Conditional Probability Mass Function Research Articles

Comparing Dependent Undirected Gaussian Networks

A Bayesian approach is proposed, which unifies network constructions and comparisons between two longitudinal undirected Gaussian networks on their differentiation status (identical or differential) for data collected at two time points. Utilizing the concept of modeling repeated measures, we construct a joint likelihood of networks. The conditional posterior probability mass function for network differentiation is derived and its asymptotic proposition is theoretically assessed. An alternative approach built upon latent rather than manifest data is proposed to significantly reduce computing burden. Simulations are used to demonstrate and compare the two methods and compare them with existing approaches. Based on epigenetic data collected at different ages, the proposed methods are demonstrated on their ability to detect dependent network differentiations. Our theoretical assessment, simulations, and real data applications support the effectiveness of the proposed methods, although the approach relying on latent data is less efficient.

The Novel Bivariate Distribution: Statistical Properties and Real Data Applications

This article proposes a novel class of bivariate distributions that are completely defined by stating their conditionals as Poisson exponential distributions. Numerous statistical properties of this distribution are also examined here, including the conditional probability mass function (PMF) and moments of the new class. The techniques of maximum likelihood and pseudolikelihood are used to estimate the model parameters. Additionally, the effectiveness of the bivariate Poisson exponential conditional (BPEC) distribution is compared to that of the bivariate Poisson conditional (BPC), the bivariate Poisson (BP), the bivariate Poisson–Lindley (BPL), and the bivariate negative binomial (BNB) distributions using a real-world dataset. The findings of Akaike information criterion (AIC) and Bayesian information criterion (BIC) reveal that the BPEC distribution performs better than the other distributions considered in this study. As a result, the authors claim that this distribution may be used to fit dependent and overspread count data.

On Filtering Methods for State-Space Systems having Binary Output Measurements

Abstract In this paper we develop two filtering algorithms for state-space systems with binary outputs. We approximate the conditional probability mass function of the output signal given the state by using a Gaussian quadrature rule. This approximation naturally leads to a Gaussian Sum structure for the a posteriori density function. Our first algorithm is based on Gaussian Mixture models, and the second algorithm is based on Particle Filtering. Finally, we present numerical examples to illustrate the effectiveness of our proposal.

Gaussian Bayesian network comparisons with graph ordering unknown

A Bayesian approach is proposed that unifies Gaussian Bayesian network constructions and comparisons between two networks (identical or differential) for data with graph ordering unknown. When sampling graph ordering, to escape from local maximums, an adjusted single queue equi-energy algorithm is applied. The conditional posterior probability mass function for network differentiation is derived and its asymptotic proposition is theoretically assessed. Simulations are used to demonstrate the approach and compare with existing methods. Based on epigenetic data at a set of DNA methylation sites (CpG sites), the proposed approach is further examined on its ability to detect network differentiations. Findings from theoretical assessment, simulations, and real data applications support the efficacy and efficiency of the proposed method for network comparisons.

Structural Results for Average‐Cost Inventory Models with Markov‐Modulated Demand and Partial Information

We consider a discrete‐time infinite‐horizon inventory system with non‐stationary demand, full backlogging, and deterministic replenishment lead time. Demand arrives according to a probability distribution conditional on the state of the world that undergoes Markovian transitions over time. But the actual state of the world can only be imperfectly estimated based on past demand data. We model the inventory replenishment problem for this system as a Markov decision process (MDP) with an uncountable state space consisting of both the inventory position and the most recent belief, a conditional probability mass function, about the actual state of the world. Assuming that the state of the world evolves as an ergodic Markov chain, using the vanishing discount method along with a coupling argument, we prove the existence of an optimal average cost that is independent of the initial system state. For our linear cost structure, we also establish the average‐cost optimality of a belief‐dependent base‐stock policy. We then discretize the uncountable belief space into a regular grid and observe that the average cost under our discretization converges to the optimal average cost as the number of grid points grows large. Finally, we conduct numerical experiments to evaluate the use of a myopic belief‐dependent base‐stock policy as a heuristic for our MDP with the uncountable state space. On a test bed of 108 instances, the average cost obtained from the myopic policy deviates by no more than a few percent from the best lower bound on the optimal average cost obtained from our discretization.

Noncausal Count Processes

We introduce noncausal processes to the count time series literature. These processes are defined by time-reversing an INAR(1) process, a non-INAR(1) Markov affine count process, or a random coefficient INAR(1) [RCINAR(1)] process. In the special cases of INAR(1) and RCINAR(1), the causal process and its noncausal counterpart are closely related through a same queuing system with different stochastic specifications. The noncausal processes we introduce are generically time irreversible and have some unique calendar time dynamic properties that are unreplicable by existing causal models. In particular they allow for locally bubble-like explosion, while at the same time remaining stationary. These processes have closed form calendar time conditional probability mass function, which facilitates nonlinear forecasting.

Structured random forest for label distribution learning

Abstract Label distribution learning (LDL) has proven effective in many machine learning applications. Previous LDL methods have focused on learning a non-linear conditional probability mass function by maximizing entropy or minimizing the Kullback–Leibler (K-L) divergence. In order to make full use of the structural information among different classes, a method called structured random forest (StructRF) regression is used which has been applied to semantic image labeling and edge detection. It is a general LDL model that treats the distribution as an integral whole. In StructRF, all label distributions are mapped to a discrete space at each split node in a random forest. In this way, standard information gain measures can be evaluated. Then the predicted distribution can be obtained directly without calculating the probability of each class individually during the test. StructRF is proved to be fast in training and it reaches higher accuracies and lower standard deviations among different measurements. Besides, we propose an adaptive variable step method that can speed up the training process and reduce the calculations of information gain significantly. It is suitable for the most decision tree based models.

Bayesian inference on group differences in multivariate categorical data

Multivariate categorical data are common in many fields. An illustrative example is provided by election polls studies assessing evidence of changes in voters’ opinions with their candidates preferences in the 2016 United States Presidential primaries or caucuses. Similar goals arise in routine applications, but current literature lacks a general methodology which combines flexibility, efficiency, and tractability in testing for group differences in multivariate categorical data at different – potentially complex – scales. This contribution addresses such goal by leveraging a Bayesian representation, which factorizes the joint probability mass function for the group variable and the multivariate categorical data as the product of the marginal probabilities for the groups and the conditional probability mass function of the multivariate categorical data, given the group membership. To enhance flexibility, the conditional probability mass function of the multivariate categorical data is defined via a group-dependent mixture of tensor factorizations which facilitates dimensionality reduction and borrowing of information, while providing tractable procedures for computation, and accurate tests assessing global and local group differences. The proposed methods are compared with popular competitors, and the improved performance is outlined in simulations and in American election polls studies.

Bayesian Inference and Testing of Group Differences in Brain Networks

Network data are increasingly collected along with other variables of interest. Our motivation is drawn from neurophysiology studies measuring brain connectivity networks for a sample of individuals along with their membership to a low or high creative reasoning group. It is of paramount importance to develop statistical methods for testing of global and local changes in the structural interconnections among brain regions across groups. We develop a general Bayesian procedure for inference and testing of group differences in the network structure, which relies on a nonparametric representation for the conditional probability mass function associated with a network-valued random variable. By leveraging a mixture of low-rank factorizations, we allow simple global and local hypothesis testing adjusting for multiplicity. An efficient Gibbs sampler is defined for posterior computation. We provide theoretical results on the flexibility of the model and assess testing performance in simulations. The approach is applied to provide novel insights on the relationships between human brain networks and creativity.

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.

Supervised-Learning-Aided Communication Framework for MIMO Systems With Low-Resolution ADCs

This paper considers a multiple-input-multiple-output (MIMO) system with low-resolution analog-to-digital converters (ADCs). In this system, we propose a novel communication framework that is inspired by supervised learning. The key idea of the proposed framework is to learn the non-linear input-output system, formed by the concatenation of a wireless channel and a quantization function used at the ADCs, for data detection. In this framework, a conventional channel estimation process is replaced by a system learning process, in which the conditional probability mass functions (PMFs) of the nonlinear system are empirically learned by sending the repetitions of all possible data signals as pilot signals. Then the subsequent data detection process is performed based on the empirical conditional PMFs obtained during the system learning. To reduce both the training overhead and the detection complexity, we also develop a supervised-learning-aided successive-interference-cancellation method. In this method, a data signal vector is divided into two subvectors with reduced dimensions. Then these two subvectors are successively detected based on the conditional PMFs that are learned using artificial noise signals and an estimated channel. For the case of one-bit ADCs, we derive an analytical expression for vector-error-rate of the proposed framework under perfect channel knowledge at the receiver. Simulations demonstrate the detection error reduction of the proposed framework compared to conventional detection techniques that are based on channel estimation.

