A Bayesian network (from now on BN) of a set of variables X = {X1, . . . , Xn} represents a joint probability distribution over those variables. It consists of a network structure that encodes assertions of conditional independence in the distribution and a set of conditional probability distributions corresponding to that structure. It is graphically represented by directed acyclic graphs, who...

To motivate the Dirichlet process, let us consider a simple density estimation problem: modeling the height of UBC students. We are going to take a Bayesian approach to this problem, considering the parameters as random variables. In one of the most basic models, one would define a mean and variance random parameter θ = (μ, σ), and a height random variable normally distributed conditionally on ...

157 HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm A two-parameter family of random permutations of [n] is introduced, with distribution conditionally uniform given the counts of upper and lower records. The family interpolates between two versions of Ewens' distribution. A distinguished role of the family is determined by the fact that every sequence of coherent p...

The classification of subjects with unknown distribution in a small sample size often involves order-restricted constraints in multivariate parameter setups. Those problems make the optimality of a conventional likelihood ratio based statistical inferences not feasible. Fortunately, Roy (1953) introduced union-intersection principle(UIP) which provides an alternative avenue. Multivariate linear...

The estimation of a monotone trend that has been obscured by stationary fluctuations is considered and illustrated by global temperature anomalies. At an interior point, the rescaled isotonic estimators are shown to converge in distribution to Chernoff’s distribution under minimal conditions on the stationary errors; and two modifications for estimating the value at an end point are compared. T...

We provide a reason for Bayesian updating, in the Bernoulli case, even when it is assumed observations are independent and identically distributed with fixed but unknown parameter θ0. The motivation relies on the use of loss functions and asymptotics. Such a justification is important due to the recent interest and focus on Bayesian consistency which indeed assumes the observations are independ...

We propose a latent Markov quantile regression model for longitudinal data with non-informative drop-out. The observations, conditionally on covariates, are modeled through an asymmetric Laplace distribution. Random effects are assumed to be time-varying and to follow a first order latent Markov chain. This latter assumption is easily interpretable and allows exact inference through an ad-hoc E...

We compare multivariate aging notions of conditionally independent lifetimes with corresponding univariate aging notions of their one-dimensional marginal distributions. Let G(t|θ) (t ≥ 0) be a one-dimensional survival function indexed by θ; for simplicity’s sake, we assume here that θ is a non-negative scalar quantity; let, furthermore, Π be a probability distribution over [0,∞) and consider t...

We have focused mainly on linear models for signals, in particular the subspace model x = Uθ, whereU is a n× k matrix and θ ∈ Rk is a vector of k < n parameters describing the signal x. The subspace model is useful because it reduces the number of parameters or degrees of freedom in the model from n to k. While applicable to many real-world problems, this is not the only way of modeling signals...

Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter....