New algorithm of exact sampling with directional threshold

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods. Supplementary materials for this article are available online.

In Monte Carlo Markov Chain theory, a particle on a Markov chain is run for a ’long time’ until the particle is nearly in the stationary distribution. Coupling from the past is a methodology which allows samples to be taken exactly from the stationary distribution of the chain, but which requires that we have a completely coupling chain. One chain often used for the ferromagnetic Ising model is the Swendsen-Wang chain. We give the first method for running Swendsen-Wang as a completely coupling chain, thereby giving a procedure for taking exact samples from the ferromagnetic Ising model using Swendsen-Wang. Moreover, this chain completely couples in polynomial time when a parameter of the chain is inversely proportional to the maximum degree of the graph, or when this parameter is close to 1. This has two implications. First, the Swendsen-Wang chain is rapidly mixing over this set of parameter values, a fact recently (and independently) shown by Cooper and Frieze using path coupling. Second, over this set of parameter values the exact sampling procedure will have an expected running time which is polynomial. ∗This work was partially supported by an Office of Naval Research Fellowship and NSF grants CCR-9307391, CCR-9700029, and DMS-9505155, and ONR grant N00014-96-1-00500

SummaryMarkov Chain Monte Carlo (MCMC) methods are used to sample from complicated multivariate distributions with normalizing constants that may not be computable in practice and from which direct sampling is not feasible. A fundamental problem is to determine convergence of the chains. Propp & Wilson (1996) devised a Markov chain algorithm called Coupling From The Past (CFTP) that solves this problem, as it produces exact samples from the target distribution and determines automatically how long it needs to run. Exact sampling by CFTP and other methods is currently a thriving research topic. This paper gives a review of some of these ideas, with emphasis on the CFTP algorithm. The concepts of coupling and monotone CFTP are introduced, and results on the running time of the algorithm presented. The interruptible method of Fill (1998) and the method of Murdoch & Green (1998) for exact sampling for continuous distributions are presented. Novel simulation experiments are reported for exact sampling from the Ising model in the setting of Bayesian image restoration, and the results are compared to standard MCMC. The results show that CFTP works at least as well as standard MCMC, with convergence monitored by the method of Raftery & Lewis (1992, 1996).

The famous Propp and Wilson (Random Structures and Algorithms 9: 223–252, 1996, Journal of the American Statistical Association 90: 558–566, 1998) protocol called coupling from the past (CFTP) allows exact sampling from steady-state distribution of a Markov chain. When the Markov chain is stiff (i.e. existence of rarely visited states), CFTP spends a prohibitive time to reach stationarity. To reduce this time we propose to combine the variance reduction technique Importance Sampling (IS) with CFTP. Also we propose another technique, based on the power of the Markov chain kernel, to reduce the CFTP simulation time in standard case. When the period δ of the simulated Markov chain is greater than one (δ > 1), the stopping condition of CFTP is not satisfied. To break the deadlock of CFTP in this case, we propose to transform the studied chain on δ subchains that are aperiodic and for which CFTP can be applied. Some numerical examples are presented to bring the utility of the proposed simulation techniques.

We propose to perform statistical probabilistic model checking by using perfect simulation in order to verify steady-state and time unbounded until formulas over Markov chains. The model checking of probabilistic models by statistical methods has received increased attention in the last years since it provides an interesting alternative to numerical model checking which is poorly scalable with the increasing model size. In previous statistical model checking works, unbounded until formulas could not be efficiently verified, and steady-state formulas had not been considered due to the burn-in time problem to detect the steady-state. Perfect simulation is an extension of Markov Chain Monte Carlo (MCMC) methods that allows us to obtain exact steady-state samples of the underlying Markov chain, and thus it avoids the burn-in time problem to detect the steady-state. Therefore we suggest to verify time unbounded until and steady-state dependability properties for large Markov chains through statistical model checking by combining perfect simulation and statistical hypothesis testing.

The pointwise sampling algorithm has been used in adaptive contrast enhancement (ACE) for nearly three decades. To overcome its computational overhead, various stepwise sampling algorithms have been proposed, including the highly overlapped interpolation contrast enhancement (HOICE) algorithm that addresses the artefacts, such as image distortion and blocking effects, of other stepwise sampling algorithms while keeping the advantage of speed. However, HOICE's internal mechanism has not been compared with traditional pointwise algorithms, which may affect the scope of its future application. The authors establish a bridge between traditional pointwise sampling and HOICE: they first mathematically analyse the internal mechanism of two typical sampling algorithms (pointwise and HOICE); and then quantitatively demonstrate the similarities between these two kinds of algorithms. Moreover, the results of quantitative measurement suggest that HOICE has an advantage over the pointwise algorithm in reducing the distortion along image edges. Therefore HOICE has the advantages of both pointwise and other stepwise algorithms making it a reliable sampling algorithm for ACE.

