Computational Aspects of Optional P\'{o}lya Tree

Optional P\'{o}lya Tree (OPT) is a flexible non-parametric Bayesian model for density estimation. Despite its merits, the computation for OPT inference is challenging. In this paper we present time complexity analysis for OPT inference and propose two algorithmic improvements. The first improvement, named Limited-Lookahead Optional P\'{o}lya Tree (LL-OPT), aims at greatly accelerate the computation for OPT inference. The second improvement modifies the output of OPT or LL-OPT and produces a continuous piecewise linear density estimate. We demonstrate the performance of these two improvements using simulations

Asymptotic unbiased density estimator

International audienceThis paper introduces a computationally tractable density estimator that has the same asymptotic variance as the classical Nadaraya-Watson density estimator but whose asymptotic bias is zero. We achieve this result using a two stage estimator that applies a multiplicative bias correction to an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator. Mathematics Subject Classification. 62G07, 62G20

Distinguishing DDoS attacks from flash crowds using probability metrics

Both Flash crowds and DDoS (Distributed Denial-of-Service) attacks have very similar properties in terms of internet traffic, however Flash crowds are legitimate flows and DDoS attacks are illegitimate flows, and DDoS attacks have been a serious threat to internet security and stability. In this paper we propose a set of novel methods using probability metrics to distinguish DDoS attacks from Flash crowds effectively, and our simulations show that the proposed methods work well. In particular, these mathods can not only distinguish DDoS attacks from Flash crowds clearly, but also can distinguish the anomaly flow being DDoS attacks flow or being Flash crowd flow from Normal network flow effectively. Furthermore, we show our proposed hybrid probability metrics can greatly reduce both false positive and false negative rates in detection.<br /

Bayesian adaptation

In the need for low assumption inferential methods in infinite-dimensional settings, Bayesian adaptive estimation via a prior distribution that does not depend on the regularity of the function to be estimated nor on the sample size is valuable. We elucidate relationships among the main approaches followed to design priors for minimax-optimal rate-adaptive estimation meanwhile shedding light on the underlying ideas.Comment: 20 pages, Propositions 3 and 5 adde

Weakly Convergent Nonparametric Forecasting of Stationary Time Series

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression and classification are discussed

Predictive Inference Based on Markov Chain Monte Carlo Output

In Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture‐of‐parameters approximations that exploit the parametric structure of Bayesian models perform particularly well. Under the continuous ranked probability score, the empirical distribution function is a simple and appealing alternative option