Exact Probability Distribution versus Entropy

The problem addressed concerns the determination of the average numberof successive attempts of guessing a word of a certain length consisting of letters withgiven probabilities of occurrence. Both first- and second-order approximations to a naturallanguage are considered. The guessing strategy used is guessing words in decreasing orderof probability. When word and alphabet sizes are large, approximations are necessary inorder to estimate the number of guesses. Several kinds of approximations are discusseddemonstrating moderate requirements regarding both memory and central processing unit(CPU) time. When considering realistic sizes of alphabets and words (100), the numberof guesses can be estimated within minutes with reasonable accuracy (a few percent) andmay therefore constitute an alternative to, e.g., various entropy expressions. For manyprobability distributions, the density of the logarithm of probability products is close to anormal distribution. For those cases, it is possible to derive an analytical expression for theaverage number of guesses. The proportion of guesses needed on average compared to thetotal number decreases almost exponentially with the word length. The leading term in anasymptotic expansion can be used to estimate the number of guesses for large word lengths.Comparisons with analytical lower bounds and entropy expressions are also provided.

Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

Analysis on the connotation of probability in the artificial intelligence algorithm for economic decision making.

This paper analyzes the inner relations between classical sub-scheme probability and statistic probability, subjective probability and objective probability, prior probability and posterior probability, transition probability and probability of utility, and further analysis the goal, method, and its practical economic purpose which represent by these various probability from the perspective of mathematics, so as to deeply understand there connotation and its relation with economic decision making, thus will pave the route for scientific predication and decision making.

Model Probability in Self-organising Maps

Growing models have been widely used for clustering or topology learning. Traditionally these models work on stationary environments, grow incrementally and adapt their nodes to a given distribution based on global parameters. In this paper, we present an enhanced unsupervised self-organising network for the modelling of visual objects. We first develop a framework for building non-rigid shapes using the growth mechanism of the self-organising maps, and then we define an optimal number of nodes without overfitting or underfitting the network based on the knowledge obtained from information-theoretic considerations. We present experimental results for hands and we quantitatively evaluate the matching capabilities of the proposed method with the topographic product.

Probability and Common-Sense: Tandem Towards Robust Robotic Object Recognition in Ambient Assisted Living

The suitable operation of mobile robots when providing Ambient Assisted Living (AAL) services calls for robust object recognition capabilities. Probabilistic Graphical Models (PGMs) have become the de-facto choice in recognition systems aiming to e ciently exploit contextual relations among objects, also dealing with the uncertainty inherent to the robot workspace. However, these models can perform in an inco herent way when operating in a long-term fashion out of the laboratory, e.g. while recognizing objects in peculiar con gurations or belonging to new types. In this work we propose a recognition system that resorts to PGMs and common-sense knowledge, represented in the form of an ontology, to detect those inconsistencies and learn from them. The utilization of the ontology carries additional advantages, e.g. the possibility to verbalize the robot's knowledge. A primary demonstration of the system capabilities has been carried out with very promising results.

Estudo das propriedades críticas do processo epidêmico por par com difusão de pares

The pair contact process - PCP is a nonequilibrium stochastic model which, like the basic contact process - CP, exhibits a phase transition to an absorbing state. While the absorbing state CP corresponds to a unique configuration (empty lattice), the PCP process infinitely many. Numerical and theoretical studies, nevertheless, indicate that the PCP belongs to the same universality class as the CP (direct percolation class), but with anomalies in the critical spreading dynamics. An infinite number of absorbing configurations arise in the PCP because all process (creation and annihilation) require a nearest-neighbor pair of particles. The diffusive pair contact process - PCPD) was proposed by Grassberger in 1982. But the interest in the problem follows its rediscovery by the Langevin description. On the basis of numerical results and renormalization group arguments, Carlon, Henkel and Schollwöck (2001), suggested that certain critical exponents in the PCPD had values similar to those of the party-conserving - PC class. On the other hand, Hinrichsen (2001), reported simulation results inconsistent with the PC class, and proposed that the PCPD belongs to a new universality class. The controversy regarding the universality of the PCPD remains unresolved. In the PCPD, a nearest-neighbor pair of particles is necessary for the process of creation and annihilation, but the particles to diffuse individually. In this work we study the PCPD with diffusion of pair, in which isolated particles cannot move; a nearest-neighbor pair diffuses as a unit. Using quasistationary simulation, we determined with good precision the critical point and critical exponents for three values of the diffusive probability: D=0.5 and D=0.1. For D=0.5: PC=0.89007(3), β/v=0.252(9), z=1.573(1), =1.10(2), m=1.1758(24). For D=0.1: PC=0.9172(1), β/v=0.252(9), z=1.579(11), =1.11(4), m=1.173(4)