On Binomial Observations of Continuous-Time Markovian Population Models

In this paper we consider a class of stochastic processes based on binomial observations of continuous-time, Markovian population models. We derive the conditional probability mass function of the next binomial observation given a set of binomial observations. For this purpose, we first find the conditional probability mass function of the underlying continuous-time Markovian population model, given a set of binomial observations, by exploiting a conditional Bayes' theorem from filtering, and then use the law of total probability to find the former. This result paves the way for further study of the stochastic process introduced by the binomial observations. We utilize our results to show that binomial observations of the simple birth process are non-Markovian.

On Binomial Observations of Continuous-Time Markovian Population Models

In this paper we consider a class of stochastic processes based on binomial observations of continuous-time, Markovian population models. We derive the conditional probability mass function of the next binomial observation given a set of binomial observations. For this purpose, we first find the conditional probability mass function of the underlying continuous-time Markovian population model, given a set of binomial observations, by exploiting a conditional Bayes' theorem from filtering, and then use the law of total probability to find the former. This result paves the way for further study of the stochastic process introduced by the binomial observations. We utilize our results to show that binomial observations of the simple birth process are non-Markovian.

Yield and quality grade outcomes as affected by molecular breeding values for commercial beef cattle.

Although genetic tests for many economically important beef cattle traits are commercially available, additional information is needed to help the industry better understand how the results from these tests translate into phenotypic outcomes. This information has important implications for marker-assisted management. The objective of this research was to determine the relationship between molecular breeding values (MBV) characterizing yield grade and marbling and distributions of phenotypic outcomes for yield grade and quality grade. Using data collected from commercially-fed cattle, mixed-model regression equations were estimated for yield grade and quality grade outcomes for both the full sample of commercial cattle (n= 8,995) and a subsample of black-hided steers (n = 4,790). Significant yield grade (P < 0.01) and marbling (P < 0.01) MBV effects were found. However, the yield grade MBV held up better at predicting phenotypic outcomes than the marbling MBV. Estimated conditional probability mass functions of yield and quality grade outcomes for the general population and black-hided steers were similar. Since distributions for black-hided steers were expected to be more applicable from a management perspective, we focused our analysis on these animals. For example, black-hided steers with "low" genetic potential for yield grade and marbling had about a 29% chance of achieving the base price or better on a price grid (yield grade 3 or lower and quality grade Low Choice or better). Increasing genetic potential for marbling increased the likelihood of achieving this same outcome. However, increasing genetic potential for yield grade was unlikely to increase overall carcass quality given its large deleterious effect on quality grade outcomes. Instead, simultaneous improvements in genetic potential for yield grade and marbling offered much more reliable improvements in overall carcass quality. For example, an animal with "moderate" genetic potential for both yield grade and marbling had a 46% chance of achieving the base price or better on a price grid, and an animal with the maximum observed MBV for these traits had a 66% chance of achieving this same outcome. Although the yield grade MBV held up better at predicting phenotypic outcomes, shifts in the distributions of yield grade and quality grade suggested that these increases were largely determined by the marbling MBV.

Intraday Stock Price Dependence Using Dynamic Discrete Copula Distributions

We investigate the intraday dependence pattern between tick data of stock price changes using a new time-varying model for discrete copulas. We let parameters of both the marginal models and the copula vary over time using an observation driven autoregressive updating scheme based on the score of the conditional probability mass function with respect to the time-varying parameters. We apply the model to high-frequency stock price changes expressed as discrete tick-size multiples for four liquid U.S. financial stocks. Our modeling framework is based on Skellam densities for the marginals and a range of different copula functions. We find evidence of intraday time-variation in the dependence structure. After the opening and before the close of the stock market, dependence levels are lower. We attribute this finding to more idiosyncratic trading at these times. The introduction of score driven dynamics in the dependence structure significantly increases the likelihood values of the time-varying copula model. By contrast, a fixed daily seasonal dependence pattern clearly fits the data less well.

