Variance Reduction Techniques in Monte Carlo Methods

Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo

SMCTC : sequential Monte Carlo in C++

Sequential Monte Carlo methods are a very general class of Monte Carlo methods for sampling from sequences of distributions. Simple examples of these algorithms are used very widely in the tracking and signal processing literature. Recent developments illustrate that these techniques have much more general applicability, and can be applied very effectively to statistical inference problems. Unfortunately, these methods are often perceived as being computationally expensive and difficult to implement. This article seeks to address both of these problems. A C++ template class library for the efficient and convenient implementation of very general Sequential Monte Carlo algorithms is presented. Two example applications are provided: a simple particle filter for illustrative purposes and a state-of-the-art algorithm for rare event estimation

Cross-entropy optimisation of importance sampling parameters for statistical model checking

Statistical model checking avoids the exponential growth of states associated with probabilistic model checking by estimating properties from multiple executions of a system and by giving results within confidence bounds. Rare properties are often very important but pose a particular challenge for simulation-based approaches, hence a key objective under these circumstances is to reduce the number and length of simulations necessary to produce a given level of confidence. Importance sampling is a well-established technique that achieves this, however to maintain the advantages of statistical model checking it is necessary to find good importance sampling distributions without considering the entire state space. Motivated by the above, we present a simple algorithm that uses the notion of cross-entropy to find the optimal parameters for an importance sampling distribution. In contrast to previous work, our algorithm uses a low dimensional vector of parameters to define this distribution and thus avoids the often intractable explicit representation of a transition matrix. We show that our parametrisation leads to a unique optimum and can produce many orders of magnitude improvement in simulation efficiency. We demonstrate the efficacy of our methodology by applying it to models from reliability engineering and biochemistry.Comment: 16 pages, 8 figures, LNCS styl

Unbiased and Consistent Nested Sampling via Sequential Monte Carlo

We introduce a new class of sequential Monte Carlo methods called Nested Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested Sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. This new framework allows convergence results to be obtained in the setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood estimates are unbiased. In contrast to NS, the analysis of NS-SMC does not require the (unrealistic) assumption that the simulated samples be independent. As the original NS algorithm is a special case of NS-SMC, this provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels in an automated manner via a preliminary pilot run, and present a new method for appropriately choosing the number of MCMC repeats at each iteration. Finally, a numerical study is conducted where the performance of NS-SMC and temperature-annealed SMC is compared on several challenging and realistic problems. MATLAB code for our experiments is made available at https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio

Monte Carlo sampling and variance reduction techniques

Simulation

SMCTC: Sequential Monte Carlo in C++

Sequential Monte Carlo methods are a very general class of Monte Carlo methods for sampling from sequences of distributions. Simple examples of these algorithms are used very widely in the tracking and signal processing literature. Recent developments illustrate that these techniques have much more general applicability, and can be applied very effectively to statistical inference problems. Unfortunately, these methods are often perceived as being computationally expensive and difficult to implement. This article seeks to address both of these problems. A C++ template class library for the efficient and convenient implementation of very general Sequential Monte Carlo algorithms is presented. Two example applications are provided: a simple particle filter for illustrative purposes and a state-of-the-art algorithm for rare event estimation.

Forward Flux Sampling for rare event simulations

Rare events are ubiquitous in many different fields, yet they are notoriously difficult to simulate because few, if any, events are observed in a conventiona l simulation run. Over the past several decades, specialised simulation methods have been developed to overcome this problem. We review one recently-developed class of such methods, known as Forward Flux Sampling. Forward Flux Sampling uses a series of interfaces between the initial and final states to calculate rate constants and generate transition paths, for rare events in equilibrium or nonequilibrium systems with stochastic dynamics. This review draws together a number of recent advances, summarizes several applications of the method and highlights challenges that remain to be overcome.Comment: minor typos in the manuscript. J.Phys.:Condensed Matter (accepted for publication

Importance sampling large deviations in nonequilibrium steady states. I

Large deviation functions contain information on the stability and response of systems driven into nonequilibrium steady states, and in such a way are similar to free energies for systems at equilibrium. As with equilibrium free energies, evaluating large deviation functions numerically for all but the simplest systems is difficult, because by construction they depend on exponentially rare events. In this first paper of a series, we evaluate different trajectory-based sampling methods capable of computing large deviation functions of time integrated observables within nonequilibrium steady states. We illustrate some convergence criteria and best practices using a number of different models, including a biased Brownian walker, a driven lattice gas, and a model of self-assembly. We show how two popular methods for sampling trajectory ensembles, transition path sampling and diffusion Monte Carlo, suffer from exponentially diverging correlations in trajectory space as a function of the bias parameter when estimating large deviation functions. Improving the efficiencies of these algorithms requires introducing guiding functions for the trajectories.Comment: Published in JC