SAMPLING BIAS IN EVALUATING THE PROBABILITY OF SEISMICALLY INDUCED SOIL LIQUEFACTION WITH SPT & CPT CASE HISTORIES

Several deterministic and probabilistic methods are used to evaluate the probability of seismically induced liquefaction of a soil. The probabilistic models usually possess some uncertainty in that model and uncertainties in the parameters used to develop that model. These model uncertainties vary from one statistical model to another. Most of the model uncertainties are epistemic, and can be addressed through appropriate knowledge of the statistical model. One such epistemic model uncertainty in evaluating liquefaction potential using a probabilistic model such as logistic regression is sampling bias. Sampling bias is the difference between the class distribution in the sample used for developing the statistical model and the true population distribution of liquefaction and non-liquefaction instances. Recent studies have shown that sampling bias can significantly affect the predicted probability using a statistical model. To address this epistemic uncertainty, a new approach was developed for evaluating the probability of seismically-induced soil liquefaction, in which a logistic regression model in combination with Hosmer-Lemeshow statistic was used. This approach was used to estimate the population (true) distribution of liquefaction to non-liquefaction instances of standard penetration test (SPT) and cone penetration test (CPT) based most updated case histories. Apart from this, other model uncertainties such as distribution of explanatory variables and significance of explanatory variables were also addressed using KS test and Wald statistic respectively. Moreover, based on estimated population distribution, logistic regression equations were proposed to calculate the probability of liquefaction for both SPT and CPT based case history. Additionally, the proposed probability curves were compared with existing probability curves based on SPT and CPT case histories.

On the universality class of monopole percolation in scalar QED

We study the critical properties of the monopole-percolation transition in U(1) lattice gauge theory coupled to scalars at infinite (β = 0) gauge coupling. We find strong scaling corrections in the critical exponents that must be considered by means of an infinite-volume extrapolation. After the extrapolation, our results are as precise as the obtained for the four dimensional site-percolation and, contrary to previously stated, fully compatible with them.

Relevant Aspects on Teaching Probability and Statistics in Basic General Education

Probability and Statistics were included in the Basic General Education curricula by the Ministry of Public Education (Costa Rica), since 1995. To analyze the teaching reality in these fields, a research was conducted in two educational regions of the country: Heredia and Pérez Zeledón. The survey included university training and updating processes of teachers teaching Statistics and Probability in the schools. The research demonstrated the limited university training in these fields, the dissatisfaction of teachers about it, and the poor support of training institutions to their professional exercise.

Statistical learning of random probability measures

The study of random probability measures is a lively research topic that has attracted interest from different fields in recent years. In this thesis, we consider random probability measures in the context of Bayesian nonparametrics, where the law of a random probability measure is used as prior distribution, and in the context of distributional data analysis, where the goal is to perform inference given avsample from the law of a random probability measure. The contributions contained in this thesis can be subdivided according to three different topics: (i) the use of almost surely discrete repulsive random measures (i.e., whose support points are well separated) for Bayesian model-based clustering, (ii) the proposal of new laws for collections of random probability measures for Bayesian density estimation of partially exchangeable data subdivided into different groups, and (iii) the study of principal component analysis and regression models for probability distributions seen as elements of the 2-Wasserstein space. Specifically, for point (i) above we propose an efficient Markov chain Monte Carlo algorithm for posterior inference, which sidesteps the need of split-merge reversible jump moves typically associated with poor performance, we propose a model for clustering high-dimensional data by introducing a novel class of anisotropic determinantal point processes, and study the distributional properties of the repulsive measures, shedding light on important theoretical results which enable more principled prior elicitation and more efficient posterior simulation algorithms. For point (ii) above, we consider several models suitable for clustering homogeneous populations, inducing spatial dependence across groups of data, extracting the characteristic traits common to all the data-groups, and propose a novel vector autoregressive model to study of growth curves of Singaporean kids. Finally, for point (iii), we propose a novel class of projected statistical methods for distributional data analysis for measures on the real line and on the unit-circle.

