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This study investigates the dynamics of three discrete independent events occurring randomly and repeatedly within the interval [0,T]. Each event spans a predetermined fraction γ of the total interval length T before concluding. Three independent continuous random variables represent the starting times of these events, uniformly distributed over the time interval [0,T]. By employing a geometric probability approach, we derive a rigorous closed-form expression for the probability of the joint occurrence of these three events, taking into account various values of the fraction γ. Additionally, we determine the expected value of the intersection duration of the three events within the time interval [0,T]. Furthermore, we provide a comprehensive solution for evaluating the expected number of trials required for the simultaneous occurrence of these events. Numerous numerical examples support the theoretical analysis presented in this paper, further validating our findings.

By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the χ-products 〈·〉[sub.χ] and [·][sub.χ] are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained.

The contextual bandit literature has traditionally focused on algorithms that address the exploration–exploitation tradeoff. In particular, greedy algorithms that exploit current estimates without any exploration may be suboptimal in general. However, exploration-free greedy algorithms are desirable in practical settings where exploration may be costly or unethical (e.g., clinical trials). Surprisingly, we find that a simple greedy algorithm can be rate optimal (achieves asymptotically optimal regret) if there is sufficient randomness in the observed contexts (covariates). We prove that this is always the case for a two-armed bandit under a general class of context distributions that satisfy a condition we term covariate diversity . Furthermore, even absent this condition, we show that a greedy algorithm can be rate optimal with positive probability. Thus, standard bandit algorithms may unnecessarily explore. Motivated by these results, we introduce Greedy-First, a new algorithm that uses only observed contexts and rewards to determine whether to follow a greedy algorithm or to explore. We prove that this algorithm is rate optimal without any additional assumptions on the context distribution or the number of arms. Extensive simulations demonstrate that Greedy-First successfully reduces exploration and outperforms existing (exploration-based) contextual bandit algorithms such as Thompson sampling or upper confidence bound. This paper was accepted by J. George Shanthikumar, big data analytics.

This study investigates the B[sub.t] -transformation of probability measures within the framework of free probability. A primary focus is the invariance under this transformation of two fundamental families: the free Meixner family and the free analog of the Letac–Mora class. In addition, we introduce novel characteristics associated with the B[sub.t] -transformation, offering refined analytical tools to probe its structural and functional properties. These tools allow us to uncover new and significant properties of several distributions in free probability, including the semicircle, the Marchenko–Pastur, and the free Gamma laws, yielding explicit invariance results and stability conditions. Our findings extend the theoretical understanding of the B[sub.t] -transformation and provide practical methods for analyzing the dynamics and stability of classical free distributions under this operator.

The authors use recent probabilistic theories of neural computation to argue that confidence and certainty are not identical concepts. They propose precise mathematical definitions for both of these concepts and discuss putative neural representations. When facing uncertainty, adaptive behavioral strategies demand that the brain performs probabilistic computations. In this probabilistic framework, the notion of certainty and confidence would appear to be closely related, so much so that it is tempting to conclude that these two concepts are one and the same. We argue that there are computational reasons to distinguish between these two concepts. Specifically, we propose that confidence should be defined as the probability that a decision or a proposition, overt or covert, is correct given the evidence, a critical quantity in complex sequential decisions. We suggest that the term certainty should be reserved to refer to the encoding of all other probability distributions over sensory and cognitive variables. We also discuss strategies for studying the neural codes for confidence and certainty and argue that clear definitions of neural codes are essential to understanding the relative contributions of various cortical areas to decision making.

In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ -divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on φ -divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with φ -divergence uncertainty is tractable for most of the choices of φ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach. This paper was accepted by Gérard P. Cachon, optimization.

pLogo visualizes sequence motifs according to their statistical significance rather than their frequency. Methods for visualizing protein or nucleic acid motifs have traditionally relied upon residue frequencies to graphically scale character heights. We describe the pLogo, a motif visualization in which residue heights are scaled relative to their statistical significance. A pLogo generation tool is publicly available at http://plogo.uconn.edu/ and supports real-time conditional probability calculations and visualizations.

The Asymmetric Simple Inclusion Process (ASIP) is an n-site tandem stochastic network with a Poisson arrival influx into the first site. Each site has an unlimited buffer with a gate in front of it. Each gate opens, independently of all other gates, following a site-dependent Exponential inter-opening time. When a site’s gate opens, all particles occupying the site move simultaneously to the next site. In this paper, a Generalized ASIP network is analyzed where the influx is to all sites, while gate openings are determined by a general renewal process. A compact matrix approach—instead of the conventional (and tedious) successive substitution method—is constructed for the derivation of the multidimensional probability-generating function (PGF) of the site occupancies. It is shown that the set of (2nn) linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of (2n−1n) equations, and then further reduced to a set of 2[sup.n] −(n+1) equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an (n+1)-site network can be derived from the corresponding PGF of an n-site system. Explicit results for networks with n=3 and n=4 sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site k(k=1,2,…,n) is occupied. We show that, in the case where arrivals occur to the first site only, these probabilities are functions of both the site’s index and the arrival flux and not solely of the site’s index. Consequently, refined formulas for the latter probabilities and for the mean conditional site occupancies are derived. We further show that in the case where the arrival process to the first site is Poisson with rate λ, the following interesting property holds: P(sitekisoccupied|λ=1)=P(sitek+1isoccupied|λ→∞). The case where the inter-gate opening intervals are Gamma distributed is investigated and explicit formulas are obtained. Mean site occupancy and mean total load of the first k sites are calculated. Numerical results are presented.

While causal reasoning is a core facet of our cognitive abilities, its time-course has not received proper attention. As the duration of reasoning might prove crucial in understanding the underlying cognitive processes, we asked participants in two experiments to make probabilistic causal inferences while manipulating time pressure. We found that participants are less accurate under time pressure, a speed-accuracy-tradeoff, and that they respond more conservatively. Surprisingly, two other persistent reasoning errors—Markov violations and failures to explain away—appeared insensitive to time pressure. These observations seem related to confidence: Conservative inferences were associated with low confidence, whereas Markov violations and failures to explain were not. These findings challenge existing theories that predict an association between time pressure and all causal reasoning errors including conservatism. Our findings suggest that these errors should not be attributed to a single cognitive mechanism and emphasize that causal judgements are the result of multiple processes.

This paper investigates the set stability of generalized asynchronous probabilistic Boolean networks (GAPBNs) with impulsive effects. To this end, an efficient algorithm is designed to determine the largest invariant set of a given set. A necessary and sufficient criterion is then derived to determine set stability of GAPBNs with impulsive effects. Subsequently, the global stability and synchronization of GAPBNs with impulsive effects were verified by selecting different sets. Finally, examples are given to illustrate the results on set stability and synchronization.