Mark Huber

Perfect sampling using bounding chains

In Monte Carlo simulation, samples are drawn from a distribution to estimate properties of the distribution that are too difficult to compute analytically. This has applications in numerous fields, including optimization, statistics, statistical mechanics, genetics, and the design of approximation algorithms. In the Monte Carlo Markov chain method, a Markov chain is constructed which has the target distribution as its stationary distribution. After running the Markov chain “long enough”, the distribution of the final state will be close to the stationary distribution of the chain. Unfortunately, for most Markov chains, the time needed to converge to the stationary distribution (the mixing time) is completely unknown. Here we develop several new techniques for dealing with unknown mixing times. First we introduce the idea of a bounding chain, which delivers a wealth of information about the chain. Once a bounding chain is created for a particular chain, it is possible to empirically estimate the mixing time of the chain. Using ideas such as coupling from the past and the Fill-Murdoch-Rosenthal algorithm, bounding chains can also become the basis of perfect sampling algorithms. Unlike traditional Monte Carlo Markov chain methods, these algorithms draw samples which are exactly distributed according to the stationary distribution. We develop bounding chains for several Markov chains of practical interest, chains from statistical mechanics like the Swendsen-Wang chain for the Ising model, the Dyer-Greenhill chain for the discrete hard core gas model, and the continuous Widom-Rowlinson mixture model with more than three components in the mixture. We also give techniques for sampling from weighted permutations which have applications in database access and nonparametric statistical tests. In addition, we present here bounding chains for a variety of Markov chains of theoretical interest, such as the k coloring chain, the sink free orientation chain, and the antiferromagnetic Potts model with more than three colors. Finally we develop new Markov chains (and bounding chains) for the continuous hard core gas model and the Widom-Rowlinson model which are provably faster in practice.

Conditions for rapid and torpid mixing of parallel and simulated tempering on multimodal distributions

We obtain upper bounds on the convergence rates of Markov chains constructed by parallel and simulated tempering. These bounds are used to provide a set of sucien t conditions for torpid mixing of both techniques. We apply these conditions to show torpid mixing of parallel and simulated tempering for three examples: a normal mixture model with unequal covariances in R M and the mean-eld Potts model with q 3, regardless of the number and choice of temperatures, and the meaneld Ising model when an insucien t set of temperatures is chosen. The latter result contrasts with the rapid mixing of parallel and simulated tempering on the meaneld Ising model with a linearly increasing set of temperatures as shown previously.

A bounding chain for Swendsen-Wang

The greatest drawback of Monte Carlo Markov chain methods is lack of knowledge of the mixing time of the chain. The use of bounding chains solves this difficulty for some chains by giving theoretical and experimental upper bounds on the mixing time. Moreover, when used with methodologies such as coupling from the past, bounding chains allow the user to take samples drawn exactly from the stationary distribution without knowledge of the mixing time. Here we present a bounding chain for the Swendsen-Wang process. The Swendsen-Wang bounding chain allow us to efficiently obtain exact samples from the ferromagnetic Q-state Potts model for certain classes of graphs. Also, by analyzing this bounding chain, we will show that Swendsen-Wang is rapidly mixing over a slightly larger range of parameters than was known previously.

Likelihood-based inference for Matérn type-III repulsive point processes

In a repulsive point process, points act as if they are repelling one another, leading to underdispersed configurations when compared to a standard Poisson point process. Such models are useful when competition for resources exists, as in the locations of towns and trees. Bertil Matérn introduced three models for repulsive point processes, referred to as types I, II, and III. Matérn used types I and II, and regarded type III as intractable. In this paper an algorithm is developed that allows for arbitrarily accurate approximation of the likelihood for data modeled by the Matérn type-III process. This method relies on a perfect simulation method that is shown to be fast in practice, generating samples in time that grows nearly linearly in the intensity parameter of the model, while the running times for more naive methods grow exponentially.

Fast perfect sampling from linear extensions

In this paper, we study the problem of sampling (exactly) uniformly from the set of linear extensions of an arbitrary partial order. Previous Markov chain techniques have yielded algorithms that generate approximately uniform samples. Here, we create a bounding chain for one such Markov chain, and by using a non-Markovian coupling together with a modified form of coupling from the past, we build an algorithm for perfectly generating samples. The expected running time of the procedure is O(n^3lnn), making the technique as fast as the mixing time of the Karzanov/Khachiyan chain upon which it is based.

Force distributions in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces. For periodic boundary conditions corresponding to isotropic compressive stress, we find that the probability distribution for single-contact forces decays faster than exponentially. This superexponential decay persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges in the force distribution, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.

Spatial Point Processes

Assume that the function fX : R → [0,∞) is a probability density function (pdf). Suppose that g : R → R is a function of interest and that we want to know the value of EfXg = ∫ Rp g(x)fX(x) dx, but this integral cannot be computed analytically. There are many ways of approximating such intractable integrals and these include numerical integration, analytical approximations and Monte Carlo methods. In this chapter, we will describe a Markov chain Monte Carlo (MCMC) method called the data augmentation (DA) algorithm.

Exact Sampling from Perfect Matchings of Dense Regular Bipartite Graphs

AbstractWe present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain results obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is Θ(n10 (ln n)2). Other algorithms (such as Kasteleyns O(n3) algorithm for planar graphs) concentrated on restricted versions of the problem. Our algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregmans theorem. For graphs with 2n nodes, where the degree of every node is γn for a constant γ, the expected running time is O(n1.5 + .5/γ). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in Θ(n4.5 + .5/γ ln n) time, making our algorithm significantly faster on this class of graphs. The problem of counting the number of perfect matchings in these types of graphs is # P complete. In addition, we give a 1 + σ approximation algorithm for finding the permanent of 0–1 matrices with identical row and column sums that runs in O(n1.5 + .5/γ (1/σ2) log (1/δ))

Approximation algorithms for the normalizing constant of Gibbs distributions

, where the probability that the output is within 1 + \sigma

Using TPA to count linear extensions

of the permanent is at least 1 – δ.