We propose Simple Propagation (SP) as a new join tree propagation algorithm for exact inference in discrete Bayesian networks. We establish the correctness of SP. The striking feature of SP is that its message construction exploits the factorization of potentials at a sending node, but without the overhead of building and examining graphs as done in Lazy Propagation (LP). Experimental results on numerous benchmark Bayesian networks show that SP is often faster than LP.

The majority of Bayesian networks learning and inference algorithms rely on the assumption that all random variables are discrete, which is not necessarily the case in real-world problems. In situations where some variables are continuous, a trade-off between the expressive power of the model and the computational complexity of inference has to be done: on one hand, conditional Gaussian models are computationally efficient but they lack expressive power; on the other hand, mixtures of exponentials (MTE), bases or polynomials are expressive but this comes at the expense of tractability. In this paper, we propose an alternative model that lies in between. It is composed of a "discrete" Bayesian network (BN) combined with a set of monodimensional conditional truncated densities modeling the uncertainty over the continuous random variables given their discrete counterpart resulting from a discretization process. We show that inference computation times in this new model are close to those in discrete BNs. Experiments confirm the tractability of the model and highlight its expressive power by comparing it with MTE.

A continuous time Bayesian network is a graphical model capable of describing discrete state systems that evolve in continuous time. Unfortunately, the number of parameters required for each node in the graph is exponential in the number of parents of the node, which can be prohibitively large for many real-world systems. To mitigate this problem, we propose a Noisy-OR model for continuous time Bayesian networks, which can reduce the number of required parameters from exponential to linear. We describe the model, as well as the process required to compute the remaining unspecified parameters. Finally, we experimentally validate the correctness of the proposed Noisy-OR formulation.

Consider the problem of estimating the expected number of distinct eligible voters among the authors of a set of electronic signatures gathered for a petition (or citizen initiative) that has to pass legally required thresholds. We formalize this problem and propose an extension to the Pretty Good Privacy Web Of Trust, a mechanism for reciprocally certifying identities between peers. The extension (a) enables agents to certify additional relevant statements about others, and (b) gives agents opportunities for negative authentication statements (e.g., on ineligibility of an identity). A Bayesian Network model enables inferences on the data provided by the proposed PGP extension. Simulations and an agent-based platform are used to validate the concepts.

Testing independencies in Bayesian networks (BNs) is a fundamental task in probabilistic reasoning. In this paper, we propose inaugural-separation (i-separation) as a new method for testing independencies in BNs. We establish the correctness of i-separation. Our method has several theoretical and practical advantages. There are at least five ways in which i-separation is simpler than d-separation, the classical method for testing independencies in BNs, of which the most important is that "blocking" works in an intuitive fashion. In practice, our empirical evaluation shows that i-separation tends to be faster than d-separation in large BNs.

A general method of building a predictive model requires least square estimation at first. Then we need work on the residuals, find the confidence interval of parameters and test how well the model fits the data which are based on the normally distributed assumption of the residuals (or noises). But unfortunately the assumption is not guaranteed. Most of the time, you will have a graph of residuals that looks like another distribution rather than the normal. At this moment you could add one more factor term to your model so as to filter out the non-normal distributed noise, and then calculate the LSE again.

We consider a distributed learning setup where a network of agents sequentially access realizations of a set of random variables with unknown distributions. The network objective is to find a parametrized distribution that best describes their joint observations in the sense of the Kullback-Leibler divergence. Apart from recent efforts in the literature, we analyze the case of countably many hypotheses and the case of a continuum of hypotheses. We provide non-asymptotic bounds for the concentration rate of the agents' beliefs around the correct hypothesis in terms of the number of agents, the network parameters, and the learning abilities of the agents. Additionally, we provide a novel motivation for a general set of distributed Non-Bayesian update rules as instances of the distributed stochastic mirror descent algorithm.

Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes' rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose a simple yet provable algorithm, \emph{Particle Mirror Descent} (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, with $m$ particles, PMD provides a posterior density estimator that converges in terms of $KL$-divergence to the true posterior in rate $O(1/\sqrt{m})$. We demonstrate competitive empirical performances of PMD compared to several approximate inference algorithms in mixture models, logistic regression, sparse Gaussian processes and latent Dirichlet allocation on large scale datasets.

Something with BUGS or JAGS or (best of all) STAN code examples would be greatly appreciated.

In an Estimation problem, looking at a data to derive any inference about a'characteristic' of a Population, this approach mainly uses a sample taken at'random' from a collection of these similar items. An'estimate' of that characteristic (also known as a parameter) of the collection (or Universe, Population), is computed from that sample. This estimate is then tested to find out how close it might be to the original parameter, which is usually unknown. Graphical methods such EDA (Exploratory Data Analysis) are also used to study and guess the nature of the characteristic in the population, based on the data from the sample. Sampling is repeated or replicated several times, to reduce the error in the estimate.