We study the planar maximum coverage location problem (MCLP) with rectilinear distance and rectangular demand zones in the case where “partial coverage” is allowed in its true sense, i.e., when covering part of a demand zone is allowed and the coverage accrued as a result of this is proportional to the demand of the covered part only. We pose the problem in a slightly more general form by allowing service zones to be rectangular instead of squares, thereby addressing applications in camera view-frame selection as well. More specifically, our problem, referred to as PMCLP-PCR (planar MCLP with partial coverage and rectangular demand and service zones), is to position a given number of rectangular service zones (SZs) on the two-dimensional plane to (partially) cover a set of existing (possibly overlapping) rectangular demand zones (DZs) such that the total covered demand is maximized. Previous studies on (planar) MCLP have assumed binary coverage, even when nonpoint objects such as lines or polygons have been used to represent demand. Under the binary coverage assumption, the problem can be readily formulated and solved as a binary linear program; whereas, partial coverage, although much more realistic, cannot be efficiently handled by binary linear programming, making PMCLP-PCR much more challenging to solve. In this paper, we first prove that PMCLP-PCR is NP-hard if the number of SZs is part of the input. We then present an improved algorithm for the single-SZ PMCLP-PCR, which is at least two times faster than the existing exact plateau vertex traversal algorithm. Next, we study multi-SZ PMCLP-PCR for the first time and prove theoretical properties that significantly reduce the search space for solving this problem, and we present a customized branch-and-bound exact algorithm to solve it. Our computational experiments show that this algorithm can solve relatively large instances of multi-SZ PMCLP-PCR in a short time. We also propose a fast polynomial time heuristic algorithm. Having optimal solutions from our exact algorithm, we benchmark the quality of solutions obtained from our heuristic algorithm. Our results show that for all the random instances solved to optimality by our exact algorithm, our heuristic algorithm finds a solution in a fraction of a second, where its objective value is at least 91% of the optimal objective in 90% of the instances (and at least 69% of the optimal objective in all the instances).

A of a Markov chain is a larger chain obtained by replacing each state of the original chain by a set of states, with transition probabilities defined in such a way that the lifted chain projects down exactly to the original one. It is well known that lifting can potentially speed up the mixing time substantially. Essentially all known examples of efficiently implementable liftings have required a high degree of symmetry in the original chain. Addressing an open question of Chen, Lovasz and Pak, we present the first example of a successful lifting for a complex Markov chain that has been used in sampling algorithms. This chain, first introduced by Sinclair and Jerrum, samples a leaf uniformly at random in a large tree, given approximate information about the number of leaves in any subtree, and has applications to the theory of approximate counting and to importance sampling in Statistics. Our lifted version of the chain (which, unlike the original one, is non-reversible) gives a significant speedup over the original version whenever the error in the leaf counting estimates is o(1). Our lifting construction, based on flows, is systematic, and we conjecture that it may be applicable to other Markov chains used in sampling algorithms.

In this article a new algorithm is presented for the propagation of probabilities in junction trees. It is based on a hybrid methodology. Given a junction tree, some of the nodes carry out an exact calculation, and the other an approximation by Monte Carlo methods. For the exact calculation we will use Shafer/Shenoy method and for the Monte Carlo estimation a general class of importance sampling algorithms is used. We briefly study how to apply this sampler on the clusters in a junction tree. The basic algorithm and some of its variations are presented, depending on the family of functions to which we apply the importance sampler: potentials or/and messages in the tree. An experimental evaluation is carried out, comparing their performance with the well-known likelihood weighting approximated algorithm. This family of methods shows a very promising performance. © 1997 John Wiley & Sons, Inc.

Recent work in sensor databases has focused extensively on distributed query problems, notably distributed computation of aggregates. Existing methods for computing aggregates broadcast queries to all sensors and use in-network aggregation of responses to minimize messaging costs. In this work, we focus on uniform random sampling across nodes, which can serve both as an alternative building block for aggregation and as an integral component of many other useful randomized algorithms. Prior to our work, the best existing proposals for uniform random sampling of sensors involve contacting all nodes in the network. We propose a practical method which is only approximately uniform, but contacts a number of sensors proportional to the diameter of the network instead of its size. The approximation achieved is tunably close to exact uniform sampling, and only relies on well-known existing primitives, namely geographic routing, distributed computation of Voronoi regions and von Neumann's rejection method. Ultimately our sampling algorithm has the same worst-case asymptotic cost as routing a point-to-point message, and thus it is asymptotically optimal among request/reply-based sampling methods. We provide experimental results demonstrating the effectiveness of our algorithm on both synthetic and real sensor topologies.

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.A study is made of the joint distribution of the allele frequencies in two samples taken timetapart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at timetapart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2tapart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

We present a signal compression scheme based on coding linear segments approximating the signal. Although the approach is useful for many types of signals, we focus in this paper on compression of electrocardiogram (EGG) signals. The ECG signal compression has traditionally been tackled by heuristic approaches. However, it has been demonstrated that exact optimization algorithms outclass these heuristic approaches by a wide margin with respect to the reconstruction error. The exact optimization algorithm extracts signal samples from the original signal by formulating the sample selection problem as a graph theory problem. Thus known optimization theory can be applied in order to yield optimal compression. This paper generalizes the exact optimization scheme by removing the interpolation restriction when applying piecewise linear approximation. This guarantees a lower reconstruction error with respect to the number of extracted signal samples. The method shows superior performance compared to traditional ECG compression methods.

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming–Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming–Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identified as a mixture of Pólya urns, whereby the observations can be values from the baseline distribution or copies of previous draws collected at the same time as in the usual Pólya urn, or can be sampled from a random subset of the data collected at previous times. We characterize the time-dependent weights of the mixture which select such subsets and discuss the asymptotic regimes. We describe the induced partition by means of a Chinese restaurant process metaphor with a conveyor belt, whereby new customers who do not sit at an occupied table open a new table by picking a dish either from the baseline distribution or from a time-varying offer available on the conveyor belt. We lay out explicit algorithms for exact and approximate posterior sampling of both observations and partitions, and illustrate our results on predictive problems with synthetic and real data.