A partially collapsed Gibbs sampler for Bayesian quantile regression

We introduce a set of new Gibbs sampler for Bayesian analysis of quantile re-gression model. The new algorithm, which partially collapsing an ordinary Gibbs sampler, is called Partially Collapsed Gibbs (PCG) sampler. Although the Metropolis-Hastings algorithm has been employed in Bayesian quantile regression, including median regression, PCG has superior convergence properties to an ordinary Gibbs sampler. Moreover, Our PCG sampler algorithm, which is based on a theoretic derivation of an asymmetric Laplace as scale mixtures of normal distributions, requires less computation than the ordinary Gibbs sampler and can significantly reduce the computation involved in approximating the Bayes Factor and marginal likelihood. Like the ordinary Gibbs sampler, the PCG sample can also be used to calculate any associated marginal and predictive distributions. The quantile regression PCG sampler is illustrated by analysing simulated data and the data of length of stay in hospital. The latter provides new insight into hospital perfor-mance. C-code along with an R interface for our algorithms is publicly available on request from the first author. JEL classification: C11, C14, C21, C31, C52, C53

Record statistics for biased random walks, with an application to financial data

We consider the occurrence of record-breaking events in random walks with asymmetric jump distributions. The statistics of records in symmetric random walks was previously analyzed by Majumdar and Ziff and is well understood. Unlike the case of symmetric jump distributions, in the asymmetric case the statistics of records depends on the choice of the jump distribution. We compute the record rate $P_n(c)$, defined as the probability for the $n$th value to be larger than all previous values, for a Gaussian jump distribution with standard deviation $\sigma$ that is shifted by a constant drift $c$. For small drift, in the sense of $c/\sigma \ll n^{-1/2}$, the correction to $P_n(c)$ grows proportional to arctan$(\sqrt{n})$ and saturates at the value $\frac{c}{\sqrt{2} \sigma}$. For large $n$ the record rate approaches a constant, which is approximately given by $1-(\sigma/\sqrt{2\pi}c)\textrm{exp}(-c^2/2\sigma^2)$ for $c/\sigma \gg 1$. These asymptotic results carry over to other continuous jump distributions with finite variance. As an application, we compare our analytical results to the record statistics of 366 daily stock prices from the Standard & Poors 500 index. The biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices, and after detrending the number of upper records is in good agreement with the symmetric random walk. However the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified.Comment: 16 pages, 7 figure

From Vicious Walkers to TASEP

We propose a model of semi-vicious walkers, which interpolates between the totally asymmetric simple exclusion process and the vicious walkers model, having the two as limiting cases. For this model we calculate the asymptotics of the survival probability for $m$ particles and obtain a scaling function, which describes the transition from one limiting case to another. Then, we use a fluctuation-dissipation relation allowing us to reinterpret the result as the particle current generating function in the totally asymmetric simple exclusion process. Thus we obtain the particle current distribution asymptotically in the large time limit as the number of particles is fixed. The results apply to the large deviation scale as well as to the diffusive scale. In the latter we obtain a new universal distribution, which has a skew non-Gaussian form. For $m$ particles its asymptotic behavior is shown to be $e^{-\frac{y^{2}}{2m^{2}}}$ as $y\to -\infty$ and $e^{-\frac{y^{2}}{2m}}y^{-\frac{m(m-1)}{2}}$ as $y\to \infty$.Comment: 37 pages, 4 figures, Corrected reference

Statistical mechanics of the mixed majority-minority game with random external information

We study the asymptotic macroscopic properties of the mixed majority-minority game, modeling a population in which two types of heterogeneous adaptive agents, namely ``fundamentalists'' driven by differentiation and ``trend-followers'' driven by imitation, interact. The presence of a fraction f of trend-followers is shown to induce (a) a significant loss of informational efficiency with respect to a pure minority game (in particular, an efficient, unpredictable phase exists only for f<1/2), and (b) a catastrophic increase of global fluctuations for f>1/2. We solve the model by means of an approximate static (replica) theory and by a direct dynamical (generating functional) technique. The two approaches coincide and match numerical results convincingly.Comment: 19 pages, 3 figure

Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project