Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Our sampling algorithm is a probabilistic interpretation of the ordinary generating function for these objects, derived from the symbolic method of analytic combinatorics. Using the Khintchine-Meinardus probabilistic method to bound the rejection rate of our Boltzmann sampler through singularity analysis of Dirichlet generating functions, we offer an alternative approach to analyze Boltzmann samplers for objects with multiplicative structure.Comment: 20 pages, 1 figur

Infinite Boltzmann Samplers and Applications to Branching Processes

National audienceIn this short note, we extend the Boltzmann model for combinatorial random sampling [8] to allow for infinite size objects; in particular, this extension now fully includes Galton-Watson processes. We then illustrate our idea with two examples, one of which is the generation of prefixes of infinite Cayley trees

Polynomial tuning of multiparametric combinatorial samplers

Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tuning procedure. In this paper, we propose an efficient polynomial-time, with respect to the number of tuned parameters, tuning algorithm based on convex optimisation techniques. Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and P\'olya structures including polyomino tilings with prescribed tile frequencies, planar trees with a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures, colours. Implementation and examples are available at [1] https://github.com/maciej-bendkowski/boltzmann-brain [2] https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler

Beta-Negative Binomial Process and Exchangeable Random Partitions for Mixed-Membership Modeling

The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clusters the data points of each group into a random number of exchangeable partitions, which are shared across all the groups. A fully collapsed Gibbs sampler is developed for the BNBP, leading to a novel nonparametric Bayesian topic model that is distinct from existing ones, with simple implementation, fast convergence, good mixing, and state-of-the-art predictive performance.Comment: in Neural Information Processing Systems (NIPS) 2014. 9 pages + 3 page appendi

Bayesian Poisson process partition calculus with an application to Bayesian L\'evy moving averages

This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12 (1984) 351-357] and [Ann. Inst. Statist. Math. 41 (1989) 227-245]. In order to illustrate the flexibility of the approach, large classes of random probability measures and random hazards or intensities which can be expressed as functionals of Poisson random measures are described. We describe a unified posterior analysis of classes of discrete random probability which identifies and exploits features common to all these models. The analysis circumvents many of the difficult issues involved in Bayesian nonparametric calculus, including a combinatorial component. This allows one to focus on the unique features of each process which are characterized via real valued functions h. The applicability of the technique is further illustrated by obtaining explicit posterior expressions for L\'evy-Cox moving average processes within the general setting of multiplicative intensity models.Comment: Published at http://dx.doi.org/10.1214/009053605000000336 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gamma Processes, Stick-Breaking, and Variational Inference

While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process, the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric prior in its own right. Current inference schemes for models involving the gamma process are restricted to MCMC-based methods, which limits their scalability. In this paper, we present a variational inference framework for models involving gamma process priors. Our approach is based on a novel stick-breaking constructive definition of the gamma process. We prove correctness of this stick-breaking process by using the characterization of the gamma process as a completely random measure (CRM), and we explicitly derive the rate measure of our construction using Poisson process machinery. We also derive error bounds on the truncation of the infinite process required for variational inference, similar to the truncation analyses for other nonparametric models based on the Dirichlet and beta processes. Our representation is then used to derive a variational inference algorithm for a particular Bayesian nonparametric latent structure formulation known as the infinite Gamma-Poisson model, where the latent variables are drawn from a gamma process prior with Poisson likelihoods. Finally, we present results for our algorithms on nonnegative matrix factorization tasks on document corpora, and show that we compare favorably to both sampling-based techniques and variational approaches based on beta-Bernoulli priors

Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average

In this work we show that for every $d < \infty$ and the Ising model defined on $G(n,d/n)$, there exists a $\beta_d > 0$, such that for all $\beta < \beta_d$ with probability going to 1 as $n \to \infty$, the mixing time of the dynamics on $G(n,d/n)$ is polynomial in $n$. Our results are the first polynomial time mixing results proven for a natural model on $G(n,d/n)$ for $d > 1$ where the parameters of the model do not depend on $n$. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than n \polylog(n). Our proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex $v$ of the graph has a neighborhood $N(v)$ of radius $O(\log n)$ in which the induced sub-graph is a tree union at most $O(\log n)$ edges and where for each simple path in $N(v)$ the sum of the vertex degrees along the path is $O(\log n)$. Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for $G(n,d/n)$ it applies for all external fields and $\beta < \beta_d$, where $d \tanh(\beta_d) = 1$ is the critical point for decay of correlation for the Ising model on $G(n,d/n)$.Comment: Corrected proof of Lemma 2.

Support Size Estimation: The Power of Conditioning

We consider the problem of estimating the support size of a distribution $D$. Our investigations are pursued through the lens of distribution testing and seek to understand the power of conditional sampling (denoted as COND), wherein one is allowed to query the given distribution conditioned on an arbitrary subset $S$. The primary contribution of this work is to introduce a new approach to lower bounds for the COND model that relies on using powerful tools from information theory and communication complexity. Our approach allows us to obtain surprisingly strong lower bounds for the COND model and its extensions. 1) We bridge the longstanding gap between the upper ($O(\log \log n + \frac{1}{\epsilon^2})$) and the lower bound $\Omega(\sqrt{\log \log n})$ for COND model by providing a nearly matching lower bound. Surprisingly, we show that even if we get to know the actual probabilities along with COND samples, still $\Omega(\log \log n + \frac{1}{\epsilon^2 \log (1/\epsilon)})$ queries are necessary. 2) We obtain the first non-trivial lower bound for COND equipped with an additional oracle that reveals the conditional probabilities of the samples (to the best of our knowledge, this subsumes all of the models previously studied): in particular, we demonstrate that $\Omega(\log \log \log n + \frac{1}{\epsilon^2 \log (1/\epsilon)})$ queries are necessary