iterative methods for the simultaneous approximation of multiple zeros of algebraic polynomials is proposed taking various forms of a function involved in the iterative formula a number of different simultaneous methods can be obtained it is proved that the order of convergence is 4 5 or 6 depending of the type of employed corrective approximations two numerical examples are given to demonstrate the convergence properties of the proposed family of simultaneous methods displayed trajectories of the sequences of approximations point to global characteristics of the proposed family of iterative methods

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

a note on some improvements of the simultaneous methods for determination of polynomial zeros abstract applying gauss seidel approach to the improvements of two simultaneous methods for finding polynomial zeros presented in 9 two iterative methods with faster convergence are obtained the lower bounds of the r order of convergence for the accelerated methods are given the improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations the considered methods are illustrated numerically in the example of an algebraic equation

maximizing information in unreliable sensor networks under deadline and energy constraints we study the problem of maximizing the information in a wireless sensor network with unreliable links we consider a sensor network with a tree topology where the root corresponds to the sink and the rest of the network detects an event and transmits data to the sink we formulate a combinatorial optimization problem that maximizes the information that reaches the sink under deadline energy and interference constraints this framework allows using a variety of error recovery schemes to tackle link unreliability we show that this optimization problem is np hard in the strong sense when the input is the maximum node degree of the tree we then propose a dynamic programming framework for solving the problem exactly which involves solving a special case of the job interval selection problem jisp at each node our solution has a polynomial time complexity when the maximum node degree is o logn in a tree with n nodes for trees with higher node degrees we further develop a suboptimal solution which has low complexity and allows distributed implementation we investigate tree structures for which this solution is optimal to the original problem the efficiency of the suboptimal solution is further demonstrated through numerical results on general trees

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

asymptotic theory of information theoretic experimental design we discuss an idea for collecting data in a relatively efficient manner our point of view is bayesian and information theoretic on any given trial we want to adaptively choose the input in such a way that the mutual information between the unknown state of the system and the stochastic output is maximal given any prior information including data collected on any previous trials we prove a theorem that quantifies the effectiveness of this strategy and give a few illustrative examples comparing the performance of this adaptive technique to that of the more usual nonadaptive experimental design in particular we calculate the asymptotic efficiency of the information maximization strategy and demonstrate that this method is in a well defined sense never less efficient and is generically more efficient than the nonadaptive strategy for example we are able to explicitly calculate the asymptotic relative efficiency of the staircase method widely employed in psychophysics research and to demonstrate the dependence of this efficiency on the form of the psychometric function underlying the output responses

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

submodular utility maximization for deadline constrained data collection in sensor networks we study the utility maximization problem for data collection in a wireless sensor network subject to a deadline constraint where the data on a selected subset of nodes are collected through a routing tree subject to the 1 hop interference model our problem is closely related to the traditional utility maximization problems in networking and communications however instead of a separable concave form of utility functions commonly seen in this area we consider the class of monotone submodular utility functions defined on subsets of nodes which is more appropriate for the applications we consider while submodular maximization subject to a cardinality constraint has been well understood our problem is more challenging due to the multi hop data forwarding nature even under the simple interference model we have derived efficient approximation solutions to this problem both for raw data collection and when in network data aggregation is applied

an exact algorithm for maximum entropy sampling we study the experimental design problem of selecting a most informative subset having prespecified size from a set of correlated random variables the problem arises in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy in the gaussian case the problem is recast as that of maximizing the determinant of the covariance matrix of the chosen subset we demonstrate that this problem is np hard we establish an upper bound for the entropy based on the eigenvalue interlacing property and we incorporate this bound in a branch and bound algorithm for the exact solution of the problem we present computational results for estimated covariance matrices that correspond to sets of environmental monitoring stations in the united states

shrinkage estimators for covariance matrices estimation of covariance matrices in small samples has been studied by many authors standard estimators like the unstructured maximum likelihood estimator ml or restricted maximum likelihood reml estimator can be very unstable with the smallest estimated eigenvalues being too small and the largest too big a standard approach to more stably estimating the matrix in small samples is to compute the ml or reml estimator under some simple structure that involves estimation of fewer parameters such as compound symmetry or independence however these estimators will not be consistent unless the hypothesized structure is correct if interest focuses on estimation of regression coefficients with correlated or longitudinal data a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure with large matrices however the inefficiency of the sandwich estimator becomes worrisome we consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients the first involves shrinking the eigenvalues of the unstructured ml or reml estimator the second involves shrinking an unstructured estimator toward a structured estimator for both cases the data determine the amount of shrinkage these estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples the final estimator chosen includes a combination of both shrinkage approaches i e shrinking the eigenvalues and then shrinking toward structure we illustrate our approach on a sleep eeg study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen we recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

