Bayesian Updating, Model Class Selection and Robust Stochastic Predictions of Structural Response

A fundamental issue when predicting structural response by using mathematical models is how to treat both modeling and excitation uncertainty. A general framework for this is presented which uses probability as a multi-valued conditional logic for quantitative plausible reasoning in the presence of uncertainty due to incomplete information. The fundamental probability models that represent the structure’s uncertain behavior are specified by the choice of a stochastic system model class: a set of input-output probability models for the structure and a prior probability distribution over this set that quantifies the relative plausibility of each model. A model class can be constructed from a parameterized deterministic structural model by stochastic embedding utilizing Jaynes’ Principle of Maximum Information Entropy. Robust predictive analyses use the entire model class with the probabilistic predictions of each model being weighted by its prior probability, or if structural response data is available, by its posterior probability from Bayes’ Theorem for the model class. Additional robustness to modeling uncertainty comes from combining the robust predictions of each model class in a set of competing candidates weighted by the prior or posterior probability of the model class, the latter being computed from Bayes’ Theorem. This higherlevel application of Bayes’ Theorem automatically applies a quantitative Ockham razor that penalizes the data-fit of more complex model classes that extract more information from the data. Robust predictive analyses involve integrals over highdimensional spaces that usually must be evaluated numerically. Published applications have used Laplace's method of asymptotic approximation or Markov Chain Monte Carlo algorithms

Coherent Price Systems and Uncertainty-Neutral Valuation

We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem, see Harrison and Kreps (1979). We establish a microeconomic foundation of sublinear price systems and present an extension result. In this context we introduce a prior dependent notion of marketed spaces and viable price systems. We associate this extension with a canonically altered concept of equivalent symmetric martingale measure sets, in a dynamic trading framework under absence of prior depending arbitrage. We prove the existence of such sets when volatility uncertainty is modeled by a stochastic differential equation, driven by Peng's G-Brownian motions

The distance and luminosity probability distributions derived from parallax and flux with their measurement errors with application to the millisecond pulsar PSR J0218+4232

We use a Bayesian approach to derive the distance probability distribution for one object from its parallax with measurement uncertainty for two spatial distribution priors, viz. a homogeneous spherical distribution and a galactocentric distribution - applicable for radio pulsars - observed from Earth. We investigate the dependence on measurement uncertainty, and show that a parallax measurement can underestimate or overestimate the actual distance, depending on the spatial distribution prior. We derive the probability distributions for distance and luminosity combined, and for each separately, when a flux with measurement error for the object is also available, and demonstrate the necessity of and dependence on the luminosity function prior. We apply this to estimate the distance and the radio and gamma-ray luminosities of PSR J0218+4232. The use of realistic priors improves the quality of the estimates for distance and luminosity, compared to those based on measurement only. Use of a wrong prior, for example a homogeneous spatial distribution without upper bound, may lead to very wrong results.Comment: 10 pages, 9 figures, accepted 27-04-2016 to Astronomy and Astrophysic

Using the Bayesian information criterion to develop two-stage model-robust and model-sensitive designs.

In this paper, we investigate use of the Bayesian Information Criterion (BIC) in the development of Bayesian two-stage designs robust to model uncertainty. The BIC is particularly appealing in this situation as it avoids the necessity of prior specification on the model parameters and can readily be computed from the output of standard statistical software packages.Bias; BIC; Design; Information; Integrated likelihood; Lack-of-fit; Model; Model-sensitive; Posterior probabilities; Prior probabilities; Probability; Software; Software packages; Two-stage procedures; Uncertainty;