Robust Coin Flipping

Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias $\alpha$, as a function of data she receives from $p$ sources; she seeks privacy from any coalition of $r$ of them. We show: If $p/2 \leq r < p$, the bias can be any rational number and nothing else; if $0 < r < p/2$, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the probabilistic method. Our results improve on those laid out by Yao, who asserts one direction of the $r=1$ case in his seminal paper [Yao82]. We also provide an application to secure multiparty computation.Comment: 22 pages, 1 figur

Function Approximation Using Probabilistic Fuzzy Systems

We consider function approximation by fuzzy systems. Fuzzy systems are typically used for approximating deterministic functions, in which the stochastic uncertainty is ignored. We propose probabilistic fuzzy systems i

Conditional Density Models Integrating Fuzzy and Probabilistic Representations of Uncertainty

__Abstract__ Conditional density estimation is an important problem in a variety of areas such as system identification, machine learning, artificial intelligence, empirical economics, macroeconomic analysis, quantitative finance and risk management. This work considers the general problem of conditional density estimation, i.e., estimating and predicting the density of a response variable as a function of covariates. The semi-parametric models proposed and developed in this work combine fuzzy and probabilistic representations of uncertainty, while making very few assumptions regarding the functional form of the response variable's density or changes of the functional form across the space of covariates. These models possess sufficient generalization power to approximate a non-standard density and the ability to describe the underlying process using simple linguistic descriptors despite the complexity and possible non-linearity of this process. These novel models are applied to real world quantitative finance and risk management problems by analyzing financial time-series data containing non-trivial statistical properties, such as fat tails, asymmetric distributions and changing variation over time

Estimation of flexible fuzzy GARCH models for conditional density estimation

In this work we introduce a new flexible fuzzy GARCH model for conditional density estimation. The model combines two different types of uncertainty, namely fuzziness or linguistic vagueness, and probabilistic uncertainty. The probabilistic uncertainty is modeled through a GARCH model while the fuzziness or linguistic vagueness is present in the antecedent and combination of the rule base system. The fuzzy GARCH model under study allows for a linguistic interpretation of the gradual changes in the output density, providing a simple understanding of the process. Such a system can capture different properties of data, such as fat tails, skewness and multimodality in one single model. This type of models can be useful in many fields such as macroeconomic analysis, quantitative finance and risk management. The relation to existing similar models is discussed, while the properties, interpretation and estimation of the proposed model are provided. The model performance is illustrated in simulated time series data exhibiting complex behavior and a real data application of volatility forecasting for the S&P 500 daily returns series

An algebraic/numerical formalism for one-loop multi-leg amplitudes

We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.Comment: 71 pages, 7 figures, formulas for rank 6 pentagons added in Appendix