We introduce the concept of Maximal Conditional Posterior Distribution (MCPD) to assess the uncertainty of model parameters in a Bayesian framework. Although, Markov Chains Monte Carlo (MCMC) methods are particularly suited for this task, they become challenging with highly parameterized nonlinear models. The MCPD represents the conditional probability distribution function of a given parameter...

We present a computationally efficient method to generate random variables from a univariate conditional probability density function (PDF) derived from a multivariate α-sub-Gaussian (αSG) distribution. The approach may be used to sequentially generate variates for sliding-window models that constrain immediately adjacent samples to be αSG random vectors. We initially derive and establish vario...

Discriminative learning when training and test data belong to different distributions is a challenging and complex task. Often times we have very few or no labeled data from the test or target distribution but may have plenty of labeled data from multiple related sources with different distributions. The difference in distributions may be both in marginal and conditional probabilities. Most of ...

We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable ∆ is influenced by a set of covariates that can be partitioned as (X,Z) where Z (real valued) is the covariate of primary interest and X (vector va...

This technical report summarizes a number of results for the multivariate t distribution which can exhibit heavier tails than the Gaussian distribution. It is shown how t random variables can be generated, the probability density function (pdf) is derived, and marginal and conditional densities of partitioned t random vectors are presented. Moreover, a brief comparison with the multivariate Gau...

Vulnerability assessment of structures encounter many uncertainties like seismic excitations intensity and response of structures. The most common approach adopted to deal with these uncertainties is vulnerability assessment through fragility functions. Fragility functions exhibit the probability of exceeding a state namely performance-level as a function of seismic intensity. A common approach...

Bayesian network (BN) A directed graph whose nodes represent variables, and edges represent influences. Together with conditional probability distributions, a Bayesian network represents the joint probability distribution of its variables. Conditional probability distribution Assignment of probabilities to all instances of a set of variables when the value of one or more variables is known. Con...

Probability theory first reached its modern form in the work of Kolmogorov.1 Along with his famous axioms for unconditional probability, he gave a formula for calculating conditional probabilities. “If P (A) > 0, then the quotient PA(B) = P (AB) P (A) is defined to be the conditional probability of the event B under the condition A.”2 (Throughout this paper, I will use the more standard notatio...

Suppose the variable X to be predicted and the learning sample Y" that was observed have a joint distribution, which depends on an unknown parameter 0. The parameter 0 can be finite or infinite dimensional. A prediction region Dn for X is a random set, depending on Yn, that contains X with prescribed probability a. This paper studies methods for controlling simultaneously the conditional covera...