Bridging Probability and Possibility via Bayesian Theorem

The connection between probability and possibility has been studied for decades. Several studies have intended to provide a unified interpretation for these concepts; other studies attempt to discover their transformation relationship. This paper addresses these problems using a unified perspective. By extending the viewpoint of interpreting the grade of membership as a conditional probability, we introduce the conditional probability mass function and likelihood mass function to describe randomness and fuzziness, respectively. We draw the conclusion that conditional probability is undetermined itself and can be used for describing either randomness or fuzziness, depending on how it is interpreted. A fuzzy Bayesian theorem is presented based on the fuzziness interpretation of conditional probability. Additionally, a probability-possibility conversion is derived by assigning different interpretations for conditional probability of the Bayesian theorem, which is notably similar to Klir's normalized transformation. An example of target recognition demonstrates that the fuzzy Bayesian classifier outperforms the usual Bayesian classifier.

Random rounded integer-valued autoregressive conditional heteroskedastic process

The statistical literature on the analysis of discrete variate time series has concentrated mainly on parametric models, that is the conditional probability mass function is assumed to belong to a parametric family. Generally, these parametric models impose strong assumptions on the relationship between the conditional mean and variance. To generalize these implausible assumptions, this paper instead considers a more realistic semiparametric model, called random rounded integer-valued autoregressive conditional heteroskedastic (RRINARCH) model, where there are essentially no assumptions on the relationship between the conditional mean and variance. The new model has several advantages: (a) it provides a coherent semiparametric framework for discrete variate time series, in which the conditional mean and variance can be modeled separately; (b) it allows negative values both for the series and its autocorrelation function; (c) its autocorrelation structure is the same as that of a standard autoregressive (AR) process; (d) standard software for its estimation is directly applicable. For the new model, conditions for stationarity, ergodicity and the existence of moments are established and the consistency and asymptotic normality of the conditional least squares estimator are proved. Simulation experiments are carried out to assess the performance of the model. The analyses of real data sets illustrate the flexibility and usefulness of the RRINARCH model for obtaining more realistic forecast means and variances.

A bivariate INAR(1) process with application

The study of time series models for count data has become a topic of special interest during the last years. However, while research on univariate time series for counts now flourishes, the literature on multivariate time series models for count data is notably more limited. In the present paper, a bivariate integer-valued autoregressive process of order 1 (BINAR(1)) is introduced. Emphasis is placed on models with bivariate Poisson and bivariate negative binomial innovations. We discuss properties of the BINAR(1) model and propose the method of conditional maximum likelihood for the estimation of its unknown parameters. Issues of diagnostics and forecasting are considered and predictions are produced by means of the conditional forecast distribution. Estimation uncertainty is accommodated by taking advantage of the asymptotic normality of maximum likelihood estimators and constructing appropriate confidence intervals for the fe-step-ahead conditional probability mass function. The proposed model is applied to a bivariate data series concerning daytime and nighttime road accidents in the Netherlands.

Negative binomial time series models based on expectation thinning operators

The study of count data time series has been active in the past decade, mainly in theory and model construction. There are different ways to construct time series models with a geometric autocorrelation function, and a given univariate margin such as negative binomial. In this paper, we investigate negative binomial time series models based on the binomial thinning and two other expectation thinning operators, and show how they differ in conditional variance or heteroscedasticity. Since the model construction is in terms of probability generating functions, typically, the relevant conditional probability mass functions do not have explicit forms. In order to do simulations, likelihood inference, graphical diagnostics and prediction, we use a numerical method for inversion of characteristic functions. We illustrate the numerical methods and compare the various negative binomial time series models for a real data example.

A Discrete Stress-Strength Interference Model With Stress Dependent Strength

In structural reliability engineering, one often encounters situations where the strength of a structure is influenced by the stress, but the stress is irrelevant to the strength. This phenomenon can be called a unilateral dependency of strength on stress. To evaluate structural reliability in such cases, the stress on a structure is proposed to be a discrete random variable, and the stress dependent strength is represented by a discrete random variable that has different conditional probability mass functions under different stress amplitudes. Then a discrete stress-strength interference model with stress dependent strength is presented based on the universal generating function technique. Finally, the effectiveness of this model is demonstrated by an illustrative example.