A Bernstein-Von Mises Theorem for discrete probability distributions

We investigate the asymptotic normality of the posterior distribution in the discrete setting, when model dimension increases with sample size. We consider a probability mass function $\theta_0$ on \mathbbm{N}\setminus \{0\} and a sequence of truncation levels $(k_n)_n$ satisfying $k_n^3\leq n\inf_{i\leq k_n}\theta_0(i).$ Let $\hat{\theta}$ denote the maximum likelihood estimate of $(\theta_0(i))_{i\leq k_n}$ and let $\Delta_n(\theta_0)$ denote the $k_n$-dimensional vector which $i$-th coordinate is defined by \sqrt{n} (\hat{\theta}_n(i)-\theta_0(i)) for $1\leq i\leq k_n.$ We check that under mild conditions on $\theta_0$ and on the sequence of prior probabilities on the $k_n$-dimensional simplices, after centering and rescaling, the variation distance between the posterior distribution recentered around $\hat{\theta}_n$ and rescaled by $\sqrt{n}$ and the $k_n$-dimensional Gaussian distribution $\mathcal{N}(\Delta_n(\theta_0),I^{-1}(\theta_0))$ converges in probability to $0.$ This theorem can be used to prove the asymptotic normality of Bayesian estimators of Shannon and R\'{e}nyi entropies. The proofs are based on concentration inequalities for centered and non-centered Chi-square (Pearson) statistics. The latter allow to establish posterior concentration rates with respect to Fisher distance rather than with respect to the Hellinger distance as it is commonplace in non-parametric Bayesian statistics.Comment: Published in at http://dx.doi.org/10.1214/08-EJS262 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sparse adaptive Dirichlet-multinomial-like processes

Online estimation and modelling of i.i.d. data for short sequences over large or complex ''alphabets'' is a ubiquitous (sub)problem in machine learning, information theory, data compression, statistical language processing, and document analysis. The Dirichlet-Multinomial distribution (also called Polya urn scheme) and extensions thereof are widely applied for online i.i.d. estimation. Good a-priori choices for the parameters in this regime are difficult to obtain though. I derive an optimal adaptive choice for the main parameter via tight, data-dependent redundancy bounds for a related model. The 1-line recommendation is to set the 'total mass' = 'precision' = 'concentration' parameter to m/2ln[(n+1)/m], where n is the (past) sample size and m the number of different symbols observed (so far). The resulting estimator is simple, online, fast, and experimental performance is superb

Analysis of pivot sampling in dual-pivot Quicksort: A holistic analysis of Yaroslavskiy's partitioning scheme

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-015-0041-7The new dual-pivot Quicksort by Vladimir Yaroslavskiy-used in Oracle's Java runtime library since version 7-features intriguing asymmetries. They make a basic variant of this algorithm use less comparisons than classic single-pivot Quicksort. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample. Surprisingly, dual-pivot Quicksort then needs more comparisons than a corresponding version of classic Quicksort, so it is clear that counting comparisons is not sufficient to explain the running time advantages observed for Yaroslavskiy's algorithm in practice. Consequently, we take a more holistic approach and give also the precise leading term of the average number of swaps, the number of executed Java Bytecode instructions and the number of scanned elements, a new simple cost measure that approximates I/O costs in the memory hierarchy. We determine optimal order statistics for each of the cost measures. It turns out that the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, we finally have a convincing explanation for the success of Yaroslavskiy's algorithm in practice: compared with corresponding versions of classic single-pivot Quicksort, dual-pivot Quicksort needs significantly less I/Os, both with and without pivot sampling.Peer ReviewedPostprint (author's final draft