Arbitrary Probability Distribution Research Articles

Formula for the Laplace transform of the projection of a distribution on the positive semiaxis and some of its applications

We obtain a formula for the Laplace transform of the restriction of an arbitrary probability distribution on the positive semiaxis in the form of a Cauchy-type integral in infinite limits of the characteristic function of this distribution. Using this result and the estimates of the concentration function of the sum of independent random variables, we derive a representation for the Laplace transform of the restriction of the harmonic measure on the positive semiaxis. In conclusion, we present an estimate of the lower ladder height distribution for the case in which the distribution of the value of the jump in a random walk is normal.

Stochastic scheduling to minimize expected maximum lateness

This paper is concerned with the problems in scheduling a set of jobs associated with random due dates on a single machine so as to minimize the expected maximum lateness in stochastic environment. This is a difficult problem and few efforts have been reported on its solution in the literature. In this paper, we first derive a deterministic equivalent to the expected maximum lateness and then propose a dynamic programming algorithm to obtain the optimal solutions. The procedures to compute optimal solutions are initially developed in the case of deterministic processing times, and then extended to stochastic processing times following arbitrary probability distributions. Moreover, several heuristic rules are suggested to compute near-optimal solutions, which are shown to be highly efficient and accurate by computer-based experiments.

Production lot sizing with process deterioration and machine breakdown

The paper presents a generalized economic manufacturing quantity model for an unreliable production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time (when it starts producing defective items) and may ultimately break down afterwards. If a machine breakdown occurs during a production run, then corrective repair is done; otherwise, preventive repair is performed at the end of the production run to enhance the system reliability. The proposed model is formulated assuming that the time to machine breakdown, corrective and preventive repair times follow arbitrary probability distributions. However, the criteria for the existence and uniqueness of the optimal production time are derived under general breakdown and uniform repair time (corrective and preventive) distributions. The optimal production run time is determined numerically and the joint effect of process deterioration, machine breakdowns and repairs (corrective and preventive) on the optimal decisions is investigated for a numerical example.

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.

Modifying the Hoede-Bakker Index to the Shapley-Shubik and Holler-Packel Indices

In the paper, we present some modifications of the Hoede-Bakker index defined in a social network in which players may influence each other. Due to influences of the other actors, the final decision of a player may be different from his original inclination. These modifications are defined for an arbitrary probability distribution over all inclination vectors. In particular, they concern a situation in which the inclination vectors may be not equally probable. Furthermore, by assuming special probability distributions over all inclination vectors, we construct modifications of the Hoede-Bakker index that coincide with the Shapley-Shubik index and with the Holler-Packel index, respectively.

Prospect theory for continuous distributions

We extend the original form of prospect theory by Kahneman and Tversky from finite lotteries to arbitrary probability distributions, using an approximation method based on weak-⋆ convergence. The resulting formula is computationally easier than the corresponding formula for cumulative prospect theory and makes it possible to use prospect theory in future applications in economics and finance. Moreover, we suggest a method how to incorporate a crucial step of the “editing phase” into prospect theory and to remove in this way the discontinuity of the original model.

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.

The learnability of quantum states

Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n . But using ideas from computational learning theory, we show that one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with n . This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice.

STUDY ON LONGER AND SHORTER BOUNDARY DURATION VECTORS WITH ARBITRARY DURATION AND COST VALUES

We use the project network in AOA form (arrows denote the activities and nodes denote the status of the project) to represent a large-scale project. The activity duration and the activity cost are both random variables which take arbitrary integer values with the arbitrary probability distribution. Under the project time (the deadline to complete the project) constraint and the budget constraint, this paper studies how to schedule all activity durations of the project. Two algorithms are proposed to generate all upper and lower boundary vectors for the project, respectively. All feasible activity durations of the project are among such upper and lower boundary vectors.

Real symmetric random matrices and replicas

Various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large- N limit. A result on the Laplacian random matrix with Wigner-rescaling is generalized to arbitrary probability distribution.

