MULTIVARIATE COMPOSITE COPULAS

In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

The Markovian Pattern of Social Deprivation for Mexicans with Diabetes

This paper aims to determine the Markovian pattern of the factors influencing social deprivation in Mexicans with Type 2 diabetes mellitus (DM2). To this end, we develop a methodology to meet the theoretical and practical considerations involved in applying a Hidden Markov Model that uses non-panel data. After estimating the latent states and ergodic vectors for diabetic and non-diabetic populations, we find that the long-term state-dependent probabilities for people with DM2 show a darker perspective of impoverishment than the rest of the Mexican population. In the absence of extreme events that modify the present probability structure, the Markovian pattern confirms that people with DM2 will most likely become the poorest of Mexico’s poor.

Newcomb’s problem isn’t a choice dilemma

Newcomb’s problem involves a decision-maker faced with a choice and a predictor forecasting this choice. The agents’ interaction seems to generate a choice dilemma once the decision-maker seeks to apply two basic principles of rational choice theory (RCT): maximize expected utility (MEU); adopt the dominant strategy (ADS). We review unsuccessful attempts at pacifying the dilemma by excluding Newcomb’s problem as an RCT-application, by restricting MEU and ADS, and by allowing for backward causation. A probability approach shows that Newcomb’s original problem-formulation lacks causal information. This makes it impossible to specify the probability structure of Newcomb’s univocally. Once missing information is added, Newcomb’s problem and RCT re-align, thus explaining Newcomb’s problem as a seeming dilemma. Building on Wolpert and Benford (Synthese 190(9):1637–1646, 2013), we supply additional details and offer a crucial correction to their formal proof.

Implicit anticipation of probabilistic regularities: Larger CNV emerges for unpredictable events

Anticipation of upcoming events plays a crucial role in automatic behaviors. It is, however, still unclear whether the event-related brain potential (ERP) markers of anticipation could track the implicit acquisition of probabilistic regularities that can be considered as building blocks of automatic behaviors. Therefore, in a four-choice reaction time (RT) task performed by young adults (N = 36), the contingent negative variation (CNV) as an ERP marker of anticipation was measured from the onset of a cue stimulus until the presentation of a target stimulus. Due to the probability structure of the task, target stimuli were either predictable or unpredictable, but this was unknown to participants. The cue did not contain predictive information on the upcoming target. Results showed that the CNV amplitude during response preparation was larger before the unpredictable than before the predictable target stimuli. In addition, although RTs increased, the P3 amplitude decreased for the unpredictable as compared with the predictable target stimuli, possibly due to the stronger response preparation that preceded stimulus presentation. These results suggest that enhanced attentional resources are allocated to the implicit anticipation and processing of unpredictable events. This possibly results from the formation of internal models on the probabilistic regularities of the stimulus stream, favoring predictable events. Overall, we provide ERP evidence for the implicit anticipation of probabilistic regularities, confirming the role of predictive processes in learning and memory.

Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities

This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.

On the S-instability and degeneracy of discrete deep learning models

A probability model exhibits instability if small changes in a data outcome result in large and, often unanticipated, changes in probability. This instability is a property of the probability model, given by a distributional form and a given configuration of parameters. For correlated data structures found in several application areas, there is increasing interest in identifying such sensitivity in model probability structure. We consider the problem of quantifying instability for general probability models defined on sequences of observations, where each sequence of length $N$ has a finite number of possible values that can be taken at each point. A sequence of probability models, indexed by $N$, and an associated parameter sequence result to accommodate data of expanding dimension. Model instability is formally shown to occur when a certain log probability ratio under such models grows faster than $N$. In this case, a one component change in the data sequence can shift probability by orders of magnitude. Also, as instability becomes more extreme, the resulting probability models are shown to tend to degeneracy, placing all their probability on potentially small portions of the sample space. These results on instability apply to large classes of models commonly used in random graphs, network analysis and machine learning contexts.

A robust likelihood approach to inference about the difference between two multinomial distributions in paired designs

Pairing serves as a way of lessening heterogeneity but pays the price of introducing more parameters to the model. This complicates the probability structure and makes inference more intricate. We employ the simpler structure of the parallel design to develop a robust score statistic for testing the equality of two multinomial distributions in paired designs. This test incorporates the within-pair correlation in a data-driven manner without a full model specification. In the paired binary data scenario, the robust score statistic turns out to be the McNemar’s test. We provide simulations and real data analysis to demonstrate the advantage of the robust procedure.