Spacecraft Collision Probability Calculation from Conjunction Data Message and High Precision Orbit Propagator

The increasing number of Resident Space Objects (RSOs) is a threat to spaceflight operations. Conjunction Data Messages (CDMs) are sent to satellite operators to warn for possible future collision and their probabilities. The research project described herein pushed forward an algorithm that is able to update the collision probability directly on-board starting from CDMs and the state vector of the hosting satellite which is constantly updated thanks to an onboard GNSS receiver. A large set of methods for computing the collision probability was analyzed in order to find the best ones for this application. The selected algorithm was then tested to assess and improve its performance. Finally, parts of the algorithm and external software were implemented on a Raspberry Pi 3B+ board to demonstrate the compatibility of this approach with computational resources similar to those typically available onboard modern spacecraft.

Development And Characterization Of Microsatellite Loci For Ocotea Species (lauraceae) Threatened With Extinction.

The Atlantic rainforest species Ocotea catharinensis, Ocotea odorifera, and Ocotea porosa have been extensively harvested in the past for timber and oil extraction and are currently listed as threatened due to overexploitation. To investigate the genetic diversity and population structure of these species, we developed 8 polymorphic microsatellite markers for O. odorifera from an enriched microsatellite library by using 2 dinucleotide repeats. The microsatellite markers were tested for cross-amplification in O. catharinensis and O. porosa. The average number of alleles per locus was 10.2, considering all loci over 2 populations of O. odorifera. Observed and expected heterozygosities for O. odorifera ranged from 0.39 to 0.93 and 0.41 to 0.92 across populations, respectively. Cross-amplification of all loci was successfully observed in O. catharinensis and O. porosa except 1 locus that was found to lack polymorphism in O. porosa. Combined probabilities of identity in the studied Ocotea species were very low ranging from 1.0 x 10-24 to 7.7 x 10-24. The probability of exclusion over all loci estimated for O. odorifera indicated a 99.9% chance of correctly excluding a random nonparent individual. The microsatellite markers described in this study have high information content and will be useful for further investigations on genetic diversity within these species and for subsequent conservation purposes.

The Inadequacy Of The Y-axis Of Growth (sngn) For The Vertical Pattern Assessment In Patients With Sagittal Discrepancies.

The aim of this cephalometric study was to evaluate the influence of the sagittal skeletal pattern on the 'Y-axis of growth' measurement in patients with different malocclusions. Lateral head films from 59 patients (mean age 16y 7m, ranging from 11 to 25 years) were selected after a subjective analysis of 1630 cases. Sample was grouped as follows: Group 1 - class I facial pattern; group 2 - class II facial pattern; and Group 3 - class III facial pattern. Two angular measurements, SNGoGn and SNGn, were taken in order to determine skeletal vertical facial pattern. A logistic regression with errors distributed according to a binomial distribution was used to test the influence of the sagittal relationship (Class I, II, III facial patterns) on vertical diagnostic measurement congruence (SNGoGn and SNGn). RESULTS show that the probability of congruence between the patterns SNGn and SNGoGn was relatively high (70%) for group 1, but for groups II (46%) and III (37%) this congruence was relatively low. The use of SNGn appears to be inappropriate to determine the vertical facial skeletal pattern of patients, due to Gn point shifting throughout sagittal discrepancies. Clinical Significance: Facial pattern determined by SNGn must be considered carefully, especially when severe sagittal discrepancies are present.