Randomizing Functions: Simulation of a Discrete Probability Distribution Using a Source of Unknown Distribution

In this paper, we characterize functions that simulate independent unbiased coin flips from independent coin flips of unknown bias. We call such functions randomizing. Our characterization of randomizing functions enables us to identify the functions that generate the largest average number of fair coin flips from a fixed number of biased coin flips. We show that these optimal functions are efficiently computable. Then we generalize the characterization, and we present a method to simulate an arbitrary rational probability distribution optimally (in terms of the average number of output digits) and efficiently (in terms of computational complexity) from outputs of many-faced dice of unknown distribution. We also study randomizing functions on exhaustive prefix-free sets.

Bayesian inference with probabilistic population codes

Recent psychophysical experiments indicate that humans perform near-optimal Bayesian inference in a wide variety of tasks, ranging from cue integration to decision making to motor control. This implies that neurons both represent probability distributions and combine those distributions according to a close approximation to Bayes' rule. At first sight, it would seem that the high variability in the responses of cortical neurons would make it difficult to implement such optimal statistical inference in cortical circuits. We argue that, in fact, this variability implies that populations of neurons automatically represent probability distributions over the stimulus, a type of code we call probabilistic population codes. Moreover, we demonstrate that the Poisson-like variability observed in cortex reduces a broad class of Bayesian inference to simple linear combinations of populations of neural activity. These results hold for arbitrary probability distributions over the stimulus, for tuning curves of arbitrary shape and for realistic neuronal variability.

Efficient Integer Coding for Arbitrary Probability Distributions

Performance bounds on the average codeword length of Golomb codes for arbitrary probability distributions are derived in terms of the mean. Then based on Golomb codes, a class of extended gamma codes which are the generalizations of Elias gamma code is constructed and proved to be universal. Optimal decision rules for choosing parameters of Golomb codes and extended gamma codes for arbitrary probability distributions are also derived. Finally, the concept of maximum entropy codes is introduced and the existence of such codes is investigated for source classes with linear constraints. Golomb codes with optimal parameters turn out to be maximum entropy codes for sources with a fixed mean

Determination of hyperfine field distributions in amorphous magnets

We present an overview of two leading methods of determining probabilitydistributions from Mössbauer spectra, using the model amorphous magnetFe80B20. A comparison is made between the maximum-entropy method, which permitsanalysis using truly arbitrary parameter probability distributions, and aVoigtian-based analysis, which uses a sum of Gaussian components to createparameter distributions of pseudo-arbitrary shape. Our results indicate that, inFe80B20, a Gaussian distribution of magnetic hyperfine fields is a very good approximation,although small deviations from a Gaussian shape are evident. We find that the apparentexistence of correlations between the isomer shift and magnetic hyperfine field parameters,as found using Voigt-based analyses, may be an artefact of imposing a Gaussian shape onthe parameter distributions. We conclude that maximum entropy and Voigtian analysestogether provide a very powerful means of characterizing magnetic materials with Mössbauerspectroscopy.

The 1-center problem in the plane with independent random weights

The 1-center problem in the plane with random weights has only been studied for a few specific probability distributions and distance measures. In this paper we deal with this problem in a general framework, where weights are supposed to be independent random variables with arbitrary probability distributions and distances are measured by any norm function. Two objective functions are considered to evaluate the performance of any location. The first is defined as the probability of covering all demand points within a given threshold, the second is the threshold for which the probability of covering is bounded from below by a given value. We first present some properties related to the corresponding optimization problems, assuming random weights with both discrete and absolutely continuous probability distributions. For weights with discrete distributions, enumeration techniques can be used to solve the problems. For weights continuously distributed, interval branch and bound algorithms are proposed to solve the problems whatever the probability distributions are. Computational experience using the uniform and the gamma probability distributions is reported.

Computing in Finite Time

If a message can have n different values and all values are equally probable, then the entropy of the message is log (n). In the present paper, we discuss the expectation value of the entropy, for an arbitrary probability distribution. We introduce a mixture of all possible probability distributions. We assume that the mixing function is uniform. • either in flat probability space, i.e. the unitary n-dimensional hypertriangle, • or in Bhattacharyya's spherical statistical space, i.e. the unitary n-dimensional hyperoctant. A computation is a manipulation of an incoming message, i.e. a mapping in probability space: • either a reversible mapping, i.e. a symmetry operation (rotation or reflection) in n-dimensional space, • or an irreversible mapping, i.e. a projection operation from n-dimensional to lower-dimensional space. During a reversible computation, no isentropic path in the probability space can be found. Therefore we have to conclude that a computation cannot be represented by a message which merely follows a path in n-dimensional probability space. Rather, the point representing the mixing function travels along a path in an infinite-dimensional Hilbert space.

Estimating Lymphocyte Division and Death Rates from CFSE Data

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] have pioneered the quantitative analysis of CFSE data. We confirm and extend their data analysis approach using simple mathematical models. We employ the extended Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] method to estimate the time to first division, the fraction of cells recruited into division, the cell cycle time, and the average death rate from CFSE data on T cells stimulated under different concentrations of IL-2. The same data is also fitted with a simple mathematical model that we derived by reformulating the numerical model of Deenick et al. [J. Immunol. 170:4963-4972, 2003]. By a non-linear fitting procedure we estimate parameter values and confidence intervals to identify the parameters that are influenced by the IL-2 concentration. We obtain a significantly better fit to the data when we assume that the T cell death rate depends on the number of divisions cells have completed. We provide an outlook on future work that involves extending the Deenick et al. [J. Immunol. 170:4963-4972, 2003] model into the classical smith-martin model, and into a model with arbitrary probability distributions for death and division through subsequent divisions.

A law of large numbers for weighted majority

Consider an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate is p > 1 / 2 . Condorcet's Jury Theorem which he derived from the weak law of large numbers asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. The notion of influence of a voter or its voting power is relevant for extensions of the weak law of large numbers for voting rules which are more general than simple majority. In this paper we point out two different ways to extend the classical notions of voting power and influences to arbitrary probability distributions. The extension relevant to us is the “effect” of a voter, which is a weighted version of the correlation between the voter's vote and the election's outcomes. We prove an extension of the weak law of large numbers to weighted majority games when all individual effects are small and show that this result does not apply to any voting rule which is not based on weighted majority.

On the toric algebra of graphical models

We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley–Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.

Bayesian parameter estimation with informative priors for nonlinear systems

AbstractThe estimation of parameters for nonlinear process models is often accomplished through optimization routines that search for a global optimum with respect to a least squares or weighted least squares criterion. While such an approach is often reasonable, it fails to account for all of the information that is available from the data and practitioner. Here we focus on the inclusion of prior knowledge in the estimation of parameters in nonlinear dynamic systems. We use the Bayesian paradigm to define the probability distribution over process model parameters, called the Bayesian posterior. The quantities associated with this posterior distribution (e.g., credible regions, means, modes) are estimated via Markov Chain Monte Carlo (MCMC) integration. We first give a short introduction to Bayesian parameter estimation and the role of MCMC in evaluating arbitrary probability distributions. Bayesian parameter estimation (via MCMC) is then applied to three case studies. The first case study shows the basic methodology of assigning and evaluating a Bayesian posterior to a simple problem that consists of estimating the mean and variance of a sample. The second case study uses prior information that specifies a preference for a particular type of reaction mechanism over another for a simulated fermentation system; the inclusion of such prior information is shown to improve the estimated values of the model parameters in situations where data are sparse or noisy (compared to the more common weighted least squares approach). The third case study develops a hybrid semi‐parametric neural network (NN) model to predict time‐dependent observed state variables (cell and protein concentration) in Escherichia coli fermentations. An integral step in the development of this hybrid model is parameter estimation of a nonlinear dynamic model. A hybrid model developed from the Bayesian parameter estimates is shown to outperform a hybrid model developed from the weighted least squares parameter estimates for predicting the final protein yield of a test set. © 2005 American Institute of Chemical Engineers AIChE J, 2006