Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Prospective Mathematics Teachers Understanding of Classical and Frequentist Probability

Strengthening the teaching of probability requires an adequate training of prospective teachers, which should be based on the prior assessment of their knowledge. Consequently, the aim of this study was to analyse how 139 prospective Spanish mathematics teachers relate the classical and frequentist approaches to probability. To achieve this goal, content analysis was used to categorize the prospective teachers’ answers to a questionnaire with open-ended tasks in which they had to estimate and justify the composition of an urn, basing their answers on the results of 1000 extractions from the urn. Most of the sample proposed an urn model consistent with the data provided; however, the percentage that adequately justified the construction was lower. Although the majority of the sample correctly calculated the probability of an event in a new extraction and chose the urn giving the highest probability, a large proportion of the sample forgot the previously constructed urn model, using only the frequency data. Difficulties, such as equiprobability bias or not perceiving independence of trials in replacement sampling, were also observed for a small part of the sample. These results should be considered in the organisation of probabilistic training for prospective teachers.

APPLICATIONS OF TWO NON-CENTRAL HYPERGEOMETRIC DISTRIBUTIONS OF BIASED SAMPLING STATISTICAL MODELS

Statistical models of biased sampling of two non-central hypergeometric distributions Wallenius' and Fisher's distribution has been extensively used in the literature, however, not many of the logic of hypergeometric distribution have been investigated by different techniques. This research work examined the procedure of the two non-central hypergeometric distributions and investigates the statistical properties which includes the mean and variance that were obtained. The parameters of the distribution were estimated using the direct inversion method of hyper simulation of biased urn model in the environment of R statistical software, with varying odd ratios (w) and group sizes (mi). It was discovered that the two non - central hypergeometric are approximately equal in mean, variance and coefficient of variation and differ as odds ratios (w) becomes higher and differ from the central hypergeometric distribution with ω = 1. Furthermore, in univariate situation we observed that Fisher distribution at (ω = 0.2, 0.5, 0.7, 0.9) is more consistent than Wallenius distribution, although central hypergeometric is more consistent than any of them. Also, in multinomial situation, it was observed that Fisher distribution is more consistent at (ω = 0.2, 0.5), Wallenius distribution at (ω = 0.7, 0.9) and central hypergeometric at (ω = 0.2)

Random additions in urns of integers

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

A model for the Twitter sentiment curve

Twitter is among the most used online platforms for the political communications, due to the concision of its messages (which is particularly suitable for political slogans) and the quick diffusion of messages. Especially when the argument stimulate the emotionality of users, the content on Twitter is shared with extreme speed and thus studying the tweet sentiment if of utmost importance to predict the evolution of the discussions and the register of the relative narratives. In this article, we present a model able to reproduce the dynamics of the sentiments of tweets related to specific topics and periods and to provide a prediction of the sentiment of the future posts based on the observed past. The model is a recent variant of the Pólya urn, introduced and studied in Aletti and Crimaldi (2019, 2020), which is characterized by a “local” reinforcement, i.e. a reinforcement mechanism mainly based on the most recent observations, and by a random persistent fluctuation of the predictive mean. In particular, this latter feature is capable of capturing the trend fluctuations in the sentiment curve. While the proposed model is extremely general and may be also employed in other contexts, it has been tested on several Twitter data sets and demonstrated greater performances compared to the standard Pólya urn model. Moreover, the different performances on different data sets highlight different emotional sensitivities respect to a public event.

Bayesian modeling of skewed X inactivation in genetically diverse mice identifies a novelXceallele associated with copy number changes

Female mammals are functional mosaics of their parental X-linked gene expression due to X chromosome inactivation (XCI). This process inactivates one copy of the X chromosome in each cell during embryogenesis and that state is maintained clonally through mitosis. In mice, the choice of which parental X chromosome remains active is determined by the X chromosome controlling element (Xce), which has been mapped to a 176-kb candidate interval. A series of functional Xce alleles has been characterized or inferred for classical inbred strains based on biased, or skewed, inactivation of the parental X chromosomes in crosses between strains. To further explore the function structure basis and location of the Xce, we measured allele-specific expression of X-linked genes in a large population of F1 females generated from Collaborative Cross (CC) strains. Using published sequence data and applying a Bayesian “Pólya urn” model of XCI skew, we report two major findings. First, inter-individual variability in XCI suggests mouse epiblasts contain on average 20–30 cells contributing to brain. Second, CC founder strain NOD/ShiLtJ has a novel and unique functional allele, Xceg, that is the weakest in the Xce allelic series. Despite phylogenetic analysis confirming that NOD/ShiLtJ carries a haplotype almost identical to the well-characterized C57BL/6J (Xceb), we observed unexpected patterns of XCI skewing in females carrying the NOD/ShiLtJ haplotype within the Xce. Copy number variation is common at the Xce locus and we conclude that the observed allelic series is a product of independent and recurring duplications shared between weak Xce alleles.

Estimating red deer (Cervus elaphus) population size based on non-invasive genetic sampling

Some deer species are of conservation concern; others are officially managed as a food source or for their trophies, whereas in many regions, deer are regarded as overabundant or even as a nuisance causing damages. Regardless of local management issues, in most cases, reliable data on deer population sizes and sex ratios are lacking. Non-invasive genetic approaches are promising tools for the estimation of population size and structure. We developed and tested a non-invasive genetic approach for red deer (Cervus elaphus) population size and density estimation based on faeces collected from three free-ranging red deer populations in south-western Germany. Altogether, we genotyped 2762 faecal samples, representing 1431 different individuals. We estimated population density for both sexes separately using two different approaches: spatially explicit capture-recapture (SECR) approach and a single-session urn model (CAPWIRE). The estimated densities of both approaches were similar for all three study areas, ranging between total densities of 3.3 (2.5–4.4) and 8.5 (6.4–11.3) red deer/km2. The estimated sex ratios differed significantly between the studied populations (ranging between 1:1.1 and 1:1.7), resulting in considerable consequences for management. In further research, the issues of population closure and approximation of the effectively sampled area for density estimation should be addressed. The presented approach can serve as a valuable tool for the management of deer populations, and to our knowledge, it represents the only sex-specific approach for estimation of red deer population size and density.

Subjective logic reasoning: an urn model intuition and application to connected automated driving

Subjective Logic (SL) is a powerful extend of classical probability theory that can handle small sample sizes and, with that, the resulting statistical uncertainty. However, SL is a quite abstract theory and has found limited attention in the field of automation so far. In this work, we present a new urn model intuition to SL that connects SL with the Pólya urn scheme. The application of SL-based reliability estimation in automation is demonstrated on two examples from the domain of connected automated driving: first to assess external information for motion planning on-board the vehicle and second to rate connected vehicles as agents within a large-scale multi-agent system.