Copula-like Variational Inference

This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity $\mathcal{O}(d \log d)$. We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.Comment: 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canad

GeD spline estimation of multivariate Archimedean copulas

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so called Geometrically Designed splines (GeD splines), recently introduced by Kaishev et al. (2006 a,b) [10] and [11], to represent the cdf of a random variable Wµ, obtained through the probability integral transform of an Archimedean copula with parameter µ. Sufficient conditions for the GeD spline estimator to posses the properties of the underlying theoretical cdf, K(µ; t), of Wµ, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure,finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem, are separated. Thus, the resulting spline estimator ^K (^µ; t) is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d ¸ 2, as illustrated by the numerical examples presented

Types of dependence and time-dependent association between two lifetimes in single parameter copula models

Most publications on modeling insurance contracts on two lives, assuming dependence of the two lifetimes involved, focus on the time of inception of the contract. The dependence between the lifetimes is usually modeled through a copula and the effect of this dependence on the pricing of a joint life policy is measured. This paper investigates the effect of association at the outset on the mortality in the future. The conditional law of mortality of an individual, given his survival and given the life status of the partner is derived. The conditional joint survival distribution of a couple at any duration, given that the two lives are then alive, is also derived. We analyze how the degree of dependence between the two members of a couple varies throughout the duration of a contract. We will do that for (mainly Archimedean) copula models, with one parameter for the degree of dependence. The conditional distributions hence derived provide the basis for the calculation of prospective provisions

Excess of loss reinsurance under joint survival optimality

Explicit expressions for the probability of joint survival up to time x of the cedent and the reinsurer, under an excess of loss reinsurance contract with a limiting and a retention level are obtained, under the reasonably general assumptions of any non-decreasing premium income function, Poisson claim arrivals and continuous claim amounts, modelled by any joint distribution. By stating appropriate optimality problems, we show that these results can be used to set the limiting and the retention levels in an optimal way with respect to the probability of joint survival. Alternatively, for fixed retention and limiting levels, the results yield an optimal split of the total premium income between the two parties in the excess of loss contract. This methodology is illustrated numerically on several examples of independent and dependent claim severities. The latter are modelled by a copula function. The effect of varying its dependence parameter and the marginals, on the solutions of the optimality problems and the joint survival probability, has also been explored

Decentralized learning with budgeted network load using Gaussian copulas and classifier ensembles

We examine a network of learners which address the same classification task but must learn from different data sets. The learners cannot share data but instead share their models. Models are shared only one time so as to preserve the network load. We introduce DELCO (standing for Decentralized Ensemble Learning with COpulas), a new approach allowing to aggregate the predictions of the classifiers trained by each learner. The proposed method aggregates the base classifiers using a probabilistic model relying on Gaussian copulas. Experiments on logistic regressor ensembles demonstrate competing accuracy and increased robustness in case of dependent classifiers. A companion python implementation can be downloaded at https://github.com/john-klein/DELC

Modelling stochastic bivariate mortality

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining increasing reputation as a way to represent mortality risk. This paper represents a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. On the theoretical side, we extend to couples the Cox processes set up, i.e. the idea that mortality is driven by a jump process whose intensity is itself a stochastic process, proper of a particular generation within each gender. Dependence between the survival times of the members of a couple is captured by an Archimedean copula. On the calibration side, we fit the joint survival function by calibrating separately the (analytical) copula and the (analytical) margins. First, we select the best fit copula according to the methodology of Wang and Wells (2000) for censored data. Then, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the analytical marginal survival functions. Coupling the best fit copula with the calibrated margins we obtain, on a sample generation, a joint survival function which incorporates the stochastic nature of mortality improvements and is far from representing independency.On the contrary, since the best fit copula turns out to be a Nelsen one, dependency is increasing with age and long-term dependence exists