signal acquisition via polarization modulation in single photon sources a simple model system is introduced for demonstrating how a single photon source might be used to transduce classical analog information the theoretical scheme results in measurements of analog source samples that are i quantized in the sense of analog to digital conversion and ii corrupted by random noise that is solely due to the quantum uncertainty in detecting the polarization state of each photon this noise is unavoidable if more than 1 bit per sample is to be transmitted and we show how it may be exploited in a manner inspired by suprathreshold stochastic resonance the system is analyzed information theoretically as it can be modeled as a noisy optical communication channel although unlike classical poisson channels the detector s photon statistics are binomial previous results on binomial channels are adapted to demonstrate numerically that the classical information capacity and thus the accuracy of the transduction increases logarithmically with the square root of the number of photons n although the capacity is shown to be reduced when an additional detector nonideality is present the logarithmic increase with n remains

learning from minimum entropy queries in a large committee machine in supervised learning the redundancy contained in random examples can be avoided by learning from queries using statistical mechanics we study learning from minimum entropy queries in a large tree committee machine the generalization error decreases exponentially with the number of training examples providing a significant improvement over the algebraic decay for random examples the connection between entropy and generalization error in multi layer networks is discussed and a computationally cheap algorithm for constructing queries is suggested and analysed

simultaneous factorization of a polynomial by rational approximation in this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously our approach gives methods of arbitrary order grau s method carstensen 1992 grau 1971 is obtained as the second order method which is durand kerner s method when we have linear factors for linear factors our approach yields the simultaneous methods introduced in sakurai et al 1991 we prove local convergence and estimate the r order of the total step version as well as the single step version of the methods we derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted

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

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

optimal stimulus and noise distributions for information transmission via suprathreshold stochastic resonance suprathreshold stochastic resonance ssr is a form of noise enhanced signal transmission that occurs in a parallel array of independently noisy identical threshold nonlinearities including model neurons unlike most forms of stochastic resonance the output response to suprathreshold random input signals of arbitrary magnitude is improved by the presence of even small amounts of noise in this paper the information transmission performance of ssr in the limit of a large array size is considered using a relationship between shannon s mutual information and fisher information a sufficient condition for optimality i e channel capacity is derived it is shown that capacity is achieved when the signal distribution is jeffrey s prior as formed from the noise distribution or when the noise distribution depends on the signal distribution via a cosine relationship these results provide theoretical verification and justification for previous work in both computational neuroscience and electronics

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

capacity of quantum channels using product measurements the capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input and whether product or entangled measurements are used at the output we show that when product measurements are used the capacity of the channel is achieved with product input states so that entangled inputs do not increase capacity we show that this result continues to hold if sequential measurements are allowed whereby the choice of successive measurements may depend on the results of previous measurements we also present a new simplified expression which gives an upper bound for the shannon capacity of a channel and which bears a striking resemblance to the well known holevo bound

string submodular functions with curvature constraints consider the problem of choosing a string of actions to optimize an objective function that is string submodular it was shown in previous papers that the greedy strategy consisting of a string of actions that only locally maximizes the step wise gain in the objective function achieves at least a 1 e 1 approximation to the optimal strategy this paper improves this approximation by introducing additional constraints on curvature namely total backward curvature total forward curvature and elemental forward curvature we show that if the objective function has total backward curvature sigma then the greedy strategy achieves at least a 1 sigma 1 e sigma approximation of the optimal strategy if the objective function has total forward curvature epsilon then the greedy strategy achieves at least a 1 epsilon approximation of the optimal strategy moreover we consider a generalization of the diminishing return property by defining the elemental forward curvature we also introduce the notion of string matroid and consider the problem of maximizing the objective function subject to a string matroid constraint we investigate two applications of string submodular functions with curvature constraints 1 choosing a string of actions to maximize the expected fraction of accomplished tasks and 2 designing a string of measurement matrices such that the information gain is maximized

balanced convergence of iterative methods to a multiple zero of a complex polynomial abstract we deal with the durand kerner iterative method for a complex polynomial equation having multiple zeros we will give the detailed proofs that the durand kerner approximations to a multiple zero locate equidistant points on a sufficiently small circle when they are close to the exact zeros and furthermore that the durand kerner mapping whose definition is given in the present paper is contractive this fact supports the balanced convergence of iteration observed in the author s preceding paper this journal 1989

minimax redundancy for the class of memoryless sources let x sup n x sub 1 x sub n be a memoryless source with unknown distribution on a finite alphabet of size k we identify the asymptotic minimax coding redundancy for this class of sources and provide a sequence of asymptotically minimax codes equivalently we determine the limiting behavior of the minimax relative entropy min sub qxn max sub pxn d p sub xn spl par q sub xn where the maximum is over all independent and identically distributed i i d source distributions and the minimum is over all joint distributions we show in this paper that the minimax redundancy minus k 1 2 log n 2 spl pi e converges to log spl int spl radic det i spl theta d spl theta log spl gamma 1 2 sup k spl gamma k 2 where i spl theta is the fisher information and the integral is over the whole probability simplex the bayes strategy using jeffreys prior is shown to be asymptotically maximin but not asymptotically minimax in our setting the boundary risk using jeffreys prior is higher than that of interior points we provide a sequence of modifications of jeffreys prior that put some prior mass near the boundaries of the probability simplex to pull down that risk to the asymptotic minimax level in the limit

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints